Definite Sums of Hypergeometric Terms and Limits of P-Recursive Sequences
EEingereicht von
Hui Huang
Angefertigt am
Institut für Algebra
Betreuer undErstbeurteiler
Univ.-Prof. Dr.Manuel Kauers
Zweitbeurteiler
Prof. Dr. Ziming Li
Mitbetreuung
Prof. Dr. Ziming Li
Januar 2017
JOHANNES KEPLERUNIVERSITÄT LINZ
Definite Sums ofHypergeometric Termsand Limits ofP-Recursive Sequences
Dissertation zur Erlangung des akademischen Grades
Doktorin der Naturwissenschaften im Doktoratsstudium
Naturwissenschaften a r X i v : . [ c s . S C ] O c t efinite Sums ofHypergeometric TermsandLimits of P-Recursive Sequences Hui Huang
Doctoral ThesisInstitute for AlgebraJohannes Kepler University Linzadvised by
Univ.-Prof. Dr. Manuel KauersProf. Dr. Ziming Li examined by
Univ.-Prof. Dr. Manuel KauersProf. Dr. Ziming LiThe research was partially funded by the Austrian Science Fund (FWF):W1214-N15, project DK13. idesstattliche Erklärung
Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig undohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittelnicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solchekenntlich gemacht habe.Die vorliegende Dissertation ist mit dem elektronisch übermittelten Textdoku-ment identisch.Linz, im Januar 2017 Hui Huang bstract
The ubiquity of the class of D-finite functions and P-recursive sequences in symboliccomputation is widely recognized. This class is defined in terms of linear differentialand difference equations with polynomial coefficients. In this thesis, the presentedwork consists of two parts related to this class.In the first part, we generalize the reduction-based creative telescoping algo-rithms to the hypergeometric setting. This generalization allows to deal with definitesums of hypergeometric terms more quickly.The Abramov-Petkovšek reduction computes an additive decomposition of agiven hypergeometric term, which extends the functionality of Gosper’s algorithmfor indefinite hypergeometric summation. We modify this reduction so as to decom-pose a hypergeometric term as the sum of a summable term and a non-summableone. Properties satisfied by the output of the original reduction carry over to ourmodified version. Moreover, the modified reduction does not solve any auxiliarylinear difference equation explicitly.Based on the modified reduction, we design a new algorithm to compute minimaltelescopers for bivariate hypergeometric terms. This new algorithm can avoid thecostly computation of certificates, and outperforms the classical Zeilberger algorithmno matter whether certificates are computed or not according to the computationalexperiments.We further employ a new argument for the termination of the above new algo-rithm, which enables us to derive order bounds for minimal telescopers. Comparedto the known bounds in the literature, our bounds are sometimes better, and neverworse than the known ones.In the second part of the thesis, we study the class of D-finite numbers, whichis closely related to D-finite functions and P-recursive sequences. It consists of thelimits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definitionof the class of D-finite numbers depends on two subrings of the field of complexnumbers. We investigate how different choices of these two subrings affect the class.Moreover, we show that D-finite numbers over the Gaussian rational field are essen-tially the same as the values of D-finite functions at non-singular algebraic numberarguments (so-called the regular holonomic constants). This result makes it easierto recognize certain numbers as belonging to this class. i usammenfassung Die Allgegenwart der Klasse der D-finiten Funktionen und der P-rekursiven Fol-gen im Gebiet des Symbolischen Rechnens ist allgemein bekannt. Diese Klasse istdefiniert durch lineare Differential- und Differenzengleichungen mit polynomiellenKoeffizienten. Die Ergebnisse dieser Arbeit bestehen aus Teilen, die mit dieser Klassezu tun haben.Im ersten Teil verallgemeinern wir die reduktions-basierten Algorithmen für cre-ative telescoping auf den hypergeometrischen Fall. Diese Verallgemeinerung erlaubteine effizientere Behandlung von definiten Summen hypergeometrischer Terme.Die Abramov-Petkovšek-Reduktion berechnet eine additive Zerlegung eines gegebe-nen hypergeometrischen Terms, durch die die Funktionalität des Gosper-Algorithmusfür indefinite hypergeometrische Summen erweitert. Wir adaptieren diese Reduk-tion so, dass sie einen hypergeometrischen Term in einen summierbaren und einennichtsummierbaren Term zerlegt. Eigenschaften des Outputs der ursprünglichen Zer-legung bleiben für unsere modifizierte Version erhalten. Darüber hinaus braucht manbei der modifizierten Reduktion keine lineare Hilfsrekurrenz explizit zu lösen.Ausgehend von der modifizierten Reduktion entwickeln wir einen neuen Al-gorithmus zur Berechnung minimaler Telescoper für bivariate hypergeometrischeTerme. Dieser neue Algorithmus can die teure Berechnung von Zertifikaten vermei-den, und gemäß unserer Experimente läuft er schneller als der klassische Zeilberger-Algorithmus, egal ob man Zertifikate mitberechnet oder nicht.Wir verwenden außerdem ein neues Argument für die Terminierung der genan-nten neuen Algorithmen, das es uns erlaubt, Schranken für die Ordnung des mini-malen Telescopers herzuleiten. Verglichen mit den bekannten Schranken in der Lit-eratur sind unsere Schranken manchmal besser und nie schlechter als die bekannten.Im zweiten Teil der Arbeit untersuchen wir die Klasse der D-finiten Zahlen, dieeng verwandt mit D-finiten Funktionen und P-rekursiven Folgen ist. Sie bestehtaus den Grenzwerten der konvergenten P-rekursiven Folgen. Typischerweise enthältdiese Klasse neben den algebraischen Zahlen viele weitere bekannte mathematischeKonstanten. Unsere Definition der Klasse der D-finiten Zahlen hängt von zwei Un-terringen des Körpers der komplexen Zahlen ab. Wir untersuchen, wie die Klasse vonder Wahl dieser zwei Unterringe abhängt. Außerdem zeigen wir, dass die D-finitenZahlen über dem Körper der Gaußschen rationalen Zahlen im wesentlichen diesel-ben Zahlen sind, die auch als Werte von D-finiten Funktionen an nicht-singulärenalgebraischen Argumenten auftreten (die sogenannten regulären holonomen Kon-stanten). Dieses Resultat erleichtert es, gewisse Zahlen als Elemente der Klasse zuerkennen. iii cknowledgments
I would like to express my deepest gratitude to my two co-supervisors: ManuelKauers and Ziming Li, for their academic guidance, constant support and sincereadvices. I thank Manuel, for giving me the opportunity to benefit from his immenseknowledge, excellent programming skills, amazing scientific insights and high enthu-siasm for math. I thank Ziming, for sharing his rigorous scientific attitude, assistingwith mathematical and other matters, training my speaking skills over and overagain with great patience, and also providing countless valuable suggestions.My special thanks go to Shaoshi Chen, from whom I profited a lot. I thank himvery much for his useful suggestions and constructive comments, which subsequentlyimproved my work considerably. Besides, I am impressed with his obsession withbooks and high enthusiasm for math.I was very lucky to be a student of the two lectures “Computer Algebra for Con-crete Mathematics” and “Algorithmic combinatorics” given by Peter Paule. I thankhim for his enlightening lessons and valuable encouragement. I also learned muchfrom discussions with Hao Du, Ruyong Feng, Christoph Koutschan and StephenMelczer. I would especially like to thank Hao Du for improving my code.I really appreciate many members of the Key Laboratory of Mathematics Mecha-nization for their help, particularly, Wen-tsun Wu for creating this beautiful subjectand Xiaoshan-Gao for making our lab a magnificent place to work in. I also appreci-ate all of my colleagues at the Institute for Algebra as well as my former colleaguesat RISC. Special thanks are due to all secretaries for their assistance, and to myfriends: Zijia, Miriam, Peng, Ronghua, Rika, Liangjie, for making my life in Linzawesome.Moreover, I wish to thank Mark Giesbrecht, George Labahn and Éric Schost, foraccepting me as a Postdoctoral Fellow in the Symbolic Computation Group at theUniversity of Waterloo.Most importantly, I would like to dedicate this work to my beloved parents, whomake all these possible. Thank you for your unbounded love and care. I thank allmy family, for their selfless support and for their understanding and appreciation ofmy work.This work was supported by the Austrian Science Fund (FWF) grant W1214-N15(project DK13), two NSFC grants (91118001, 60821002/F02) and a 973 project(2011CB302401). v ontents Abstract iZusammenfassung iii1 Introduction 1
I Definite Sums of Hypergeometric Terms 7 vii iii
Contents
II Limits of P-recursive Sequences 61
Appendices 83
Appendix A The
ShiftReductionCT
Package 85Appendix B Comparison of Memory Requirements 93Bibliography 97Notation 103Index 107 hapter 1
Introduction
Using computer instead of human thought is one of the main themes in the studyof symbolic computation for the past century. In particular, finding algorithmicsolutions for problems about special functions is one of the very popular topicsnowadays.As an especially attractive class of special functions, D-finite functions have beenrecognized long ago [59, 45, 70, 57, 46, 60]. They are interesting on the one handbecause each of them can be easily described by a finite amount of data, and efficientalgorithms are available to do exact as well as approximate computations with them.On the other hand, the class is interesting because it covers a lot of special functionswhich naturally appear in various different context, both within mathematics as wellas in applications.The defining property of a
D-finite function is that it satisfies a linear differen-tial equation with polynomial coefficients. This differential equation, together withan appropriate number of initial terms, uniquely determines the function at hand.Similarly, a sequence is called
P-recursive (or rarely,
D-finite ) if it satisfies a lin-ear recurrence equation with polynomial coefficients. Also in this case, the equationtogether with an appropriate number of initial terms uniquely determine the object.The set of P-recursive sequences covers a lot of important combinatorial se-quences, including C-finite sequences, hypergeometric terms and sequences whosegenerating functions are algebraic (called algebraic sequences in this thesis). Ratherthan talking about sequences themselves, our main interest focus on their definitesums and limits. This thesis is divided into two components. ⋆ ⋆ ⋆ ⋆ ⋆
Part I. Hypergeometric terms.
The set of hypergeometric terms is a basic andpowerful class of P-recursive sequences. It is defined to be the nonzero solutions offirst-order (partial) difference equations with polynomial coefficients. Many familiarfunctions are hypergeometric terms, for instance, nonzero rational functions, expo-nential functions, factorial terms, binomial coefficients, etc. In the study of symbolicsummation, there are mainly two kinds of problems related to hypergeometric terms. Chapter 1. Introduction
Problem 1.1 (Hypergeometric summation) . Investigate whether or not the follow-ing sum is expressible in simple “closed form”, 𝑏 ∑︁ 𝑘 = 𝑎 𝑓 ( 𝑛, 𝑘 ) , 𝑓 ( 𝑛, 𝑘 ) is a bivariate hypergeometric term in 𝑛, 𝑘, (1.1)where 𝑎, 𝑏 are fixed constants independent of all variables. By a closed form, wemean a linear combination of a fixed number of hypergeometric terms, where thefixed number must be a constant independent of all variables. Problem 1.2 (Hypergeometric identities) . Prove the following identity 𝑏 ∑︁ 𝑘 = 𝑎 𝑓 ( 𝑛, 𝑘 ) = ℎ ( 𝑛 ) , 𝑓 ( 𝑛, 𝑘 ) is a bivariate hypergeometric term in 𝑛, 𝑘, (1.2)where 𝑎, 𝑏 are fixed constants independent of all variables, and ℎ ( 𝑛 ) is a knownunivariate function.Analogous to the first fundamental theorem of calculus, Problem 1.1 could besolved in terms of indefinite summation provided that there exists a so-called “anti-difference”. More precisely, we compute a hypergeometric term 𝑔 ( 𝑛, 𝑘 ) such that 𝑓 ( 𝑛, 𝑘 ) = 𝑔 ( 𝑛, 𝑘 + 1) − 𝑔 ( 𝑛, 𝑘 ) , and then Problem 1.1 easily follows by the telescoping sum technique. To ourknowledge, the first complete algorithm for indefinite summation was designed byGosper [36] in 1978, namely the famous Gosper’s algorithm. To address the case whenGosper’s algorithm is not applicable, i.e., there exists no such 𝑔 , Wilf and Zeilbergerdeveloped a constructive theory in a series of articles [65, 66, 67, 68, 69, 70, 71]in early 1990s. This theory came to be known as Wilf-Zeilberger’s theory, whosemain idea is to construct a so-called telescoper for 𝑓 to derive a difference equationwith polynomial coefficients satisfied by (1.1), and then applying Petkovšek’s algo-rithm [53], which detects the existence of the hypergeometric terms solutions, to thisequation gives the final answer for Problem 1.1.On the other hand, Wilf-Zeilberger’s theory also works for Problem 1.2. To beprecise, after deriving a difference equation satisfied by the left-hand side of (1.2)from a telescoper as for Problem 1.1, we verify that ℎ satisfies the same equationand then (1.2) easily follows by checking the initial values.Wilf-Zeilberger’s theory not only provides an algorithmic method to solve theproblems about hypergeometric summations or identities, but also gives a con-structive way to find new combinatorial identities. In terms of algorithms, Wilf-Zeilberger’s theory is a strong fundamental tool for combinatorics and also the theoryof special functions.From the above discussion, one sees that the key step of Wilf-Zeilberger’s theoryis to construct a telescoper. This process is referred to as creative telescoping . Tobe more specific, for a bivariate hypergeometric term 𝑓 ( 𝑛, 𝑘 ), the task consists infinding some nonzero recurrence operator 𝐿 and another hypergeometric term 𝑔 suchthat 𝐿 · 𝑓 ( 𝑛, 𝑘 ) = 𝑔 ( 𝑛, 𝑘 + 1) − 𝑔 ( 𝑛, 𝑘 ) . (1.3) .1. Background and motivation 𝐿 does not contain 𝑘 or the shift operator 𝜎 𝑘 , i.e., itmust have the form 𝐿 = 𝑒 + 𝑒 𝜎 𝑛 + · · · + 𝑒 𝜌 𝜎 𝜌𝑛 for some 𝑒 , . . . , 𝑒 𝜌 that only dependon 𝑛 . If 𝐿 and 𝑔 ( 𝑛, 𝑘 ) are as above, we say that 𝐿 is a telescoper for 𝑓 ( 𝑛, 𝑘 ), and 𝑔 ( 𝑛, 𝑘 ) is a certificate for 𝐿 .As outlined in the introduction of [19], we can distinguish four generations ofcreative telescoping algorithms. The first generation [29, 70, 54, 27] dates back to the 1940s, and the algorithmswere based on elimination techniques.
The second generation [69, 11, 71, 54]started with what is now known as Zeilberger’s (fast) algorithm. The algorithms ofthis generation use the idea of augmenting Gosper’s algorithm for indefinite sum-mation (or integration) by additional parameters 𝑒 , . . . , 𝑒 𝜌 that are carried alongduring the calculation and are finally instantiated, if at all possible, such as to ensurethe existence of a certificate 𝑔 in (1.3). These algorithms have been implemented inmany computer algebra programs, for example Maple [5] and
Mathematica [52].See [54] for details about the first two generations.
The third generation [49, 12] was initiated by Apagodu and Zeilberger. In asense, they applied a second-generation algorithm by hand to a generic input andworked out the resulting linear system of equations for the parameters 𝑒 , . . . , 𝑒 𝜌 and the coefficients inside the certificate 𝑔 . Their algorithm then merely consistsin solving this system. This approach is interesting not only because it is easier toimplement and tends to run faster than earlier algorithms, but also because it iseasy to analyze. In fact, the analysis of algorithms from this family gives rise tothe best output size estimates for creative telescoping known so far [20, 21, 22].A disadvantage is that these algorithms may not always find the smallest possibleoutput. The fourth generation of the creative telescoping algorithms, so-called reduction-based algorithms, originates from [14]. The basic idea behind these algorithms is tobring each term 𝜎 𝑖𝑛 𝑓 of the left-hand side of (1.3) into some kind of normal formmodulo all terms that are differences of other terms. Then to find 𝑒 , . . . , 𝑒 𝜌 amountsto finding a linear dependence among these normal forms. The key advantage of thisapproach is that it separates the computation of the 𝑒 𝑖 from the computation of 𝑔 .This is interesting because a certificate is not always needed, and it is typically muchlarger (and thus computationally more expensive) than the telescoper, so we maynot want to compute it if we don’t have to. With previous algorithms there wasno way to obtain telescopers without also computing the corresponding certificates,but with fourth generation algorithms there is. So far this approach has only beenworked out for several instances in the differential case [14, 16, 15]. The goal of thefirst part of the present thesis is to give a fourth-generation algorithm for the shiftcase, namely for the classical setting of hypergeometric telescoping. ⋆ ⋆ ⋆ ⋆ ⋆ Part II. D-finite numbers.
In a sense, the theory of D-finite functions generalizesthe theory of algebraic functions. Many concepts that have first been introducedfor the latter have later been formulated also for the former. In particular, everyalgebraic function is D-finite (Abel’s theorem), and many properties the class ofalgebraic function enjoys carry over to the class of D-finite functions.
Chapter 1. Introduction
The theory of algebraic functions in turn may be considered as a generalization ofthe classical and well-understood class of algebraic numbers. The class of algebraicnumbers suffers from being relatively small. There are many important numbers,most prominently the numbers e and 𝜋 , which are not algebraic.Many larger classes of numbers have been proposed, let us just mention threeexamples. The first is the class of periods (in the sense of Kontsevich and Zagier [43]).These numbers are defined as the values of multivariate definite integrals of algebraicfunctions over a semi-algebraic set. In addition to all the algebraic numbers, this classcontains important numbers such as 𝜋 , all zeta constants (the Riemann zeta functionevaluated at an integer) and multiple zeta values, but it is so far not known whetherfor example e, 1 /𝜋 or Euler’s constant 𝛾 are periods (conjecturally they are not).The second example is the class of all numbers that appear as values of so-calledG-functions (in the sense of Siegel [58]) at algebraic number arguments [30, 31]. Theclass of G-functions is a subclass of the class of D-finite functions, and it inheritssome useful properties of that class. Among the values that G-functions can assumeare 𝜋 , 1 /𝜋 , values of elliptic integrals and multiple zeta values, but it is so far notknown whether for example e, Euler’s constant 𝛾 or a Liouville number are such avalue (conjecturally not).Another class of numbers is the class of holonomic constants, studied by Flajoletand Vallée [35, §4]. (We thank Marc Mezzarobba for pointing us to this reference.)A number is holonomic if it is equal to the (finite) value of a D-finite function atan algebraic point. The number is further called a regular holonomic constant ifthe evaluation point is an ordinary point of the defining differential equation of thegiven D-finite function; otherwise it is called a singular holonomic constant . Typicalexamples of the regular holonomic constants are 𝜋 , log(2), e and the polylogarithmicvalue Li (1 / 𝜁 (3), Catalan’sconstant G are of singular type.It is tempting to believe that there is a strong relation between holonomic con-stants and limits of convergent P-recursive sequences. To make this relation precise,we introduce the class of D-finite numbers in this thesis. Let 𝑅 be a subring of C and F be a subfield of C . A complex number 𝜉 is called D-finite (w.r.t. 𝑅 and F ) if itis the limit of a convergent sequence in 𝑅 N which is P-recursive over F . We denoteby 𝒟 𝑅, F the set of all D-finite numbers with respect to 𝑅 and F .It is clear that 𝒟 𝑅, F contains all the elements of 𝑅 , but it typically contains manyfurther elements. For example, let 𝑖 be the imaginary unit, then 𝒟 Q ( 𝑖 ) contains many(if not all) the periods and, as we will see below, many (if not all) the values of G-functions. In addition, it is not hard to see that e and 1 /𝜋 are D-finite numbers.According to Fischler and Rivoal’s work [31], also Euler’s constant 𝛾 and any valueof the Gamma function at a rational number are D-finite. (We thank Alin Bostanfor pointing us to this reference.)The definition of D-finite numbers given above involves two subrings of C asparameters: the ring to which the sequence terms of the convergent sequences aresupposed to belong, and the field to which the coefficients of the polynomials inthe recurrence equations should belong. Obviously, these choices matter, because wehave, for example, 𝒟 R , R = R ̸ = C = 𝒟 C , C . Also, since 𝒟 Q , Q is a countable set, wehave 𝒟 Q , Q ̸ = 𝒟 R , R . On the other hand, different choices of 𝑅 and F may lead tothe same classes. For example, we would not get more numbers by allowing F to be .2. Main results and outline
5a subring of C rather than a field, because we can always clear denominators in adefining recurrence. One of our goals is to investigate how 𝑅 and F can be modifiedwithout changing the resulting class of D-finite numbers.As a long-term goal, we hope to establish the notion of D-finite numbers as aclass that naturally relates to the class of D-finite functions in the same way as theclassical class of algebraic numbers relates to the class of algebraic functions. This section is intended to provide an outline of the thesis and the main results.In Chapter 2, we recall basic notions and facts about hypergeometric terms.In Chapter 3, our starting point is the Abramov-Petkovšek reduction for hyper-geometric terms introduced in [7, 10]. Unfortunately the reduced forms obtained bythis reduction are not sufficiently “normal” for our purpose. Therefore, we presenta modified version of the reduction process, which does not solve any auxiliarylinear difference equation explicitly like the original one and totally separates thesummable and non-summable parts of a given hypergeometric term. The outputs ofthe Abramov-Petkovšek reduction and our modified version share the same requiredproperties. According to the experimental comparison, the modified reduction is alsomore efficient than the original one.Chapter 4 is mainly used to connect univariate hypergeometric terms with bi-variate ones for later use. We explore some important properties of discrete residualforms by means of rational normal forms [10]. Furthermore, we show that the residualforms are well-behaved with respect to taking linear combinations.We translate terminology concerning univariate hypergeometric terms to bivari-ate ones in Chapter 5. Based on the modified version of Abramov-Petkovšek re-duction in Chapter 3, we present a new algorithm to compute minimal telescopersfor bivariate hypergeometric terms. This new algorithm keeps the key feature ofthe fourth generation, that is, it separates the computations of telescopers and cer-tificates. Experimental results illustrate that the new algorithm is faster than theclassical Zeilberger’s algorithm if it returns a normalized certificate; and the newalgorithm is much more efficient if it omits certificates.In Chapter 6, we present a new argument for the termination of the new al-gorithm in Chapter 5. This new argument provides an independent proof of theexistence of telescopers and even enables us to obtain upper and lower bounds forthe order of minimal telescopers for hypergeometric terms. Compared to the knownbounds in the literature, our bounds are sometimes better and never worse than theknown ones. Moreover, we present a variant of the new algorithm by combining ourbounds, which improves the new algorithm in some special cases.In Chapter 7, we review basic notions and useful properties of the class of D-finitefunctions and P-recursive sequences mainly from [34, 41].In Chapter 8, we study the class of D-finite numbers, defined as the limits ofconvergent P-recursive sequences. In general, this class is much larger than the classof algebraic numbers. The definition of the class depends on two subrings of thefield of complex numbers. We investigate the possible choices of these two subrings
Chapter 1. Introduction that keep the class unchanged. Moreover, we connect this class with the class ofholonomic constants [35] and show that D-finite numbers over the Gaussian rationalfield are essentially the same as the regular holonomic constants. With this result,certain numbers are easily recognized as belonging to this class, including manyperiods as well as many values of G-functions.
The main results in Chapters 3 – 5 are joint work with S. Chen, M. Kauers and Z.Li, which have been published in [19]. The main results in Chapter 6 were publishedin [38]. The main results in Chapter 8 are joint work with M. Kauers, and are inpreparation [39]. art I
Definite Sums ofHypergeometric Terms hapter 2
Hypergeometric Terms
In this chapter, we recall basic notions and facts on difference rings (fields) andhypergeometric terms. In addition, we review the context of summability and multi-plicative decomposition for hypergeometric terms. These topics are well-known andmore details can be found in [50, 28].
Let F be a field of characteristic zero, and F ( 𝑘 ) be the field of rational functions in 𝑘 over F . Let 𝜎 𝑘 be the automorphism that maps 𝑟 ( 𝑘 ) to 𝑟 ( 𝑘 + 1) for every rationalfunction 𝑟 ∈ F ( 𝑘 ). The pair ( F ( 𝑘 ) , 𝜎 𝑘 ) is called a difference field . A difference ringextension of ( F ( 𝑘 ) , 𝜎 𝑘 ) is a ring D containing F ( 𝑘 ) together with a distinguishedendomorphism 𝜎 𝑘 : D → D whose restriction to F ( 𝑘 ) agrees with the automorphismdefined before. An element 𝑐 ∈ D is called a constant if 𝜎 𝑘 ( 𝑐 ) = 𝑐 . It is readilyseen that all constants in D form a subring of D , denoted by 𝐶 𝜎 𝑘 , D . In particular, 𝐶 𝜎 𝑘 , D is a field whenever D is one. Moreover, we have 𝐶 𝜎 𝑘 , F ( 𝑘 ) = F according to [9,Theorem 2]. In other words, the set of all constants in F ( 𝑘 ) w.r.t. 𝜎 𝑘 is exactly thefield F .Throughout the thesis, for a polynomial 𝑝 ∈ F [ 𝑘 ], its degree and leading coeffi-cient are denoted by deg 𝑘 ( 𝑝 ) and lc 𝑘 ( 𝑝 ), respectively. For convenience, we define thedegree of zero to be −∞ . Definition 2.1.
Let D be a difference ring extension of F ( 𝑘 ) . A nonzero element 𝑇 ∈ D is called a hypergeometric term over F ( 𝑘 ) if it is invertible and 𝜎 𝑘 ( 𝑇 ) = 𝑟𝑇 forsome 𝑟 ∈ F ( 𝑘 ) . We call 𝑟 the shift-quotient of 𝑇 w.r.t. 𝑘 . In the following two chapters, whenever we mention hypergeometric terms, theyalways belong to some difference ring extension D of F ( 𝑘 ), unless specified otherwise. Example 2.2.
All nonzero rational functions are hypergeometric. Moreover, thefollowing two classes of combinatorial functions are also hypergeometric.1. (Exponential functions). 𝑇 = 𝑐 𝑘 where 𝑐 ∈ F ∖ { } . The shift-quotient of 𝑇 is 𝜎 𝑘 ( 𝑇 ) /𝑇 = 𝑐 .2. (Factorial terms). 𝑇 = ( 𝑎𝑘 )! with 𝑎 ∈ N and 𝑎 >
0. The shift-quotient of 𝑇 is 𝜎 𝑘 ( 𝑇 ) /𝑇 = ( 𝑎𝑘 + 𝑎 )( 𝑎𝑘 + 𝑎 − · · · ( 𝑎𝑘 + 1). Chapter 2. Hypergeometric Terms
One can easily show that the product of hypergeometric terms and the recip-rocal of a hypergeometric term are again hypergeometric. However, the sum ofhypergeometric terms is not necessarily hypergeometric. For example, 2 𝑘 + 1 isnot a hypergeometric term although 2 𝑘 and 1 both are; otherwise we would have(2 𝑘 +1 + 1) / (2 𝑘 + 1) ∈ F ( 𝑘 ), and then a straightforward calculation would yield that2 𝑘 ∈ F ( 𝑘 ), a contradiction.Recall [50, 54] that two hypergeometric terms 𝑇 , 𝑇 over F ( 𝑘 ) are called similar if there exists a rational function 𝑟 ∈ F ( 𝑘 ) such that 𝑇 = 𝑟𝑇 . This is an equivalencerelation and all rational functions form one equivalence class. By Proposition 5.6.2in [54], the sum of similar hypergeometric terms is either hypergeometric or zero. Analogous to indefinite integrals of elementary functions in calculus, we considerindefinite sums of hypergeometric terms in shift case. More precisely, given a hyper-geometric term 𝑇 ( 𝑘 ), we compute another hypergeometric term 𝐺 ( 𝑘 ) such that 𝑇 ( 𝑘 ) = 𝐺 ( 𝑘 + 1) − 𝐺 ( 𝑘 ) . This motivates the notion of hypergeometric summability.
Definition 2.3.
A univariate hypergeometric term 𝑇 over F ( 𝑘 ) is called hypergeo-metric summable , if there exists another hypergeometric term 𝐺 such that 𝑇 = Δ 𝑘 ( 𝐺 ) , where Δ 𝑘 denotes the difference of 𝜎 𝑘 and the identity map . We call 𝐺 an indefinite summation (or anti-difference ) of 𝑇 . If 𝑇 and 𝐺 are bothrational functions, we also say 𝑇 is rational summable . We abbreviate “hypergeometric summable” as “summable” in this thesis.
Example 2.4.
All polynomials are summable. Moreover, we see that 𝑘 · 𝑘 ! issummable since 𝑘 · 𝑘 ! = Δ 𝑘 ( 𝑘 !), but 𝑘 ! is not which will be shown in Example 3.7.To solve the problem of indefinite summation, Gosper [36] developed a firstcomplete algorithm which is known as Gosper’s algorithm. This is a deterministicprocedure. It determines whether or not the input hypergeometric term is summable,and then returns an indefinite summation if the answer is yes. The basic idea isto reduce the summation problem to finding polynomial solutions of a first-orderdifference equation with polynomial coefficients. By [7, 10], every hypergeometric term admits a multiplicative decomposition. Thisenables us to analyze a hypergeometric term by rational functions. To recall it,let us first review the notion of shift-free polynomials and shift-reduced rationalfunctions [7, §1]. .3. Multiplicative decomposition Definition 2.5.
A nonzero polynomial 𝑝 ∈ F [ 𝑘 ] is said to be shift-free if for anynonzero integer 𝑖 , we have gcd( 𝑝, 𝜎 𝑖𝑘 ( 𝑝 )) = 1 . Consequently, no two distinct roots of a shift-free polynomial differ by an inte-ger. The following lemma indicates the relation between shift-freeness and rationalsummability, whose proof can be found in [1, Proposition 1].
Lemma 2.6.
Let 𝑓 = 𝑝/𝑞 be a rational function in F ( 𝑘 ) , where 𝑝, 𝑞 ∈ F [ 𝑘 ] arecoprime and deg 𝑘 ( 𝑝 ) < deg 𝑘 ( 𝑞 ) . Further assume that 𝑞 is shift-free. If there exists arational function 𝑟 ∈ F ( 𝑘 ) such that 𝑓 = Δ 𝑘 ( 𝑟 ) , then 𝑓 = 0 . Definition 2.7.
A nonzero rational function 𝑓 = 𝑝/𝑞 ∈ F ( 𝑘 ) with 𝑝, 𝑞 ∈ F [ 𝑘 ] coprime, is said to be shift-reduced if for any integer 𝑖 , we have gcd( 𝑝, 𝜎 𝑖𝑘 ( 𝑞 )) = 1 . Some basic properties of shift-reduced rational functions are given below.
Lemma 2.8.
Let 𝑓 ∈ F ( 𝑘 ) be shift-reduced.(i) If there exists a nonzero rational function 𝑟 ∈ F ( 𝑘 ) such that 𝑓 = 𝜎 𝑘 ( 𝑟 ) /𝑟 ,then 𝑟 ∈ F and thus 𝑓 = 1 .(ii) If 𝑓 ̸ = 1 and there exists 𝑟 ∈ F [ 𝑘 ] such that 𝑓 𝜎 𝑘 ( 𝑟 ) − 𝑟 = 0 , then 𝑟 = 0 .Proof. (i) Suppose that 𝑟 = 𝑠/𝑡 ∈ F ( 𝑘 ) ∖ F , where 𝑠, 𝑡 are coprime and at leastone of them does not belong to F . W.l.o.g., we assume that 𝑠 / ∈ F . Then thereexists a nontrivial factor 𝑝 ∈ F [ 𝑘 ] of 𝑠 such that deg 𝑘 ( 𝑝 ) >
0. Let ℓ = min { 𝑘 ∈ Z : 𝜎 𝑘𝑘 ( 𝑝 ) | 𝑠 } and 𝑚 = max { 𝑘 ∈ Z : 𝜎 𝑘𝑘 ( 𝑝 ) | 𝑠 } . It follows that 𝑚, ℓ ≥ 𝜎 − ℓ𝑘 ( 𝑝 ) | 𝑠 but 𝜎 − ℓ𝑘 ( 𝑝 ) (cid:45) 𝜎 𝑘 ( 𝑠 );• 𝜎 𝑚 +1 𝑘 ( 𝑝 ) | 𝜎 𝑘 ( 𝑠 ) but 𝜎 𝑚 +1 𝑘 ( 𝑝 ) (cid:45) 𝑠 .Since 𝑠 and 𝑡 are coprime, so are 𝜎 𝑘 ( 𝑠 ) and 𝜎 𝑘 ( 𝑡 ). Note that 𝑓 = 𝜎 𝑘 ( 𝑟 ) 𝑟 = 𝜎 𝑘 ( 𝑠 ) 𝑡𝑠𝜎 𝑘 ( 𝑡 ) . Hence 𝜎 𝑚 +1 𝑘 ( 𝑝 ) is a nontrivial factor of the numerator of 𝑓 and 𝜎 − ℓ𝑘 ( 𝑝 ) is anontrivial factor of the denominator of 𝑓 , a contradiction as 𝑓 is shift-reduced.(ii) Suppose that 𝑟 ̸ = 0. Then 𝑓 = 𝑟𝜎 𝑘 ( 𝑟 ) = 𝜎 𝑘 (1 /𝑟 )1 /𝑟 . Since 𝑓 is unequal to one, 1 /𝑟 does not belong to F . It follows from ( 𝑖 ) that 𝑓 is not shift-reduced, a contradiction.2 Chapter 2. Hypergeometric Terms
According to [7, 10], every hypergeometric term 𝑇 admits a multiplicative de-composition 𝑆𝐻 , where 𝑆 is in F ( 𝑘 ) and 𝐻 is another hypergeometric term whoseshift-quotient is shift-reduced. We call the shift-quotient 𝐾 := 𝜎 𝑘 ( 𝐻 ) /𝐻 a kernel of 𝑇 w.r.t. 𝑘 and 𝑆 a corresponding shell . By Lemma 2.8 ( 𝑖 ), we know that 𝐾 = 1if and only if 𝑇 is a rational function, which is then equal to 𝑐𝑆 for some constant 𝑐 ∈ 𝐶 𝜎 𝑘 , D . Here D is a difference ring extension of F ( 𝑘 ).Let 𝑇 = 𝑆𝐻 be a multiplicative decomposition, where 𝑆 is a rational functionand 𝐻 a hypergeometric term with a kernel 𝐾 . Assume that 𝑇 = Δ 𝑘 ( 𝐺 ) for somehypergeometric term 𝐺 . A straightforward calculation shows that 𝐺 is similar to 𝑇 .So there exists 𝑟 ∈ F ( 𝑘 ) such that 𝐺 = 𝑟𝐻 . One can easily verify that 𝑆𝐻 = Δ 𝑘 ( 𝑟𝐻 ) ⇐⇒ 𝑆 = 𝐾𝜎 𝑘 ( 𝑟 ) − 𝑟. (2.1) hapter 3Additive Decomposition forHypergeometric Terms Computing an indefinite summation of a given hypergeometric term is one of the ba-sic problems in the theory of difference equations. In terms of algorithms, Gosper’salgorithm [36] is the first complete algorithm for solving this problem. However,when there exist no indefinite summations, Gosper’s algorithm is not applicableany more, but we still desire more information so as to handle definite summa-tions. As far as we know, the first description of the non-summable case was givenby Abramov. In 1975, Abramov [2] developed a reduction algorithm to compute anadditive decomposition of a given rational function, which was improved later by Pi-rastu and Strehl [55], Paule [51], and by Abramov himself [3], etc. These algorithmsdecompose a rational function into a summable part and a proper fractional partwhose denominator is shift-free and of minimal degree. We refer to it as a minimaladditive decomposition of the given rational function. According to Lemma 2.6, thefractional part is in fact non-summable. Hence a rational function is summable if andonly if the fractional part of a minimal decomposition is zero. In 2001, Abramov andPetkovšek [7, 10] generalized these ideas to the hypergeometric case. We call it theAbramov-Petkovšek reduction. It preserves the minimality of additive decomposi-tions. It loses, however, the separation of summable and non-summable parts. Moreprecisely, given a hypergeometric term 𝑇 , Abramov-Petkovšek reduction computestwo hypergeometric terms 𝑇 , 𝑇 such that 𝑇 = Δ 𝑘 ( 𝑇 ) ⏟ ⏞ summable + 𝑇 ⏟ ⏞ possibly summable , where 𝑇 is minimal in some sense. To determine the summability of 𝑇 , one needsto further solve an auxiliary difference equation [10, §4]. The discrepancy in thereductions for the rational case and the hypergeometric case is unpleasant.In this chapter, in order to obtain the consistency, we modify the Abramov-Petkovšek reduction by a shift variant of the method developed by Bostan et al. [15].The modified Abramov-Petkovšek reduction not only preserves the minimality ofthe output additive decomposition, but also decomposes a hypergeometric term asa sum of a summable part and a non-summable part. It laid a solid foundation The main results in this chapter are joint work with S. Chen, M. Kauers, Z. Li, published in [19]. Chapter 3. Additive Decomposition for Hypergeometric Terms for the new reduction-based creative telescoping algorithm in Chapter 5. Moreover,we implement the modified reduction in
Maple 18 and compare it with the built-in Maple procedure
SumDecomposition , which is based on the Abramov-Petkovšekreduction. The experimental results illustrate that the modified Abramov-Petkovšekreduction is more efficient than the original one.
In the shift case, reduction algorithms for computing minimal additive decompo-sitions of rational functions have been well-developed. More details can be foundin [1, 2, 3, 51, 55]. For this reason, we will mainly focus on irrational hypergeometricterms.The Abramov-Petkovšek reduction [7, 10] is fundamental for the first part of thisthesis, which computes a minimal additive decomposition of a given hypergeometricterm. It can not only be used to determine hypergeometric summability, but alsoprovide some description of the non-summable part when the given hypergeometricterm is not summable. In this sense, the Abramov-Petkovšek reduction is more usefulthan Gosper’s algorithm in some cases, as illustrated by the following example.
Example 3.1. Consider a definite sum ∞ ∑︁ 𝑘 =0 𝑇 ( 𝑘 ) , where 𝑇 ( 𝑘 ) = 1( 𝑘 + 𝑘 + 1) 𝑘 ! . Applying Gosper’s algorithm shows that 𝑇 is not summable, and thus we cannot eval-uate the sum in terms of indefinite summations. Applying the Abramov-Petkovšekreduction to 𝑇 , however, yields 𝑇 ( 𝑘 ) = Δ 𝑘 (︃ 𝑘 𝑘 − 𝑘 + 1) 𝑘 ! )︃ + 12 𝑘 ! . Summing over 𝑘 from zero to infinity and using the telescoping sum technique leadsto a “closed form” of the summation, ∞ ∑︁ 𝑘 =0 𝑇 ( 𝑘 ) = lim 𝑘 →∞ (︃ 𝑘 𝑘 − 𝑘 + 1) 𝑘 ! )︃ − ∞ ∑︁ 𝑘 =0 𝑘 ! = 12 𝑒. Thus the given sum in fact admits a simple form.To describe the Abramov-Petkovšek reduction concisely, we need a notationalconvention and a technical definition.
Convention 3.2.
Let 𝑇 be a hypergeometric term over F ( 𝑘 ) with a kernel 𝐾 anda corresponding shell 𝑆 . Then 𝑇 = 𝑆𝐻 , where 𝐻 is a hypergeometric term whoseshift-quotient is 𝐾 . Further write 𝐾 = 𝑢/𝑣 , where 𝑢, 𝑣 are nonzero polynomials in F [ 𝑘 ] with gcd( 𝑢, 𝑣 ) = 1 .Moreover, we let U 𝑇 be the union of { } and the set of summable hypergeometricterms that are similar to 𝑇 , and V 𝐾 = { 𝐾𝜎 𝑘 ( 𝑟 ) − 𝑟 | 𝑟 ∈ F ( 𝑘 ) } . We thank Yijun Chen for providing this example. .1. Abramov-Petkovšek reduction U 𝑇 and V 𝐾 are both F -linear vectorspaces and U 𝑇 = U 𝐻 since 𝐻 is similar to 𝑇 . Then (2.1) translates into 𝑆𝐻 ≡ 𝑘 U 𝐻 ⇐⇒ 𝑆 ≡ 𝑘 V 𝐾 . (3.1)These congruences enable us to shorten expressions. Definition 3.3.
With Convention 3.2, a nonzero polynomial 𝑝 in F [ 𝑘 ] is said to be strongly coprime with 𝐾 if gcd( 𝑝, 𝜎 − 𝑖𝑘 ( 𝑢 )) = gcd( 𝑝, 𝜎 𝑖𝑘 ( 𝑣 )) = 1 for all 𝑖 ≥ . The proof of Lemma 3 in [7] contains a reduction algorithm whose inputs andoutputs are given below.
Algorithm 3.4 (Abramov-Petkovšek Reduction) .Input : Two rational functions
𝐾, 𝑆 ∈ F ( 𝑘 ) as defined in Convention 3.2. Output : A rational function 𝑆 ∈ F ( 𝑘 ) and two polynomials 𝑏, 𝑤 ∈ F [ 𝑘 ] such that 𝑏 is shift-free and strongly coprime with 𝐾 , and the following equation holds: 𝑆 = 𝐾𝜎 𝑘 ( 𝑆 ) − 𝑆 + 𝑤𝑏 · 𝜎 − 𝑘 ( 𝑢 ) · 𝑣 . (3.2)The algorithm contained in the proof of Lemma 3 in [7] is described as pseudocode on page 4 of the same paper, in which the last ten lines make the denominatorof the rational function 𝑉 in its output minimal in some technical sense. We shallnot execute these lines. Then the algorithm will compute two rational functions 𝑈 and 𝑈 . They correspond to 𝑆 and 𝑤/ ( 𝑏 𝜎 − 𝑘 ( 𝑢 ) 𝑣 ) in (3.2), respectively.We slightly modify the output of the Abramov-Petkovšek reduction so that wecan analyze it more easily in the next section. Note that 𝐾 is shift-reduced and 𝑏 isstrongly coprime with 𝐾 . Thus, 𝑏 , 𝜎 − 𝑘 ( 𝑢 ) and 𝑣 are pairwise coprime. By partialfraction decomposition, (3.2) can be rewritten as 𝑆 = 𝐾𝜎 𝑘 ( 𝑆 ) − 𝑆 + (︃ 𝑎𝑏 + 𝑝 𝜎 − 𝑘 ( 𝑢 ) + 𝑝 𝑣 )︃ , where 𝑎, 𝑝 , 𝑝 ∈ F [ 𝑘 ]. Furthermore, set 𝑟 = 𝑝 /𝜎 − 𝑘 ( 𝑢 ) and a direct calculation yields 𝑟 = 𝐾𝜎 𝑘 ( − 𝑟 ) − ( − 𝑟 ) + 𝜎 𝑘 ( 𝑝 ) 𝑣 . Update 𝑆 to be 𝑆 − 𝑟 and set 𝑝 to be 𝜎 𝑘 ( 𝑝 ) + 𝑝 . Then 𝑆 = 𝐾𝜎 𝑘 ( 𝑆 ) − 𝑆 + (︁ 𝑎𝑏 + 𝑝𝑣 )︁ . (3.3)This modification leads to shell reduction specified below. Algorithm 3.5 (Shell Reduction) .Input : Two rational functions
𝐾, 𝑆 ∈ F ( 𝑘 ) as defined in Convention 3.2. Output : A rational function 𝑆 ∈ F ( 𝑘 ) and three polynomials 𝑎, 𝑏, 𝑝 ∈ F [ 𝑘 ] suchthat 𝑏 is shift-free and strongly coprime with 𝐾 , and that (3.3) holds.6 Chapter 3. Additive Decomposition for Hypergeometric Terms
Shell reduction provides us with a necessary condition on summability.
Proposition 3.6.
With Convention 3.2, let 𝑎, 𝑏, 𝑝 be polynomials in F [ 𝑘 ] where 𝑏 is shift-free and strongly coprime with 𝐾 . Assume further that (3.3) holds. If 𝑇 issummable, then 𝑎/𝑏 belongs to F [ 𝑘 ] .Proof. Recall that 𝑇 = 𝑆𝐻 by Convention 3.2 and it has a kernel 𝐾 and a corre-sponding shell 𝑆 . It follows from (3.1) and (3.3) that 𝑇 ≡ 𝑘 (︁ 𝑎𝑏 + 𝑝𝑣 )︁ 𝐻 mod U 𝐻 . Thus, 𝑇 is summable if and only if ( 𝑎/𝑏 + 𝑝/𝑣 ) 𝐻 is summable.Set 𝐻 ′ = (1 /𝑣 ) 𝐻 , which has a kernel 𝐾 ′ = 𝑢/𝜎 𝑘 ( 𝑣 ). Note that since 𝑏 is stronglycoprime with 𝐾 , so is 𝐾 ′ . Applying [10, Theorem 11] to ( 𝑎𝑣/𝑏 + 𝑝 ) 𝐻 ′ , which is equalto ( 𝑎/𝑏 + 𝑝/𝑣 ) 𝐻 , yields that ( 𝑎𝑣/𝑏 + 𝑝 ) is a polynomial. Thus, 𝑎/𝑏 is a polynomialbecause 𝑏 is coprime with 𝑣 .The above proposition enables us to determine hypergeometric summability di-rectly in some instances. Example 3.7.
Let 𝑇 = 𝑘 𝑘 ! / ( 𝑘 + 1). Then it has a kernel 𝐾 = 𝑘 + 1 and the shell 𝑆 = 𝑘 / ( 𝑘 + 1). Shell reduction yields 𝑆 ≡ 𝑘 − 𝑘 + 2 + 𝑘𝑣 mod V 𝐾 , where 𝑣 = 1. By Proposition 3.6, 𝑇 is not summable. By a similar argument asbefore, one sees that 𝑘 ! is indeed not summable as mentioned in Example 2.4.Note that 𝑎/𝑏 + 𝑝/𝑣 in (3.3) can be nonzero for a summable 𝑇 . Example 3.8.
Let 𝑇 = 𝑘 · 𝑘 ! whose kernel is 𝐾 = 𝑘 + 1 and shell is 𝑆 = 𝑘 . Then 𝑆 ≡ 𝑘 𝑘𝑣 mod V 𝐾 , where 𝑣 = 1. But 𝑇 is summable as it is equal to Δ 𝑘 ( 𝑘 !).The above example illustrates that neither shell reduction nor the Abramov-Petkovšek reduction can decide summability directly when 𝑎/𝑏 ∈ F [ 𝑘 ] in (3.3). Oneway to proceed is, according to [10], to find a polynomial solution of the auxiliaryfirst-order linear difference equation 𝑢𝜎 𝑘 ( 𝑧 ) − 𝑣𝑧 = 𝑎𝑣/𝑏 + 𝑝 , under the hypothesesof Algorithm 3.5. If there is a polynomial solution, say 𝑓 ∈ F [ 𝑘 ], then 𝑇 = Δ 𝑘 (( 𝑆 + 𝑓 ) 𝐻 ); otherwise 𝑇 is not summable. This method reduces the summability problemto solving a linear system over F . We show in the next section how this can beavoided so as to read out summability directly from a minimal decomposition. .2. Modified Abramov-Petkovšek reduction After the shell reduction described in (3.3), it remains to check the summability ofthe hypergeometric term ( 𝑎/𝑏 + 𝑝/𝑣 ) 𝐻 . In the rational case, i.e., when the kernel 𝐾 is one, the rational function 𝑎/𝑏 + 𝑝/𝑣 in (3.3) can be further reduced to 𝑎/𝑏 withdeg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), because all polynomials are rational summable. However, a hy-pergeometric term with a polynomial shell is not necessarily summable, for example, 𝑘 ! has a polynomial shell but it is not summable.In this section, we define the notion of discrete residual forms for rational func-tions, and present a discrete variant of the polynomial reduction for hyperexponen-tial functions given in [15]. This variant not only leads to a direct way to decidesummability, but also reduces the number of terms of 𝑝 in (3.3). With Convention 3.2, we define an F -linear map 𝜑 𝐾 : F [ 𝑘 ] → F [ 𝑘 ] 𝑝 ↦→ 𝑢𝜎 𝑘 ( 𝑝 ) − 𝑣𝑝, for all 𝑝 ∈ F [ 𝑘 ]. We call 𝜑 𝐾 the map for polynomial reduction w.r.t. 𝐾 . Lemma 3.9.
Let W 𝐾 = span F {︁ 𝑘 ℓ | ℓ ∈ N , ℓ ̸ = deg 𝑘 ( 𝑝 ) for all nonzero 𝑝 ∈ im ( 𝜑 𝐾 ) }︁ . Then F [ 𝑘 ] = im ( 𝜑 𝐾 ) ⊕ W 𝐾 .Proof. By the definition of W 𝐾 , im ( 𝜑 𝐾 ) ∩ W 𝐾 = { } . The same definition also im-plies that, for every nonnegative integer 𝑚 , there exists a polynomial 𝑓 𝑚 in im ( 𝜑 𝐾 ) ∪ W 𝐾 such that the degree of 𝑓 𝑚 is equal to 𝑚 . The set { 𝑓 , 𝑓 , 𝑓 , . . . } forms an F -basisof F [ 𝑘 ]. Thus F [ 𝑘 ] = im ( 𝜑 𝐾 ) ⊕ W 𝐾 .In view of the above lemma, we call W 𝐾 the standard complement of im( 𝜑 𝐾 ).Note that if 𝐾 = 1, then 𝜑 𝐾 = Δ 𝑘 and W 𝐾 = { } since all polynomials arerational summable. According to Lemma 3.9, every polynomial 𝑝 ∈ F can be uniquelydecomposed as 𝑝 = 𝑝 + 𝑝 where 𝑝 ∈ im ( 𝜑 𝐾 ) and 𝑝 ∈ W 𝐾 . Lemma 3.10.
With Convention 3.2, let 𝑝 be a polynomial in F [ 𝑘 ] . Then there existsa polynomial 𝑞 ∈ W 𝐾 such that 𝑝/𝑣 ≡ 𝑘 𝑞/𝑣 mod V 𝐾 .Proof. Let 𝑞 ∈ F [ 𝑘 ] be the projection of 𝑝 on W 𝐾 . Then there exists 𝑓 in F [ 𝑘 ] suchthat 𝑝 = 𝜑 𝐾 ( 𝑓 ) + 𝑞 , that is, 𝑝 = 𝑢𝜎 𝑘 ( 𝑓 ) − 𝑣𝑓 + 𝑞 . So 𝑝/𝑣 = 𝐾𝜎 𝑘 ( 𝑓 ) − 𝑓 + 𝑞/𝑣 , whichis equivalent to 𝑝/𝑣 ≡ 𝑘 𝑞/𝑣 mod V 𝐾 . Remark 3.11.
Replacing the polynomial 𝑝 in the above lemma by 𝑣𝑝 , we see that,for every polynomial 𝑝 ∈ F [ 𝑘 ], there exists 𝑞 ∈ W 𝐾 such that 𝑝 ≡ 𝑘 𝑞/𝑣 mod V 𝐾 .8 Chapter 3. Additive Decomposition for Hypergeometric Terms
By Lemma 3.10 and Remark 3.11, (3.3) implies that 𝑆 ≡ 𝑘 𝑎𝑏 + 𝑞𝑣 mod V 𝐾 , (3.4)where 𝑎, 𝑏, 𝑞 are polynomials in F [ 𝑘 ], deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), 𝑏 is shift-free and stronglycoprime with 𝐾 , and 𝑞 ∈ W 𝐾 . The congruence (3.4) motivates us to translate thenotion of (continuous) residual forms [15] into the discrete setting. Definition 3.12.
With Convention 3.2, we further let 𝑓 be a rational functionin F ( 𝑘 ) . Another rational function 𝑟 in F ( 𝑘 ) is called a (discrete) residual form of 𝑓 w.r.t. 𝐾 if there exist 𝑎, 𝑏, 𝑞 in F [ 𝑘 ] such that 𝑓 ≡ 𝑘 𝑟 mod V 𝐾 and 𝑟 = 𝑎𝑏 + 𝑞𝑣 , where deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ) , 𝑏 is shift-free and strongly coprime with 𝐾 , and 𝑞 belongsto W 𝐾 . For brevity, we just say that 𝑟 is a residual form w.r.t. 𝐾 if 𝑓 is clear fromthe context. Moreover, we call 𝑏 the significant denominator of 𝑟 if gcd( 𝑎, 𝑏 ) = 1 and 𝑏 is monic, i.e., lc 𝑘 ( 𝑏 ) = 1 . Residual forms help us to decide summability, as shown below.
Proposition 3.13.
With Convention 3.2, we further assume that 𝑟 is a nonzeroresidual form w.r.t. 𝐾 . Then the hypergeometric term 𝑟𝐻 is not summable.Proof. Suppose that 𝑟𝐻 is summable. Let 𝑟 = 𝑎/𝑏 + 𝑞/𝑣 , where 𝑎, 𝑏, 𝑞 ∈ F [ 𝑘 ],deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), 𝑏 is shift-free and strongly coprime with 𝐾 , and 𝑞 ∈ W 𝐾 . ByProposition 3.6, 𝑎/𝑏 is a polynomial. Since deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), we have 𝑎 = 0 andthus the term ( 𝑞/𝑣 ) 𝐻 is summable. It follows from (2.1) that there exists a rationalfunction 𝑤 ∈ F ( 𝑘 ) such that 𝑢𝜎 𝑘 ( 𝑤 ) − 𝑣𝑤 = 𝑞 . Thus, 𝑤 ∈ F [ 𝑘 ] by Theorem 5.2.1in [54, page 76], which implies that 𝑞 belongs to im ( 𝜑 𝐾 ). But 𝑞 also belongs to W 𝐾 .By Lemma 3.9, 𝑞 = 0 and then 𝑟 = 0, a contradiction.With Convention 3.2, let 𝑟 be a residual form of the shell 𝑆 w.r.t. 𝐾 . Then 𝑆𝐻 ≡ 𝑘 𝑟𝐻 mod U 𝐻 according to (3.1) and (3.4). By Proposition 3.13, 𝑆𝐻 is summable if and only if 𝑟 = 0. Thus, determining the summability of a hypergeometric term 𝑇 amounts tocomputing a residual form of a corresponding shell with respect to a kernel of 𝑇 ,which is studied below. With Convention 3.2, to compute a residual form of a rational function, we projecta polynomial on im( 𝜑 𝐾 ) and also its standard complement W 𝐾 , both defined in theprevious subsection. If the given term 𝑇 is a rational function, i.e., 𝐾 = 1, then thisprojection is trivial because im( 𝜑 ) = im(Δ 𝑘 ) = F [ 𝑘 ] and W 𝐾 = { } .Now we assume 𝐾 ̸ = 1 and let B 𝐾 = { 𝜑 𝐾 ( 𝑘 𝑖 ) | 𝑖 ∈ N } . Since 𝐾 ̸ = 1, the F -linearmap 𝜑 𝐾 is injective by Lemma 2.8 ( 𝑖𝑖 ). So B 𝐾 is an F -basis of im ( 𝜑 𝐾 ), which allows .2. Modified Abramov-Petkovšek reduction 𝜑 𝐾 ). By an echelon basis, we mean an F -basisin which distinct elements have distinct degrees. We can easily project a polynomialusing an echelon basis and linear elimination.To construct an echelon basis, we rewrite im( 𝜑 𝐾 ) asim( 𝜑 𝐾 ) = { 𝑢 Δ 𝑘 ( 𝑝 ) − ( 𝑣 − 𝑢 ) 𝑝 | 𝑝 ∈ F [ 𝑘 ] } . Set 𝛼 = deg 𝑘 ( 𝑢 ), 𝛼 = deg 𝑘 ( 𝑣 ), and 𝛽 = deg 𝑘 ( 𝑣 − 𝑢 ). Moreover, set 𝜏 𝐾 = lc 𝑘 ( 𝑣 − 𝑢 )lc 𝑘 ( 𝑢 ) , which is nonzero since 𝐾 ̸ = 1 and let 𝑝 be a nonzero polynomial in F [ 𝑘 ].We make the following case distinction. Case 1. 𝛽 > 𝛼 . Then 𝛽 = 𝛼 , and 𝜑 𝐾 ( 𝑝 ) = − lc 𝑘 ( 𝑣 − 𝑢 ) lc 𝑘 ( 𝑝 ) 𝑘 𝛼 +deg 𝑘 ( 𝑝 ) + lower terms . So B 𝐾 is an echelon basis of im( 𝜑 𝐾 ), in which deg 𝑘 ( 𝜑 𝐾 ( 𝑘 𝑖 )) is equal to 𝛼 + 𝑖 for all 𝑖 ∈ N . Accordingly, W 𝐾 has an echelon basis { , 𝑘, . . . , 𝑘 𝛼 − } and has dimension 𝛼 . Case 2. 𝛽 = 𝛼 . Then 𝜑 𝐾 ( 𝑝 ) = − lc 𝑘 ( 𝑣 − 𝑢 ) lc 𝑘 ( 𝑝 ) 𝑘 𝛼 +deg 𝑘 ( 𝑝 ) + lower terms . So B 𝐾 is an echelon basis of im( 𝜑 𝐾 ), in which deg 𝑘 ( 𝜑 𝐾 ( 𝑘 𝑖 )) is equal to 𝛼 + 𝑖 for all 𝑖 ∈ N . Accordingly, W 𝐾 has an echelon basis { , 𝑘, . . . , 𝑘 𝛼 − } and has dimension 𝛼 . Case 3. 𝛽 < 𝛼 −
1. If deg 𝑘 ( 𝑝 ) = 0, then 𝜑 𝐾 ( 𝑝 ) = ( 𝑢 − 𝑣 ) 𝑝 . Otherwise, we have 𝜑 𝐾 ( 𝑝 ) = deg 𝑘 ( 𝑝 ) lc 𝑘 ( 𝑢 ) lc 𝑘 ( 𝑝 ) 𝑘 𝛼 +deg 𝑘 ( 𝑝 ) − + lower terms . It follows that B 𝐾 is an echelon basis of im( 𝜑 𝐾 ), in which deg 𝑘 ( 𝜑 𝐾 (1)) = 𝛽 anddeg 𝑘 ( 𝜑 𝐾 ( 𝑘 𝑖 )) = 𝛼 + 𝑖 − 𝑖 ≥ . Accordingly, W 𝐾 has an echelon basis { , . . . , 𝑘 𝛽 − , 𝑘 𝛽 +1 , . . . , 𝑘 𝛼 − } and has dimen-sion 𝛼 − Case 4. 𝛽 = 𝛼 − 𝜏 𝐾 is not a positive integer. Then 𝜑 𝐾 ( 𝑝 ) = (deg 𝑘 ( 𝑝 ) lc 𝑘 ( 𝑢 ) − lc 𝑘 ( 𝑣 − 𝑢 )) lc 𝑘 ( 𝑝 ) 𝑘 𝛼 +deg 𝑘 ( 𝑝 ) − + lower terms . (3.5)Accordingly, B 𝐾 is an echelon basis of im( 𝜑 𝐾 ), in which deg 𝑘 ( 𝜑 𝐾 ( 𝑘 𝑖 )) = 𝛼 + 𝑖 − 𝑖 ∈ N . Accordingly, W 𝐾 has an echelon basis { , 𝑘, . . . , 𝑘 𝛼 − } and has dimension 𝛼 − Case 5. 𝛽 = 𝛼 − 𝜏 𝐾 is a positive integer. It follows from (3.5) that for 𝑖 ̸ = 𝜏 𝐾 ,we have deg 𝑘 ( 𝜑 𝐾 ( 𝑘 𝑖 )) = 𝛼 + 𝑖 −
1. Moreover, for every polynomial 𝑝 of degree 𝜏 𝐾 ,0 Chapter 3. Additive Decomposition for Hypergeometric Terms 𝜑 𝐾 ( 𝑝 ) is of degree less than 𝛼 + 𝜏 𝐾 −
1. So any echelon basis of im( 𝜑 𝐾 ) does notcontain a polynomial of degree 𝛼 + 𝜏 𝐾 −
1. Set B ′ 𝐾 = {︁ 𝜑 𝐾 ( 𝑘 𝑖 ) | 𝑖 ∈ N , 𝑖 ̸ = 𝜏 𝐾 }︁ . Reducing 𝜑 𝐾 ( 𝑘 𝜏 𝐾 ) by the polynomials in B ′ 𝐾 , we obtain a polynomial 𝑝 ′ with de-gree less than 𝛼 −
1. Since B 𝐾 is an F -basis and B ′ 𝐾 ⊂ B 𝐾 , 𝑝 ′ ̸ = 0. Hence B ′ 𝐾 ∪ { 𝑝 ′ } is an echelon basis of im( 𝜑 𝐾 ). Consequently, W 𝐾 has an echelon ba-sis { , 𝑘, . . . , 𝑘 deg 𝑘 ( 𝑝 ′ ) − , 𝑘 deg 𝑘 ( 𝑝 ′ )+1 , . . . , 𝑘 𝛼 − , 𝑘 𝛼 + 𝜏 𝐾 − } . The dimension of W 𝐾 isequal to 𝛼 − Example 3.14.
Let 𝐾 = ( 𝑘 + 1) / ( 𝑘 + 1) , which is shift-reduced. Then 𝜏 𝐾 = 4.According to Case 5, im( 𝜑 𝐾 ) has an echelon basis { 𝜑 𝐾 ( 𝑝 ) } ∪ { 𝜑 𝐾 ( 𝑘 𝑚 ) | 𝑚 ∈ N , 𝑚 ̸ = 4 } , where 𝑝 = 𝑘 + 𝑘/ / 𝜑 𝐾 ( 𝑝 ) = (5 / 𝑘 + 2 𝑘 + 4 /
3, and 𝜑 𝐾 ( 𝑘 𝑚 ) = ( 𝑚 − 𝑘 𝑚 +3 + lower terms . Therefore, W 𝐾 has a basis { , 𝑘, 𝑘 } .From the above case distinction and example one observes that, although thedegree of a polynomial in the standard complement depends on 𝜏 𝐾 , which may bearbitrarily high, the number of its terms depends merely on the degrees of 𝑢 and 𝑣 .We record this observation in the next proposition. Proposition 3.15.
With Convention 3.2, further let 𝛼 = deg 𝑘 ( 𝑢 ) , 𝛼 = deg 𝑘 ( 𝑣 ) and 𝛽 = max { , deg 𝑘 ( 𝑣 − 𝑢 ) } . Then there exists a set
𝒫 ⊂ { 𝑘 𝑖 | 𝑖 ∈ N } with |𝒫| ≤ max { 𝛼 , 𝛼 } − (cid:74) 𝛽 ≤ 𝛼 − (cid:75) such that every polynomial in F [ 𝑘 ] can be reduced modulo im( 𝜑 𝐾 ) to an F -linearcombination of the elements in 𝒫 . Note that here the expression (cid:74) 𝛽 ≤ 𝛼 − (cid:75) equals if 𝛽 ≤ 𝛼 − , otherwise it is .Proof. If 𝐾 = 1, then im( 𝜑 𝐾 ) = im(Δ 𝑘 ) = F [ 𝑘 ] and 𝛼 = 𝛼 = 𝛽 = 0. Taking 𝒫 = ∅ completes the proof. Otherwise 𝐾 ̸ = 1. By the above case distinction, the dimensionof W 𝐾 over F is no more than max { 𝛼 , 𝛼 } − (cid:74) 𝛽 ≤ 𝛼 − (cid:75) . The lemma follows.When 𝐾 ̸ = 1, the above case distinction enables one to find an infinite sequence 𝑝 , 𝑝 , . . . in F [ 𝑘 ] such that E 𝐾 = { 𝜑 𝐾 ( 𝑝 𝑖 ) | 𝑖 ∈ N } with deg 𝑘 𝜑 𝐾 ( 𝑝 𝑖 ) < deg 𝑘 𝜑 𝐾 ( 𝑝 𝑖 +1 ) , is an echelon basis of im ( 𝜑 𝐾 ). This basis allows one to project a polynomial on im ( 𝜑 𝐾 )and W 𝐾 , respectively. In the first four cases, the 𝑝 𝑖 ’s can be chosen as powers of 𝑘 . But in the last case, one of the 𝑝 𝑖 ’s is not necessarily a monomial as shown inExample 3.14. .2. Modified Abramov-Petkovšek reduction Algorithm 3.16 (Polynomial Reduction) .Input : A polynomial 𝑝 ∈ F [ 𝑘 ] and a shift-reduced rational function 𝐾 ∈ F ( 𝑘 ). Output : Two polynomials 𝑓, 𝑞 ∈ F [ 𝑘 ] such that 𝑞 ∈ W 𝐾 and 𝑝 = 𝜑 𝐾 ( 𝑓 ) + 𝑞 . If 𝑝 = 0, then set 𝑓 = 0 and 𝑞 = 0; and return. If 𝐾 = 1, then set 𝑓 = Δ − 𝑘 ( 𝑝 ) and 𝑞 = 0; and return. Set 𝑑 = deg 𝑘 ( 𝑝 ).Find the subset P = {︀ 𝑝 𝑖 , . . . , 𝑝 𝑖 𝑠 }︀ consisting of the preimages of allpolynomials in the echelon basis E 𝐾 whose degrees are at most 𝑑 . For 𝑗 = 𝑠, 𝑠 − , . . . , , perform linear elimination tofind 𝑐 𝑠 , 𝑐 𝑠 − , . . . , 𝑐 ∈ F such that 𝑝 − ∑︀ 𝑠𝑗 =1 𝑐 𝑗 𝜑 𝐾 ( 𝑝 𝑖 𝑗 ) ∈ W 𝐾 . Set 𝑓 = ∑︀ 𝑠𝑗 =1 𝑐 𝑗 𝑝 𝑖 𝑗 and 𝑞 = 𝑝 − 𝜑 𝐾 ( 𝑓 ); and return.Together with Algorithms 3.5 and 3.16, we are ready to present a modified versionof the Abramov-Petkovšek reduction, which is summarized as Algorithm 3.17. Thismodified reduction determines summability without solving any auxiliary differenceequations explicitly. Algorithm 3.17 (Modified Abramov-Petkovšek Reduction) .Input : A hypergeometric term 𝑇 over F ( 𝑘 ). Output : A hypergeometric term 𝐻 with a kernel 𝐾 and two rational functions 𝑓, 𝑟 ∈ F ( 𝑘 ) such that 𝑟 is a residual form w.r.t. 𝐾 and 𝑇 = Δ 𝑘 ( 𝑓 𝐻 ) + 𝑟𝐻. (3.6) Find a kernel 𝐾 and a corresponding shell 𝑆 of 𝑇 . Apply Algorithm 3.5, namely the shell reduction, to 𝑆 w.r.t. 𝐾 tofind three polynomials 𝑏, 𝑠, 𝑡 ∈ F [ 𝑘 ] and a rational function 𝑔 ∈ F ( 𝑘 )such that 𝑏 is shift-free and strongly coprime with 𝐾 , and 𝑇 = Δ 𝑘 ( 𝑔𝐻 ) + (︂ 𝑠𝑏 + 𝑡𝑣 )︂ 𝐻, (3.7)where 𝜎 𝑘 ( 𝐻 ) /𝐻 = 𝐾 and 𝑣 is the denominator of 𝐾 . Set 𝑝 and 𝑎 to be the quotient and remainder of 𝑠 and 𝑏 , respectively. Apply Algorithm 3.16, namely the polynomial reduction, to 𝑣𝑝 + 𝑡 tofind ℎ ∈ F [ 𝑘 ] and 𝑞 ∈ W 𝐾 such that 𝑣𝑝 + 𝑡 = 𝜑 𝐾 ( ℎ ) + 𝑞 . Set 𝑓 = 𝑔 + ℎ and 𝑟 = 𝑎/𝑏 + 𝑞/𝑣 ; and return 𝐻 , 𝑓 and 𝑟 . Theorem 3.18.
With Convention 3.2, Algorithm 3.17 computes a rational func-tion 𝑓 in F ( 𝑘 ) and a residual form 𝑟 w.r.t. 𝐾 such that (3.6) holds. Moreover, 𝑇 issummable if and only if 𝑟 = 0 . Chapter 3. Additive Decomposition for Hypergeometric Terms
Proof.
Recall that 𝑇 = 𝑆𝐻 , where 𝐻 has a kernel 𝐾 and 𝑆 is a rational function.Applying shell reduction to 𝑆 w.r.t. 𝐾 yields (3.7), which can be rewritten as 𝑇 = Δ 𝑘 ( 𝑔𝐻 ) + (︂ 𝑎𝑏 + 𝑣𝑝 + 𝑡𝑣 )︂ 𝐻, where 𝑎 and 𝑝 are given in step 3 of Algorithm 3.17. The polynomial reduction instep 4 yields that 𝑣𝑝 + 𝑡 = 𝑢𝜎 𝑘 ( ℎ ) − 𝑣ℎ + 𝑞 . Substituting this into (3.7) gives 𝑇 = Δ 𝑘 ( 𝑔𝐻 ) + ( 𝐾𝜎 𝑘 ( ℎ ) − ℎ ) 𝐻 + (︁ 𝑎𝑏 + 𝑞𝑣 )︁ 𝐻 = Δ 𝑘 (( 𝑔 + ℎ ) 𝐻 ) + 𝑟𝐻, where 𝑟 = 𝑎/𝑏 + 𝑞/𝑣 . Thus, (3.6) holds. By Proposition 3.13, 𝑇 is summable if andonly if 𝑟 is equal to zero. Example 3.19.
Let 𝑇 be the same hypergeometric term as in Example 3.7. Thenwe know 𝐾 = 𝑘 + 1 and 𝑆 = 𝑘 / ( 𝑘 + 1). Set 𝐻 = 𝑘 !. By the shell reduction inExample 3.7, 𝑇 = Δ 𝑘 (︂ − 𝑘 + 1 𝐻 )︂ + (︂ − 𝑘 + 2 + 𝑘𝑣 )︂ 𝐻 with 𝑣 = 1 . Applying the polynomial reduction to ( 𝑘/𝑣 ) 𝐻 yields ( 𝑘/𝑣 ) 𝐻 = Δ 𝑘 (1 · 𝐻 ). Combin-ing the above steps, we decompose 𝑇 as 𝑇 = Δ 𝑘 (︂ 𝑘𝑘 + 1 𝐻 )︂ − 𝑘 + 2 𝐻. So the input term 𝑇 is not summable, which is consistent with Example 3.7. Example 3.20.
Let 𝑇 be the same hypergeometric term as in Example 3.8. Thenwe know 𝐾 = 𝑘 + 1 and 𝑆 = 𝑘 . Set 𝐻 = 𝑘 !. The shell reduction in Example 3.8gives 𝑇 = Δ 𝑘 (0) + 𝑘𝑣 𝐻 with 𝑣 = 1 . By the polynomial reduction, ( 𝑘/𝑣 ) 𝐻 = Δ 𝑘 (1 · 𝐻 ) , and hence 𝑇 = Δ 𝑘 ( 𝑘 !), implyingthat 𝑇 is summable. Remark 3.21.
With the notation given in step 5 of Algorithm 3.17, we can rewrite 𝑟𝐻 as ( 𝑠 /𝑠 ) 𝐺 , where 𝑠 = 𝑎𝑣 + 𝑏𝑞 , 𝑠 = 𝑏 , and 𝐺 = 𝐻/𝑣 . It follows from the casedistinction in this subsection that the degree of 𝑠 is bounded by 𝜆 given in [7,Theorem 8]. The polynomial 𝑠 is equal to 𝑏 in (3.2) whose degree is minimal by [7,Theorem 3]. Moreover, 𝜎 𝑘 ( 𝐺 ) /𝐺 is shift-reduced because 𝜎 𝑘 ( 𝐻 ) /𝐻 is. These areexactly the same required properties of the output of the Abramov-Petkovšek re-duction [7]. In summary, the modified reduction preserves all required conditionsfor the outputs of the original reduction, namely, it also returns a minimal additivedecomposition of a given hypergeometric term.It is remarkable that the modified Abramov-Petkovšek reduction also applies toExample 3.1. Moreover, compared to the original reduction, the modified reductionnot only further decomposes a hypergeometric term into a summable part and anon-summable part, but also provides a new method for proving identities in severalexamples. .3. Implementation and timings Example 3.22.
Consider the following two famous combinatorial identities ∞ ∑︁ 𝑘 =0 (︂ 𝑛𝑘 )︂ = 2 𝑛 and ∞ ∑︁ 𝑘 =0 (︂ 𝑛𝑘 )︂ = (︂ 𝑛𝑛 )︂ . Many methods can be used to prove the above identities. In this example, we usethe modified Abramov-Petkovšek reduction.Referring to the first identity, we apply the modified reduction to the summandand get (︂ 𝑛𝑘 )︂ = Δ 𝑘 (︂ − (︂ 𝑛𝑘 )︂)︂ + 𝑛 + 12( 𝑘 + 1) (︂ 𝑛𝑘 )︂ . Summing over 𝑘 from zero to infinity and using the telescoping sum technique yields ∞ ∑︁ 𝑘 =0 (︂ 𝑛𝑘 )︂ = lim 𝑘 →∞ (︂ − (︂ 𝑛𝑘 )︂)︂ − (︂ − )︂ + ∞ ∑︁ 𝑘 =0 𝑛 + 12( 𝑘 + 1) (︂ 𝑛𝑘 )︂ = 12 + 12 ∞ ∑︁ 𝑘 =0 (︂ 𝑛 + 1 𝑘 + 1 )︂ = 12 ∞ ∑︁ 𝑘 =0 (︂ 𝑛 + 1 𝑘 )︂ . Let 𝐹 ( 𝑛 ) = ∑︀ ∞ 𝑘 =0 (︀ 𝑛𝑘 )︀ . Then the above equation can be rewritten as a first-orderdifference equation about 𝐹 ( 𝑛 ), 𝐹 ( 𝑛 + 1) − 𝐹 ( 𝑛 ) = 0 . It is readily seen that 2 𝑛 is a solution. Since 2 = 1 = 𝐹 (0), we have 𝐹 ( 𝑛 ) = 2 𝑛 ,which proves the first identity.For the second identity, applying the modified reduction to the summand yields (︂ 𝑛𝑘 )︂ = Δ 𝑘 (︃ − 𝑛 + 2 𝑘 + 12 𝑛 + 1 (︂ 𝑛𝑘 )︂ )︃ + 12 ( 𝑛 + 1) (2 𝑛 + 1)( 𝑘 + 1) (︂ 𝑛𝑘 )︂ . Along entirely similar lines as the first identity, we get a first-order difference equa-tion ( 𝑛 + 1) 𝐹 ( 𝑛 + 1) − 𝑛 + 1) 𝐹 ( 𝑛 ) = 0 , where 𝐹 ( 𝑛 ) = ∑︀ ∞ 𝑘 =0 (︀ 𝑛𝑘 )︀ . The second identity follows since (︀ 𝑛𝑛 )︀ satisfies the samedifference equation and has the same initial value at zero as 𝐹 ( 𝑛 ).However, the Abramov-Petkovšek reduction applies to neither the first identitynor the second one. We have implemented Algorithms 3.5 – 3.17 in
Maple 18 . The procedures areincluded in our Maple package
ShiftReductionCT . A detailed description of thispackage is given in Appendix A.In order to get an idea about the efficiency of our new procedures, we comparedtheir runtime and memory requirements to the performance of known algorithms.4
Chapter 3. Additive Decomposition for Hypergeometric Terms
Since the comparisons of runtime and memory requirements almost have the sameindication, we only show that of runtime in this section. One can refer to Appendix Bfor the memory requirements. All timings are measured in seconds on a Linux com-puter with 388Gb RAM and twelve 2.80GHz Dual core processors. The computationsfor this experiment did not use any parallelism. For brevity, we denote• G : the procedure Gosper in SumTools[Hypergeometric] , which is based onGosper’s algorithm;• AP : the procedure SumDecomposition in SumTools[Hypergeometric] , whichis based on the Abramov-Petkovšek reduction;• S : the procedure IsSummable in ShiftReductionCT , which determines hy-pergeometric summability in a similar way as Gosper’s algorithm;•
MAP : the procedure
ModifiedAbramovPetkovsekReduction in ShiftReductionCT ,which is based on the modified reduction.We make the following two comparisons. One is for random hypergeometric terms,while the other is for summable hypergeometric terms.
Example 3.23 (Random hypergeometric terms) . Consider hypergeometric termsof the form 𝑇 ( 𝑘 ) = 𝑓 ( 𝑘 ) 𝑔 ( 𝑘 ) 𝑔 ( 𝑘 ) 𝑘 ∏︁ ℓ = 𝑚 𝑢 ( ℓ ) 𝑣 ( ℓ ) , (3.8)where 𝑓 ∈ Z [ 𝑘 ] of degree 20, 𝑚 ∈ F is fixed, 𝑢, 𝑣 are both the product of twopolynomials in Z [ 𝑘 ] of degree one, 𝑔 𝑖 = 𝑝 𝑖 𝜎 𝜆𝑘 ( 𝑝 𝑖 ) 𝜎 𝜇𝑘 ( 𝑝 𝑖 ) with 𝑝 𝑖 ∈ Z [ 𝑘 ] of degree 10, 𝜆, 𝜇 ∈ N , and 𝛼, 𝛽 ∈ Z . For a selection of random terms of this type for differentchoices of 𝜆 and 𝜇 , Table 3.1 compares the timings of the four procedures describedabove. ( 𝜆, 𝜇 ) G AP S MAP (0 ,
0) 0.09 0.16 0.12 0.12(5 ,
5) 0.36 3.99 0.37 0.45(10 ,
10) 0.66 13.70 0.65 0.86(10 ,
20) 4.05 40.82 1.41 2.53(10 ,
30) 12.13 294.52 2.22 6.26(10 ,
40) 19.09 564.71 3.31 14.11(10 ,
50) 34.89 865.01 4.76 26.02
Table 3.1:
Timing comparison of Gosper’s algorithm, the Abramov-Petkovšek re-duction and the modified version for random hypergeometric terms (in seconds)
Example 3.24 (Summable hypergeometric terms) . Consider the summable terms 𝜎 𝑘 ( 𝑇 ) − 𝑇 , where 𝑇 is of the form (3.8). Similarly, for the same choices of 𝜆 and 𝜇 as the previous example, Table 3.2 compares the timings of the four procedures. .3. Implementation and timings 𝜆, 𝜇 ) G AP S MAP (0 ,
0) 1.13 2.34 1.27 1.26(5 ,
5) 1.86 6.44 1.59 1.59(10 ,
10) 2.22 13.78 1.63 1.63(10 ,
20) 7.09 29.76 2.09 2.10(10 ,
30) 19.61 57.63 2.34 2.33(10 ,
40) 30.83 95.31 2.49 2.49(10 ,
50) 64.69 168.72 2.69 2.69
Table 3.2:
Timing comparison of Gosper’s algorithm, the Abramov-Petkovšek re-duction and the modified version for summable hypergeometric terms (in seconds)
Notice that 𝜇 is the dispersion of 𝑔 𝑖 and itself in (3.8) (see Definition 4.13).From Table 3.1 and Table 3.2, we observe that for different procedures, the effectof dispersion is quite different. Figure 3.1 describes the effect of dispersion on theabove four procedures in Example 3.23 and Example 3.24. ● ● ● ● ● ● ●● ● ● ● ● ● ●● G ● S0 5 10 20 30 40 5005101520253035
Dispersion R unn i ngT i m e ( s ) Random terms ● ● ● ● ● ● ●● ● ● ● ● ● ●● AP ● MAP0 5 10 20 30 40 500250500750
Dispersion R unn i ngT i m e ( s ) Random terms ● ● ● ● ● ● ●● ● ● ● ● ● ●● G ● S0 5 10 20 30 40 500102030405060
Dispersion R unn i ngT i m e ( s ) Summable terms ● ● ● ● ● ● ●● ● ● ● ● ● ●● AP ● MAP0 5 10 20 30 40 500255075100125150175
Dispersion R unn i ngT i m e ( s ) Summable terms
Figure 3.1:
Comparison of the effect of dispersion on Gosper’s algorithm, theAbramov-Petkovšek reduction and the modified version for Examples 3.23 and 3.24 hapter 4
Further Properties ofResidual Forms In Chapter 3, we presented a modified version of the Abramov-Petkovšek reduction,which decomposes a univariate hypergeometric term into a summable part and anon-summable part. Moreover, the non-summable part is described by a residualform. In [15], the authors used the Hermite reduction for univariate hyperexponentialfunctions to compute telescopers for bivariate hyperexponential functions. It allowsone to separate the computation of telescopers from that of certificates. We try totranslate their idea into the hypergeometric setting.We call a bivariate nonzero term hypergeometric if its shift-quotients with re-spect to the two variables are both rational functions. Given a hypergeometric term 𝑇 ( 𝑛, 𝑘 ). Let 𝜎 𝑛 and 𝜎 𝑘 be the shift operators w.r.t. 𝑛 and 𝑘 , respectively. Applyingthe modified Abramov-Petkovšek reduction to 𝑇 as well as its shifts 𝜎 𝑛 ( 𝑇 ) , . . . , 𝜎 𝑖𝑛 ( 𝑇 )w.r.t. 𝑘 , where 𝑖 is a nonnegative integer, we obtain 𝜎 𝑗𝑛 ( 𝑇 ) ≡ 𝑘 𝑟 𝑗 𝐻 mod U 𝐾 for 𝑗 = 0 , . . . , 𝑖, where 𝐻 is another bivariate hypergeometric term whose shift-quotient 𝐾 w.r.t. 𝑘 is shift-reduced w.r.t. 𝑘 , and 𝑟 𝑗 is a residual form w.r.t. 𝐾 . For univariate rationalfunctions 𝑐 ( 𝑛 ) , 𝑐 ( 𝑛 ) , . . . , 𝑐 𝑖 ( 𝑛 ), not all zero, we have 𝑖 ∑︁ 𝑗 =0 𝑐 𝑗 𝜎 𝑗𝑛 ( 𝑇 ) ≡ 𝑘 𝑖 ∑︁ 𝑗 =0 𝑐 𝑗 𝑟 𝑗 𝐻 mod U 𝐾 . It is readily seen that ∑︀ 𝑖𝑗 =0 𝑐 𝑗 𝜎 𝑗𝑛 is a telescoper for 𝑇 w.r.t. 𝑘 if ∑︀ 𝑖𝑗 =0 𝑐 𝑗 𝑟 𝑗 = 0.Unfortunately, the converse is false. This is because ∑︀ 𝑖𝑗 =0 𝑐 𝑗 𝑟 𝑗 is not necessarilya residual form, although all the 𝑟 𝑗 ’s are. Thus Theorem 3.18 is not applicable.This situation does not occur in the differential case [15]. To make Theorem 3.18applicable, we need to find a way to make ∑︀ 𝑖𝑗 =1 𝑐 𝑗 𝑟 𝑗 a residual form.This chapter aims at connecting univariate hypergeometric terms with bivariateones for the next two chapters. In this chapter, we present further properties ofresidual forms so as to estimate the order bounds of telescopers in Chapter 6. To The main results in this chapter are joint work with S. Chen, M. Kauers, Z. Li, published in [19]. Chapter 4. Further Properties of Residual Forms make the modified reduction applicable to compute telescopers for hypergeometricterms in Chapter 5, we also show that the linear combination of residual forms iswell-behaved in terms of congruences.
In this section, we recall the notion of rational normal forms from [10] and reviewthe relation between distinct rational normal forms of a rational function.
Definition 4.1.
Two polynomials 𝑝, 𝑞 ∈ F [ 𝑘 ] are called shift-equivalent w.r.t. 𝑘 ifthere exists an integer 𝑚 such that 𝑝 = 𝜎 𝑚𝑘 ( 𝑞 ) . We denote it by 𝑝 ∼ 𝑘 𝑞 . It is readily seen that ∼ 𝑘 is an equivalence relation. We call a polynomial in F [ 𝑘 ] monic if its leading coefficient w.r.t. 𝑘 is 1. Definition 4.2.
Let 𝑓 be a rational function in F ( 𝑘 ) . A rational function pair ( 𝐾, 𝑆 ) with 𝐾, 𝑆 ∈ F ( 𝑘 ) is called a rational normal form of 𝑓 if 𝑓 = 𝐾 · 𝜎 𝑘 ( 𝑆 ) 𝑆 and 𝐾 is shift-reduced. By Theorem 1 in [10], every rational function has a rational normal form. It isnot hard to see that there is a one-to-one correspondence between multiplicativedecompositions for a given hypergeometric term and rational normal forms for thecorresponding shift-quotient. More precisely, for a hypergeometric term 𝑇 over F ( 𝑘 ),a rational function pair ( 𝐾, 𝑆 ) is a rational normal form of 𝜎 𝑘 ( 𝑇 ) /𝑇 if and only if 𝐾 is a kernel of 𝑇 and 𝑆 a corresponding shell, if and only if 𝑇 has a multiplicativedecomposition 𝑇 = 𝑆𝐻 with 𝐻 a hypergeometric term whose shift-quotient is 𝐾 .In fact, a rational function can have more than one rational normal form, asillustrated by the following example. Example 4.3 (Example 1 in [10]) . Consider a rational function 𝑓 = 𝑘 ( 𝑘 + 2)( 𝑘 − 𝑘 + 1) ( 𝑘 + 3) . It can be verified that the following rational function pairs (︂ 𝑘 + 1)( 𝑘 + 3) , ( 𝑘 − 𝑘 + 1) )︂ , (︃ 𝑘 + 1) , 𝑘 − 𝑘 + 2 )︃ , (︂ 𝑘 − 𝑘 − , 𝑘 + 1 𝑘 )︂ , (︂ 𝑘 − 𝑘 + 1) , 𝑘 ( 𝑘 + 2) )︂ . are all rational normal forms of 𝑓 .The next theorem describes a relation between two distinct rational normal formsof a rational function. .2. Uniqueness and relatedness of residual forms Theorem 4.4 (Theorem 2 in [10]) . Assume that ( 𝐾, 𝑆 ) , ( 𝐾 ′ , 𝑆 ′ ) ∈ F ( 𝑘 ) are distinctrational normal forms of a rational function in F ( 𝑘 ) . Write 𝐾 = 𝑐 𝑢𝑣 and 𝐾 ′ = 𝑐 ′ 𝑢 ′ 𝑣 ′ , where 𝑐, 𝑐 ′ ∈ F , 𝑢, 𝑢 ′ , 𝑣, 𝑣 ′ ∈ F [ 𝑘 ] are all monic, and gcd( 𝑢, 𝑣 ) = gcd( 𝑢 ′ , 𝑣 ′ ) = 1 . Then ( 𝑖 ) 𝑐 = 𝑐 ′ ; ( 𝑖𝑖 ) deg 𝑘 ( 𝑢 ) = deg 𝑘 ( 𝑢 ′ ) and deg 𝑘 ( 𝑣 ) = deg 𝑘 ( 𝑣 ′ ) ; ( 𝑖𝑖𝑖 ) there is a one-to-one correspondence 𝜑 between the multi-sets of nontrivialmonic irreducible factors of 𝑢 and 𝑢 ′ such that 𝑝 ∼ 𝑘 𝜑 ( 𝑝 ) for any nontrivialmonic irreducible factor 𝑝 of 𝑢 . ( 𝑖𝑣 ) there is a one-to-one correspondence 𝜓 between the multi-sets of nontrivialmonic irreducible factors of 𝑣 and 𝑣 ′ such that 𝑝 ∼ 𝑘 𝜑 ( 𝑝 ) for any nontrivialmonic irreducible factor 𝑝 of 𝑣 . In this section, we will present two useful properties of residual forms, which enablesus to derive order bounds in Chapter 6. For the notion of residual forms, one canrefer to Definition 3.12.Unlike the differential case, a rational function may have more than one residualform in the shift case. These residual forms, however, are related to each other insome way. To describe it precisely, we introduce the notion of shift-relatedness.
Definition 4.5.
Two shift-free polynomials 𝑝, 𝑞 ∈ F [ 𝑘 ] are called shift-related , de-noted by 𝑝 ≈ 𝑘 𝑞 , if for any nontrivial monic irreducible factor 𝑓 of 𝑝 , there exists aunique monic irreducible factor 𝑔 of 𝑞 with the same multiplicity as 𝑓 in 𝑝 such that 𝑓 ∼ 𝑘 𝑔 , and vice versa. It is readily seen that ≈ 𝑘 is an equivalence relation. The following theorem de-scribes the uniqueness of residual forms. Theorem 4.6.
Let 𝐾 ∈ F ( 𝑘 ) be a shift-reduced rational function. Assume that 𝑟 , 𝑟 are both residual forms of a same rational function in F ( 𝑘 ) w.r.t. 𝐾 . Then thesignificant denominators of 𝑟 and 𝑟 are shift-related to each other.Proof. Assume that 𝑟 , 𝑟 are of the forms 𝑟 = 𝑎 𝑏 + 𝑞 𝑣 and 𝑟 = 𝑎 𝑏 + 𝑞 𝑣 , where for 𝑖 = 1 , 𝑎 𝑖 , 𝑏 𝑖 ∈ F [ 𝑘 ], deg 𝑘 ( 𝑎 𝑖 ) < deg 𝑘 ( 𝑏 𝑖 ), gcd( 𝑎 𝑖 , 𝑏 𝑖 ) = 1, 𝑏 𝑖 is monic,shift-free and strongly coprime with 𝐾 , 𝑞 𝑖 ∈ W 𝐾 , and 𝑣 is the denominator of 𝐾 .Since 𝑟 , 𝑟 are both residual forms of the same rational function, 𝑟 ≡ 𝑘 𝑟 mod V 𝐾 ,which is equivalent to 𝑎 𝑏 ≡ 𝑘 𝑎 𝑏 + 𝑞 − 𝑞 𝑣 mod V 𝐾 . Chapter 4. Further Properties of Residual Forms
By (2.1), there exists 𝑤 ∈ F ( 𝑘 ) so that 𝑎 𝑣𝑏 = 𝑢𝜎 𝑘 ( 𝑤 ) − 𝑣𝑤 + 𝑎 𝑣𝑏 + ( 𝑞 − 𝑞 ) . (4.1)Let 𝑓 ∈ F [ 𝑘 ] be a nontrivial monic irreducible factor of 𝑏 with multiplicity 𝛼 > 𝑓 𝛼 divides 𝑏 , then we are done. Otherwise, let den( 𝑤 ) be the denominator of 𝑤 .Since 𝑏 is strongly coprime with 𝐾 , we have gcd( 𝑓 𝛼 , 𝑣 ) = 1. By (4.1) and partialfraction decomposition, 𝑓 𝛼 either divides den( 𝑤 ) or 𝜎 𝑘 (den( 𝑤 )). If 𝑓 𝛼 divides den( 𝑤 ),let 𝑚 = max { 𝑘 ∈ Z | 𝜎 𝑘𝑘 ( 𝑓 ) 𝛼 divides den( 𝑤 ) } , and then 𝑚 ≥
0. Since 𝑏 is strongly coprime with 𝐾 , gcd( 𝜎 𝑚 +1 𝑘 ( 𝑓 ) 𝛼 , 𝑢 ) = 1. Ap-parently, 𝜎 𝑚 +1 𝑘 ( 𝑓 ) 𝛼 divides 𝜎 𝑘 (den( 𝑤 )) but doesn’t divide den( 𝑤 ) as 𝑚 is maximal.Note that 𝑏 is shift-free and 𝑓 | 𝑏 , thus 𝑏 is not divisible by 𝜎 𝑚 +1 𝑘 ( 𝑓 ) 𝛼 . Hence (4.1)implies 𝜎 𝑚 +1 𝑘 ( 𝑓 ) 𝛼 is the required factor of 𝑏 . Similarly, we can show that 𝜎 ℓ𝑘 ( 𝑓 ) 𝛼 with ℓ = min { 𝑘 ∈ Z | 𝜎 𝑘𝑘 ( 𝑓 ) 𝛼 divides den( 𝑤 ) } ≤ − , is the required factor of 𝑏 , if 𝑓 𝛼 divides 𝜎 𝑘 (den( 𝑤 )).In summary, there always exists a monic irreducible factor of 𝑏 with multiplicityat least 𝛼 such that it is shift-equivalent to 𝑓 . Due to the shift-freeness of 𝑏 , thisfactor is unique. The same conclusion holds when we switch the roles of 𝑏 and 𝑏 .Therefore, 𝑏 ≈ 𝑘 𝑏 by definition.For a given hypergeometric term, the above theorem reveals the relation betweentwo residual forms of the shell with respect to a same kernel. To study the case withdifferent kernels, we need the following two lemmas. Lemma 4.7.
Let ( 𝐾, 𝑆 ) be a rational normal form of 𝑓 ∈ F ( 𝑘 ) and 𝑟 a residualform of 𝑆 w.r.t. 𝐾 . Write 𝐾 = 𝑢/𝑣 with 𝑢, 𝑣 ∈ F [ 𝑘 ] and gcd( 𝑢, 𝑣 ) = 1 . Assume that 𝑝 is a nontrivial monic irreducible factor of 𝑣 with multiplicity 𝛼 > . Then the pair ( 𝐾 ′ , 𝑆 ′ ) = (︂ 𝑢𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 , 𝑝 𝛼 𝑆 )︂ is a rational normal form of 𝑓 , in which 𝑣 ′ = 𝑣/𝑝 𝛼 . Moreover, there exists a residualform 𝑟 ′ of 𝑆 ′ w.r.t. 𝐾 ′ whose significant denominator equals that of 𝑟 .Proof. Since 𝐾 is shift-reduced, so is 𝐾 ′ . The first assertion follows by noticing 𝐾 𝜎 𝑘 ( 𝑆 ) 𝑆 = 𝑢𝑣 ′ 𝑝 𝛼 𝜎 𝑘 ( 𝑆 ) 𝑆 = 𝑢𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 𝜎 𝑘 ( 𝑝 𝛼 𝑆 ) 𝑝 𝛼 𝑆 = 𝐾 ′ 𝜎 𝑘 ( 𝑆 ′ ) 𝑆 ′ . Let 𝑟 be of the form 𝑟 = 𝑎/𝑏 + 𝑞/𝑣 , where 𝑎, 𝑏, 𝑞 ∈ F [ 𝑘 ], deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), gcd( 𝑎, 𝑏 ) =1, 𝑏 is monic, shift-free and strongly coprime with 𝐾 , and 𝑞 ∈ W 𝐾 . Then there existsa rational function 𝑔 ∈ F ( 𝑘 ) such that 𝑆 = 𝐾𝜎 𝑘 ( 𝑔 ) − 𝑔 + 𝑎𝑏 + 𝑞𝑣 ′ 𝑝 𝛼 , .2. Uniqueness and relatedness of residual forms 𝑆 ′ = 𝑝 𝛼 𝑆 = 𝑝 𝛼 𝐾𝜎 𝑘 ( 𝑔 ) − 𝑝 𝛼 𝑔 + 𝑎𝑝 𝛼 𝑏 + 𝑞𝑣 ′ = 𝑢𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 𝜎 𝑘 ( 𝑝 𝛼 𝑔 ) − 𝑝 𝛼 𝑔 + 𝑎𝑝 𝛼 𝑏 + 𝑞𝜎 𝑘 ( 𝑝 ) 𝛼 𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 = 𝐾 ′ 𝜎 𝑘 ( 𝑝 𝛼 𝑔 ) − 𝑝 𝛼 𝑔 + 𝑎𝑝 𝛼 𝑏 + 𝑞𝜎 𝑘 ( 𝑝 ) 𝛼 𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 Since 𝑏 is strongly coprime with 𝐾 and gcd( 𝑎, 𝑏 ) = 1, we have gcd( 𝑎𝑝 𝛼 , 𝑏 ) = 1.Using step 3 and step 4 in Algorithm 3.17 computes polynomials 𝑎 ′ , 𝑞 ′ ∈ F [ 𝑘 ] withdeg 𝑘 ( 𝑎 ′ ) < deg 𝑘 ( 𝑏 ), gcd( 𝑎 ′ , 𝑏 ) = 1 and 𝑞 ′ ∈ W 𝐾 ′ so that 𝑆 ′ ≡ 𝑘 𝑎 ′ 𝑏 + 𝑞 ′ 𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 mod V 𝐾 ′ . Note that 𝑏 is strongly coprime with 𝐾 , so 𝑏 is also strongly coprime with 𝐾 ′ . Since 𝑏 is shift-free, 𝑎 ′ /𝑏 + 𝑞 ′ / ( 𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 ) is a residual form of 𝑆 ′ w.r.t. 𝐾 ′ . Lemma 4.8.
Let ( 𝐾, 𝑆 ) be a rational normal form of 𝑓 ∈ F ( 𝑘 ) and 𝑟 a residualform of 𝑆 w.r.t. 𝐾 . Write 𝐾 = 𝑢/𝑣 with 𝑢, 𝑣 ∈ F [ 𝑘 ] and gcd( 𝑢, 𝑣 ) = 1 . Assume that 𝑝 is a nontrivial monic irreducible factor of 𝑢 with multiplicity 𝛼 > . Then the pair ( 𝐾 ′ , 𝑆 ′ ) = (︃ 𝑢 ′ 𝜎 − 𝑘 ( 𝑝 ) 𝛼 𝑣 , 𝜎 − 𝑘 ( 𝑝 ) 𝛼 𝑆 )︃ is a rational normal form of 𝑓 , in which 𝑢 ′ = 𝑢/𝑝 𝛼 . Moreover, there exists a residualform 𝑟 ′ of 𝑆 ′ w.r.t. 𝐾 ′ whose significant denominator equals that of 𝑟 .Proof. Similar to Lemma 4.7.
Proposition 4.9.
Let ( 𝐾, 𝑆 ) be a rational normal form of 𝑓 ∈ F ( 𝑘 ) and 𝑟 a residualform of 𝑆 w.r.t. 𝐾 . Then there exists a rational normal form ( ˜ 𝐾, ˜ 𝑆 ) of 𝑓 such that1. ˜ 𝐾 has shift-free numerator and shift-free denominator;2. there exists a residual form ˜ 𝑟 of ˜ 𝑆 w.r.t. ˜ 𝐾 whose significant denominator isequal to that of 𝑟 .Proof. Let 𝐾 = 𝑢/𝑣 with 𝑢, 𝑣 ∈ F [ 𝑘 ] and gcd( 𝑢, 𝑣 ) = 1, and 𝑏 be the significantdenominator of 𝑟 .Assume that 𝑣 is not shift-free. Then there exist two nontrivial monic irreduciblefactors 𝑝 and 𝜎 𝑚𝑘 ( 𝑝 ) ( 𝑚 >
0) of 𝑣 with multiplicity 𝛼 > 𝛽 >
0, respectively.W.l.o.g., assume further that 𝜎 ℓ𝑘 ( 𝑝 ) is not a factor of 𝑣 for all ℓ < ℓ > 𝑚 . ByLemma 4.7, 𝑓 has a rational normal form ( 𝐾 ′ , 𝑆 ′ ), in which 𝐾 ′ has a denominator ofthe form den( 𝐾 ′ ) = 𝑣 ′ 𝜎 𝑘 ( 𝑝 ) 𝛼 , where 𝑣 ′ = 𝑣/𝑝 𝛼 , and the numerator remains to be 𝑢 .Moreover, there exists a residual form of 𝑆 ′ w.r.t. 𝐾 ′ whose significant denominatoris 𝑏 . If 𝑚 = 1, then 𝜎 𝑘 ( 𝑝 ) is an irreducible factor of den( 𝐾 ′ ) with multiplicity 𝛼 + 𝛽 . Otherwise, it is an irreducible factor of den( 𝐾 ′ ) with multiplicity 𝛼 . More2 Chapter 4. Further Properties of Residual Forms importantly, 𝜎 ℓ𝑘 ( 𝑝 ) is not a factor of den( 𝐾 ′ ) for all ℓ <
1. Iteratively using theargument, we arrive at a rational normal form of 𝑓 such that 𝜎 𝑚𝑘 ( 𝑝 ) divides thedenominator of the new kernel with certain multiplicity but 𝜎 𝑖𝑘 ( 𝑝 ) does not whenever 𝑖 ̸ = 𝑚 , and the numerator remains to be 𝑢 . Moreover, there exists a residual form ofthe new shell with respect to the new kernel whose significant denominator is equalto 𝑏 . Applying the same argument to each irreducible factor, we can obtain a rationalnormal form of 𝑓 whose kernel has the numerator 𝑢 and a shift-free denominator,and whose shell has a residual form with significant denominator 𝑏 .With Lemma 4.8, one can obtain a rational normal form of 𝑓 whose kernel hasa shift-free numerator and whose shell has a residual form with significant denomi-nator 𝑏 .A nonzero rational function is said to be shift-free if it is shift-reduced and itsdenominator and numerator are both shift-free. The relatedness of residual formswith respect to different kernels is given below. Theorem 4.10.
Let ( 𝐾, 𝑆 ) , ( 𝐾 ′ , 𝑆 ′ ) be two rational normal forms of 𝑓 ∈ F ( 𝑘 ) ,and 𝑟, 𝑟 ′ residual forms of 𝑆 (w.r.t. 𝐾 ) and 𝑆 ′ (w.r.t. 𝐾 ′ ), respectively. Then thesignificant denominators of 𝑟 and 𝑟 ′ are shift-related.Proof. Let 𝑏 and 𝑏 ′ be the significant denominators of 𝑟 and 𝑟 ′ , respectively. Bythe above proposition, there exist two rational normal forms ( ˜ 𝐾, ˜ 𝑆 ) and ( ˜ 𝐾 ′ , ˜ 𝑆 ′ ) of 𝑓 such that their kernels are shift-free and their shells have residual forms whosesignificant denominators are 𝑏 and 𝑏 ′ , respectively.According to Theorem 4.4, the respective denominators ˜ 𝑣 and ˜ 𝑣 ′ of ˜ 𝐾 and ˜ 𝐾 ′ are shift-related. It follows that for a nontrivial monic irreducible factor 𝑝 of ˜ 𝑣 withmultiplicity 𝛼 >
0, there exists a unique factor 𝜎 ℓ𝑘 ( 𝑝 ) with ℓ ∈ Z of ˜ 𝑣 ′ with thesame multiplicity. W.l.o.g., we may assume ℓ ≤
0. Otherwise, we can switch theroles of two pairs ( ˜ 𝐾, ˜ 𝑆 ) and ( ˜ 𝐾 ′ , ˜ 𝑆 ′ ). If ℓ <
0, a repeated use of Lemma 4.7 leadsto a new rational normal form ( ˜ 𝐾 ′′ , ˜ 𝑆 ′′ ) from ( ˜ 𝐾 ′ , ˜ 𝑆 ′ ), such that ˜ 𝐾 ′′ is shift-freewith the same numerator as ˜ 𝐾 ′ , and 𝑝 is a factor of the denominator of ˜ 𝐾 ′′ withthe same multiplicity 𝛼 . Moreover, ˜ 𝑆 ′′ has a residual form w.r.t. ˜ 𝐾 ′′ with significantdenominator 𝑏 ′ .Applying the above argument to each irreducible factor and using Lemma 4.8 fornumerators in the same fashion, we can obtain two new rational normal forms whosekernels are equal and whose shells have respective residual forms with significantdenominators 𝑏 and 𝑏 ′ . It follows from Theorem 4.6 that 𝑏 and 𝑏 ′ are shift-related. To compute a telescoper for a given bivariate hypergeometric terms by the modifiedAbramov-Petkovšek reduction, we are confronted with the difficulty that the sum oftwo residual forms is not necessarily a residual form, as mentioned at the beginning ofthis chapter. This is because the least common multiple of two shift-free polynomialsis not necessarily shift-free.The goal of this section is to show that the sum of two residual forms is congruentto a residual form modulo V 𝐾 . .3. Sum of two residual forms Example 4.11.
Let 𝐾 = 1 /𝑘 , 𝑟 = 1 / (2 𝑘 + 1) and 𝑠 = 1 / (2 𝑘 + 3). Then both 𝑟 and 𝑠 are residual forms w.r.t. 𝐾 , but their sum is not, because the denominator(2 𝑘 + 1)(2 𝑘 + 3) is not shift-free. However, we can still find an equivalent residualform. For example, we have 𝑟 + 𝑠 ≡ 𝑘 − 𝑘 + 1) + 12 𝑘 mod V 𝐾 . Note that the residual form is not unique. Another possible choice is 𝑟 + 𝑠 ≡ 𝑘 𝑘 + 3) + 13 𝑘 mod V 𝐾 . Lemma 4.12.
With Convention 3.2, let 𝑟, 𝑠 ∈ F ( 𝑘 ) be two residual forms w.r.t. 𝐾 ,i.e., 𝑟 and 𝑠 can be written as 𝑟 = 𝑎𝑓 + 𝑝𝑣 and 𝑠 = 𝑏𝑔 + 𝑞𝑣 , where 𝑎, 𝑓, 𝑏, 𝑔 ∈ F [ 𝑘 ] , deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑓 ) , deg 𝑘 ( 𝑏 ) < deg 𝑘 ( 𝑔 ) , 𝑝, 𝑞 ∈ W 𝐾 , and 𝑓, 𝑔 are shift-free and strongly coprime with 𝐾 . Assume that gcd( 𝑎, 𝑓 ) = gcd( 𝑏, 𝑔 ) = 1 .Then for all 𝜆, 𝜇 ∈ F , 𝜆𝑟 + 𝜇𝑠 is a residual form w.r.t. 𝐾 if and only if the leastcommon multiple of 𝑓 and 𝑔 is shift-free.Proof. Let ℎ be the least common multiple of 𝑓 and 𝑔 . Then 𝜆𝑟 + 𝜇𝑠 = 𝜆𝑎 ( ℎ/𝑓 ) + 𝜇𝑏 ( ℎ/𝑔 ) ℎ + 𝜆𝑝 + 𝜇𝑞𝑣 . (4.2)We first show the sufficiency. Assume that ℎ is shift-free. It is clear thatdeg 𝑘 ( 𝜆𝑎 ( ℎ/𝑓 ) + 𝜇𝑏 ( ℎ/𝑔 )) < deg 𝑘 ( ℎ ) . Since W 𝐾 is a F -vector space, we have 𝜆𝑝 + 𝜇𝑞 ∈ W 𝐾 . Note that 𝑓 and 𝑔 are stronglycoprime with 𝐾 , so is ℎ . By definition, 𝜆𝑟 + 𝜇𝑠 is a residual form w.r.t. 𝐾 .To show the necessity, we suppose otherwise that ℎ is not shift-free. Since 𝜆𝑟 + 𝜇𝑠 is a residual form w.r.t. 𝐾 , there exist 𝑏 * , ℎ * ∈ F [ 𝑘 ] and 𝑞 * ∈ W 𝐾 with deg 𝑘 ( 𝑏 * ) < deg 𝑘 ( ℎ * ), and ℎ * shift-free and strongly coprime with 𝐾 , such that 𝜆𝑟 + 𝜇𝑠 = 𝑏 * ℎ * + 𝑞 * 𝑣 . It follows from (4.2) that( 𝜆𝑎 ( ℎ/𝑓 ) + 𝜇𝑏 ( ℎ/𝑔 )) 𝑣ℎ = 𝑏 * 𝑣ℎ * + 𝑞 * − 𝜆𝑝 − 𝜇𝑞. (4.3)Since ℎ is not shift-free and 𝑓, 𝑔 are shift-free, there exist nontrivial monic irreduciblefactors 𝑝 ′ and 𝜎 ℓ𝑘 ( 𝑝 ′ ) of ℎ such that 𝑝 ′ | 𝑓 and 𝜎 ℓ𝑘 ( 𝑝 ′ ) | 𝑔 , where ℓ is a nonzero integer.Because gcd( 𝑎, 𝑓 ) = gcd( 𝑏, 𝑔 ) = 1 and ℎ | 𝑓 𝑔 , so• 𝑝 ′ (cid:45) ( ℎ/𝑓 ) and 𝑝 ′ (cid:45) 𝑎 , but 𝑝 ′ | ( ℎ/𝑔 );• 𝜎 ℓ𝑘 ( 𝑝 ′ ) (cid:45) ( ℎ/𝑔 ) and 𝜎 ℓ𝑘 ( 𝑝 ) (cid:45) 𝑏 , but 𝜎 ℓ𝑘 ( 𝑝 ′ ) | ( ℎ/𝑓 ).4 Chapter 4. Further Properties of Residual Forms
Since ℎ is also strongly coprime with 𝐾 , 𝑝 ′ and 𝜎 ℓ𝑘 ( 𝑝 ′ ) are coprime with 𝑣 . Thusthey both divide the denominator of the left-hand side of (4.3). By partial fractiondecomposition, 𝑝 ′ and 𝜎 ℓ𝑘 ( 𝑝 ′ ) both divide ℎ * , a contradiction as ℎ * is shift-free.To describe the shift-freeness of the least common multiple of two polynomials,we introduce the following notions. Definition 4.13.
Let 𝑓 and 𝑔 be two nonzero polynomials in F [ 𝑘 ] . According to [10,§3], the dispersion of 𝑓 and 𝑔 is defined to be the largest nonnegative integer ℓ such that 𝑓 and 𝜎 ℓ𝑘 ( 𝑔 ) have a nontrivial common divisor, or − if no such ℓ exists.Moreover, we say that 𝑓 and 𝑔 are shift-coprime if gcd( 𝑓, 𝜎 ℓ𝑘 ( 𝑔 )) = 1 for all nonzerointeger ℓ . It is clear that the least common multiple of two shift-free polynomials is shift-free if and only if these two polynomials are shift-coprime. Let 𝑓 and 𝑔 be twononzero shift-free polynomials in F [ 𝑘 ]. By polynomial factorization and dispersioncomputation (see [10]), one can uniquely decompose 𝑔 = ˜ 𝑔𝜎 ℓ 𝑘 (︀ 𝑝 𝑚 )︀ · · · 𝜎 ℓ 𝜌 𝑘 (︁ 𝑝 𝑚 𝜌 𝜌 )︁ , (4.4)where ˜ 𝑔 is shift-coprime with 𝑓 , 𝑝 , . . . , 𝑝 𝜌 are pairwise distinct and monic irreduciblefactors of 𝑓 , ℓ , . . . , ℓ 𝜌 are nonzero integers, 𝑚 , . . . , 𝑚 𝜌 are multiplicities of thefactors 𝜎 ℓ 𝑘 ( 𝑝 ), . . . , 𝜎 ℓ 𝜌 𝑘 ( 𝑝 𝜌 ) in 𝑔 , respectively. We refer to (4.4) as the shift-coprimedecomposition of 𝑔 w.r.t. 𝑓 . Remark 4.14.
The factors ˜ 𝑔, 𝜎 ℓ 𝑘 (︀ 𝑝 𝑚 )︀ , . . . , 𝜎 ℓ 𝜌 𝑘 (︁ 𝑝 𝑚 𝜌 𝜌 )︁ in (4.4) are pairwise co-prime, since 𝑓 and 𝑔 are shift-free.To construct a residual form congruent to the sum of two given residual ones,we need three technical lemmas. The first one corresponds to the kernel reductionin [15]. Lemma 4.15.
With Convention 3.2, assume that 𝑝 , 𝑝 are in F [ 𝑘 ] and 𝑚 in N .Then there exist 𝑞 , 𝑞 in W 𝐾 such that 𝑝 ∏︀ 𝑚𝑖 =0 𝜎 𝑖𝑘 ( 𝑣 ) ≡ 𝑘 𝑞 𝑣 mod V 𝐾 and 𝑝 ∏︀ 𝑚𝑗 =1 𝜎 − 𝑗𝑘 ( 𝑢 ) ≡ 𝑘 𝑞 𝑣 mod V 𝐾 . Proof.
To prove the first congruence, let 𝑤 𝑚 = ∏︀ 𝑚𝑖 =0 𝜎 𝑖𝑘 ( 𝑣 ).We proceed by induction on 𝑚 . If 𝑚 = 0, then the conclusion holds by Lemma 3.10.Assume that the lemma holds for 𝑚 − 𝑚 >
0. Consider the equality 𝑝 𝑤 𝑚 = 𝐾𝜎 𝑘 (︂ 𝑠𝑤 𝑚 − )︂ − 𝑠𝑤 𝑚 − + 𝑡𝑤 𝑚 − , where 𝑠, 𝑡 ∈ F [ 𝑘 ] are to be determined. This equality holds if and only if 𝜎 𝑘 ( 𝑠 ) 𝑢 + ( 𝑡 − 𝑠 ) 𝜎 𝑚𝑘 ( 𝑣 ) = 𝑝 . .3. Sum of two residual forms 𝑢 and 𝜎 𝑚𝑘 ( 𝑣 ) are coprime, such 𝑠 and 𝑡 can be computed by the extendedEuclidean algorithm. Thus, 𝑝 /𝑤 𝑚 ≡ 𝑘 𝑡/𝑤 𝑚 − mod V 𝐾 . Consequently, 𝑝 /𝑤 𝑚 hasa required residual form by the induction hypothesis.To prove the second congruence, we use the identity 𝑝 𝜎 − 𝑘 ( 𝑢 ) = 𝐾𝜎 𝑘 (︃ − 𝑝 𝜎 − 𝑘 ( 𝑢 ) )︃ − (︃ − 𝑝 𝜎 − 𝑘 ( 𝑢 ) )︃ + 𝜎 𝑘 ( 𝑝 ) 𝑣 , which implies that 𝑝 /𝜎 − 𝑘 ( 𝑢 ) ≡ 𝑘 𝜎 𝑘 ( 𝑝 ) /𝑣 mod V 𝐾 . By Lemma 3.10, there exists apolynomial 𝑞 ∈ W 𝐾 such that 𝑞 /𝑣 is a residual form of 𝑝 /𝜎 − 𝑘 ( 𝑢 ) w.r.t. 𝐾 . Thusthe conclusion holds for 𝑚 = 0. Assume that the congruence holds for 𝑚 − 𝑚 >
0. The induction can be completed as in the proof for 𝑝 /𝑤 𝑚 .The next lemma provides us with flexibility to rewrite a rational function mod-ulo V 𝐾 . Lemma 4.16.
Let 𝐾 ∈ F ( 𝑘 ) be nonzero and shift-reduced. Then for every rationalfunction 𝑓 ∈ F ( 𝑘 ) and every positive integer ℓ , 𝑓 ≡ 𝑘 𝜎 ℓ𝑘 ( 𝑓 ) ℓ − ∏︁ 𝑖 =0 𝜎 𝑖𝑘 ( 𝐾 ) ≡ 𝑘 𝜎 − ℓ𝑘 ( 𝑓 ) ℓ ∏︁ 𝑖 =1 𝜎 − 𝑖𝑘 (︂ 𝐾 )︂ mod V 𝐾 . Proof.
Let’s show the first congruence by induction on ℓ . For ℓ = 1, the identity 𝑓 = 𝐾𝜎 𝑘 ( − 𝑓 ) − ( − 𝑓 ) + 𝜎 𝑘 ( 𝑓 ) 𝐾 implies that 𝑓 is congruent to 𝜎 𝑘 ( 𝑓 ) 𝐾 modulo V 𝐾 . Assume that it holds for ℓ − ℓ >
1. Set 𝑤 ℓ = ∏︀ ℓ − 𝑖 =0 𝜎 𝑖𝑘 ( 𝐾 ). Then by the induction hypothesis, 𝑓 ≡ 𝑘 𝜎 ℓ − 𝑘 ( 𝑓 ) 𝑤 ℓ − mod V 𝐾 . Moreover, 𝜎 ℓ − 𝑘 ( 𝑓 ) 𝑤 ℓ − ≡ 𝑘 𝜎 ℓ𝑘 ( 𝑓 ) 𝑤 ℓ mod V 𝐾 by the induction base, in which 𝑓 isreplaced with 𝜎 ℓ − 𝑘 ( 𝑓 ) 𝑤 ℓ − . Hence, 𝑓 is congruent to 𝜎 ℓ𝑘 ( 𝑓 ) 𝑤 ℓ modulo V 𝐾 .The second congruence can be shown similarly. For the base case ℓ = 1, let 𝑟 = 𝜎 − 𝑘 ( 𝑓 ) 𝜎 − 𝑘 (1 /𝐾 ). Then the identity 𝑓 = 𝐾𝜎 𝑘 ( 𝑟 ) − 𝑟 + 𝑟 implies that 𝑓 is congruentto 𝑟 modulo V 𝐾 . We can then proceed as in the proof of the first congruence. Lemma 4.17.
With Convention 3.2, let 𝑎, 𝑏 ∈ F [ 𝑘 ] with 𝑏 ̸ = 0 . Assume that 𝑏 isshift-free and strongly coprime with 𝐾 . Assume further that 𝜎 ℓ𝑘 ( 𝑏 ) is strongly coprimewith 𝐾 for some integer ℓ , then 𝑎/𝑏 has a residual form 𝑐/𝜎 ℓ𝑘 ( 𝑏 ) + 𝑞/𝑣 w.r.t. 𝐾 ,where 𝑐 ∈ F [ 𝑘 ] with deg 𝑘 ( 𝑐 ) < deg 𝑘 ( 𝑏 ) and 𝑞 ∈ W 𝐾 .Proof. First, consider the case in which ℓ ≥
0. If ℓ = 0, then there exist two polyno-mials 𝑐, 𝑝 ∈ F [ 𝑘 ] with deg 𝑘 ( 𝑐 ) < deg 𝑘 ( 𝑏 ) such that 𝑎/𝑏 = 𝑐/𝑏 + 𝑝 . The lemma followsfrom Remark 3.11. Assume that ℓ >
0. By the first congruence of Lemma 4.16, 𝑎𝑏 ≡ 𝑘 𝜎 ℓ𝑘 (︁ 𝑎𝑏 )︁ (︃ ℓ − ∏︁ 𝑖 =0 𝜎 𝑖𝑘 ( 𝐾 ) )︃ = 𝜎 ℓ𝑘 ( 𝑎 ) 𝜎 ℓ𝑘 ( 𝑏 ) ∏︀ ℓ − 𝑖 =0 𝜎 𝑖𝑘 ( 𝑢 ) ∏︀ ℓ − 𝑖 =0 𝜎 𝑖𝑘 ( 𝑣 ) mod V 𝐾 . Chapter 4. Further Properties of Residual Forms
Note that 𝜎 ℓ𝑘 ( 𝑏 ) is strongly coprime with 𝑣 by assumption. Then it is coprime withthe product 𝑣𝜎 𝑘 ( 𝑣 ) · · · 𝜎 ℓ − 𝑘 ( 𝑣 ). By partial fraction decomposition, we get 𝑎𝑏 ≡ 𝑘 ˜ 𝑎𝜎 ℓ𝑘 ( 𝑏 ) + ˜ 𝑞 ∏︀ ℓ − 𝑖 =0 𝜎 𝑖𝑘 ( 𝑣 ) mod V 𝐾 , where ˜ 𝑎, ˜ 𝑞 ∈ F [ 𝑘 ] and deg 𝑘 ˜ 𝑎 < deg 𝑘 ( 𝑏 ). By the first congruence of Lemma 4.15, thesecond summand in the right-hand side of the above congruence can be replaced bya residual form whose denominator is equal to 𝑣 . The first assertion holds.The case ℓ < Remark 4.18.
With the assumptions of the above lemma, let 𝑝 be a nontrivialfactor of 𝑏 with gcd( 𝑏 ′ , 𝑝 ) = 1 where 𝑏 ′ = 𝑏/𝑝 . Assume that 𝜎 ℓ𝑘 ( 𝑝 ) is also stronglycoprime with 𝐾 . Then by partial fraction decomposition and Lemma 4.17, thereexist 𝑐, 𝑞 ∈ F [ 𝑘 ] with deg 𝑘 ( 𝑐 ) < deg 𝑘 ( 𝑏 ) and 𝑞 ∈ W 𝐾 such that 𝑐/ ( 𝑏 ′ 𝜎 ℓ𝑘 ( 𝑝 )) + 𝑞/𝑣 isa residual form of 𝑎/𝑏 w.r.t. 𝐾 .We will refer to Lemma 4.17 and Remark 4.18 as the shifting property of signif-icant denominators . Now we are ready to present the main result of this section. Theorem 4.19.
With Convection 3.2, let 𝑟 and 𝑠 be two residual forms w.r.t. 𝐾 .Then there exists a residual form 𝑡 congruent to 𝑠 modulo V 𝐾 so that for all con-stants 𝜆, 𝜇 ∈ F , the sum 𝜆𝑟 + 𝜇𝑡 is a residual form w.r.t. 𝐾 congruent to 𝜆𝑟 + 𝜇𝑠 modulo V 𝐾 .Proof. Since 𝑟 and 𝑠 are two residual forms w.r.t. 𝐾 , they can be written as 𝑟 = 𝑎𝑓 + 𝑝𝑣 and 𝑠 = 𝑏𝑔 + 𝑞𝑣 , (4.5)where 𝑎, 𝑓, 𝑏, 𝑔 ∈ F [ 𝑘 ], deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑓 ), deg 𝑘 ( 𝑏 ) < deg 𝑘 ( 𝑔 ), 𝑝, 𝑞 ∈ W 𝐾 , and 𝑓, 𝑔 are shift-free and strongly coprime with 𝐾 .Assume that (4.4) is the shift-coprime decomposition of 𝑔 w.r.t. 𝑓 . Define 𝑃 𝑖 = 𝜎 ℓ 𝑖 𝑘 ( 𝑝 𝑖 ) for 𝑖 = 1, . . . , 𝜌 . By Remark 4.14 and partial fraction decomposition, 𝑏𝑔 = 𝑏 ˜ 𝑔 + 𝜌 ∑︁ 𝑖 =1 𝑏 𝑖 𝑃 𝑚 𝑖 𝑖 , (4.6)where 𝑏 , 𝑏 , . . . , 𝑏 𝜌 ∈ F [ 𝑘 ], deg 𝑘 ( 𝑏 ) < deg 𝑘 (˜ 𝑔 ) and deg 𝑘 ( 𝑏 𝑖 ) < 𝑚 𝑖 deg 𝑘 ( 𝑝 𝑖 ). Notethat 𝑝 𝑖 = 𝜎 − ℓ 𝑖 𝑘 ( 𝑃 𝑖 ), which is a factor of 𝑓 . Thus it is strongly coprime with 𝐾 . Sowe can apply Lemma 4.17 to each fraction 𝑏 𝑖 /𝑃 𝑚 𝑖 𝑖 in (4.6) to get 𝑏𝑔 ≡ 𝑘 𝑏 ˜ 𝑔 + 𝜌 ∑︁ 𝑖 =1 𝑏 ′ 𝑖 𝑝 𝑚 𝑖 𝑖 + 𝑞 ′ 𝑣 mod V 𝐾 , (4.7)where 𝑏 ′ , . . . , 𝑏 ′ 𝜌 ∈ F [ 𝑘 ], deg 𝑘 ( 𝑏 ′ 𝑖 ) < 𝑚 𝑖 deg 𝑘 ( 𝑝 𝑖 ) and 𝑞 ′ ∈ W 𝐾 .Let ℎ = ˜ 𝑔 ∏︀ 𝜌𝑖 =1 𝑝 𝑚 𝑖 𝑖 . Then ℎ is shift-free and strongly coprime with 𝐾 as both 𝑓 and 𝑔 are. Since 𝑓 is shift-free, all its factors are shift-coprime with 𝑓 , so are the 𝑝 𝑖 ’s, .3. Sum of two residual forms ℎ . Let 𝑡 be the sum of 𝑞/𝑣 and the rational function in the right-hand sideof (4.7). Then there exist 𝑏 * ∈ F [ 𝑘 ] with deg 𝑘 ( 𝑏 * ) < deg 𝑘 ( ℎ ) and 𝑞 * ∈ W 𝐾 such that 𝑡 = 𝑏 * ℎ + 𝑞 * 𝑣 . Since 𝑓 and ℎ are shift-coprime, their least common multiple is shift-free. There-fore, 𝜆𝑟 + 𝜇𝑡 is a residual form w.r.t. 𝐾 by Lemma 4.12, and 𝜆𝑟 + 𝜇𝑡 is congruentto 𝜆𝑟 + 𝜇𝑠 modulo V 𝐾 .The above proof contains an algorithm, which can translate a residual formproperly according to a given one, so that the resulting sum is again a residual form.We outline this algorithm as follows. Algorithm 4.20 (Translation of Discrete Residual Forms) .Input : A shift-reduced rational function 𝐾 ∈ F ( 𝑘 ), a polynomial 𝑓 ∈ F [ 𝑘 ] whichis shift-free and strongly coprime with 𝐾 , and a residual form 𝑠 w.r.t. 𝐾 of theform (4.5). Output : A rational function 𝑤 ∈ F ( 𝑘 ) and a residual form 𝑡 w.r.t. 𝐾 such that 𝑠 = 𝐾𝜎 𝑘 ( 𝑤 ) − 𝑤 + 𝑡, and the least common multiple of the given polynomial 𝑓 and the significant denom-inator of 𝑡 is shift-free. Compute the shift-coprime decomposition, say (4.4), of 𝑔 w.r.t. 𝑓 . Set 𝑃 𝑖 = 𝜎 ℓ 𝑖 𝑘 ( 𝑝 𝑖 ) for 𝑖 = 1 , . . . , 𝜌 . Compute the partial fraction decomposition (4.6) of 𝑏/𝑔 . Apply Lemma 4.17 to each 𝑏 𝑖 /𝑃 𝑚 𝑖 𝑖 to find 𝑤 𝑖 ∈ F ( 𝑘 ) and 𝑏 ′ 𝑖 , 𝑞 ′ 𝑖 ∈ F [ 𝑘 ]with deg 𝑘 ( 𝑏 ′ 𝑖 ) < 𝑚 𝑖 deg 𝑘 ( 𝑝 𝑖 ) and 𝑞 ′ 𝑖 ∈ W 𝐾 such that 𝑏 𝑖 𝑃 𝑚 𝑖 𝑖 = 𝐾𝜎 𝑘 ( 𝑤 𝑖 ) − 𝑤 𝑖 + 𝑏 ′ 𝑖 𝑝 𝑚 𝑖 𝑖 + 𝑞 ′ 𝑖 𝑣 . Set 𝑤 = ∑︀ 𝜌𝑖 =1 𝑤 𝑖 and 𝑡 = 𝑏 ˜ 𝑔 + 𝜌 ∑︁ 𝑖 =1 𝑏 ′ 𝑖 𝑝 𝑚 𝑖 𝑖 + ∑︀ 𝜌𝑖 =1 𝑞 ′ 𝑖 + 𝑞𝑣 ;and return. hapter 5 Creative Telescoping for
Hypergeometric Terms In the study of combinatorics, we often encounter problems about evaluating definitesums or proving identities of hypergeometric terms. These terms are exactly nonzerosolutions of first-order (partial) difference equations with polynomial coefficients.Traditionally [56], such problems were solved case by case using methods that donot give rise to general algorithms. Based on a series of work [65, 66, 67, 68, 69, 70, 71]in early 1990s, Wilf and Zeilberger developed a constructive theory, which is nowknown as Wilf-Zeilberger’s theory. This theory provides a systematic solution to alarge class of problems concerning hypergeometric summations and identities, andhas wide application in the areas of combinatorics and physics. The key step ofWilf-Zeilberger’s theory is to compute a telescoper for a given hypergeometric term.The efficiency of the computation determines the utility of this theory. During thepast 26 years, numerous algorithms have been developed for computing telescopers.In early 1990s, Zeilberger [70] first came up with an algorithm based on elimina-tion techniques. This algorithm was improved later by Takayama [61] and Chyzak,Salvy [27], respectively. In 1990, Zeilberger [69] developed another algorithm, knownas Zeilberger’s (fast) algorithm, based on a parametrization of Gosper’s algorithm.15 years later, Apagodu and Zeilberger designed a new algorithm which reduced theproblem to solving a linear system. The common feature of the above algorithmsis that there was no way to obtain a telescoper without also computing a certifi-cate. In many applications, however, certificates are not needed, and they typicallyrequire more storage space than telescopers do. It would be more efficient to avoidcomputing certificates if we don’t need them. To achieve this goal, Bostan et al. [14]presented a new algorithm for bivariate rational functions in the differential case,based on the Hermite reduction. This algorithm separates the computation of tele-scopers and the corresponding certificates. So far, this approach has been generalizedto several instances including rational functions in three variables [24], multivariaterational functions [16], bivariate hyperexponential functions [15] and bivariate alge-braic functions [23]. These algorithms turn out to be more efficient than the classicalalgorithms in practice. However, all these algorithms only work for the differentialcase. The main results in this chapter are joint work with S. Chen, M. Kauers, Z. Li, published in [19]. Chapter 5. Creative Telescoping for Hypergeometric Terms
In this chapter, we discuss how to translate their ideas into the hypergeomet-ric setting. Using the modified Abramov-Petkovšek reduction, we develop a newcreative telescoping algorithm. This new algorithm separates the computation oftelescopers from that of certificates. We have implemented the new algorithm in
Maple 18 and compare it to the built-in Maple procedure
Zeilberger in the pack-age
SumTools[Hypergeometric] , which is based on Zeilberger’s algorithm. Theexperimental results indicate that the new algorithm is faster than the Maple pro-cedure if it returns a normalized certificate, and the new algorithm is much moreefficient if it omits the computation of certificates.
In this section, we translate terminology concerning univariate hypergeometric termsto bivariate ones and introduce the notions of telescopers as well as certificates forbivariate hypergeometric terms. Moreover, we recall [67, 4] an existence criterion fortelescopers.Let K be a field of characteristic zero, and K ( 𝑛, 𝑘 ) be the field of rational functionsin 𝑛 and 𝑘 over K . Let 𝜎 𝑛 and 𝜎 𝑘 be the shift operators w.r.t. 𝑛 and 𝑘 , respectively,defined by 𝜎 𝑛 ( 𝑓 ( 𝑛, 𝑘 )) = 𝑓 ( 𝑛 + 1 , 𝑘 ) and 𝜎 𝑘 ( 𝑓 ( 𝑛, 𝑘 )) = 𝑓 ( 𝑛, 𝑘 + 1) , for any rational function 𝑓 ∈ K ( 𝑛, 𝑘 ). Clearly, 𝜎 𝑛 and 𝜎 𝑘 are both automorphismsof K . The pair ( K ( 𝑛, 𝑘 ) , { 𝜎 𝑛 , 𝜎 𝑘 } ) forms a partial difference field . A partial differencering extension of ( K ( 𝑛, 𝑘 ) , { 𝜎 𝑛 , 𝜎 𝑘 } ) is a ring D containing K ( 𝑛, 𝑘 ) together with twodistinguished endomorphism 𝜎 𝑛 and 𝜎 𝑘 from D to itself, whose restrictions to K ( 𝑛, 𝑘 )agree with the two automorphisms defined before, respectively.Analogous to the univariate case in Chapter 2, an element 𝑐 ∈ D is called a constant if it is invariant under the applications of 𝜎 𝑛 and 𝜎 𝑘 . It is readily seen thatall constants in D form a subring of D . Moreover, Theorem 2 in [9] yields that theset of all constants in K ( 𝑛, 𝑘 ) w.r.t. 𝜎 𝑛 and 𝜎 𝑘 is exactly the field K . Definition 5.1.
Let D be a partial difference ring extension of K ( 𝑛, 𝑘 ) . A nonzeroelement 𝑇 ∈ D is called a hypergeometric term over K ( 𝑛, 𝑘 ) if it is invertible andthere exist 𝑓, 𝑔 ∈ K ( 𝑛, 𝑘 ) such that 𝜎 𝑛 ( 𝑇 ) = 𝑓 𝑇 and 𝜎 𝑘 ( 𝑇 ) = 𝑔𝑇 . We call 𝑓 and 𝑔 the shift-quotients of 𝑇 w.r.t. 𝑛 and 𝑘 , respectively. In the rest of this chapter and also the next chapter, whenever we mention hyper-geometric terms, they always belong to some difference ring extension D of K ( 𝑛, 𝑘 ),unless specified otherwise.Let F be the field K ( 𝑛 ), and F ⟨ 𝑆 𝑛 ⟩ be the ring of linear recurrence operators in 𝑛 ,in which the commutation rule is that 𝑆 𝑛 𝑟 = 𝜎 𝑛 ( 𝑟 ) 𝑆 𝑛 for all 𝑟 ∈ F . The applicationof an operator 𝐿 = ∑︀ 𝜌𝑖 =0 ℓ 𝑖 𝑆 𝑖𝑛 ∈ F ⟨ 𝑆 𝑛 ⟩ to a hypergeometric term 𝑇 is defined as 𝐿 ( 𝑇 ) = 𝜌 ∑︁ 𝑖 =0 ℓ 𝑖 𝜎 𝑖𝑛 ( 𝑇 ) . .1. Bivariate hypergeometric terms Definition 5.2.
Let 𝑇 be a hypergeometric term over F ( 𝑘 ) . A nonzero recurrenceoperator 𝐿 ∈ F ⟨ 𝑆 𝑛 ⟩ is called a telescoper for 𝑇 w.r.t. 𝑘 if there exists a hypergeo-metric term 𝐺 such that 𝐿 ( 𝑇 ) = Δ 𝑘 ( 𝐺 ) . We call 𝐺 a certificate of 𝐿 . In contrast to the differential case, telescopers for hypergeometric terms do notalways exist. To describe the existence of telescopers concisely, we recall [4] thedefinitions of integer-linear polynomials and proper terms.
Definition 5.3.
An irreducible polynomial 𝑝 ∈ K [ 𝑛, 𝑘 ] is called integer-linear over K if there exists 𝑃 ∈ K [ 𝑧 ] and 𝜆, 𝜇 ∈ Z such that 𝑝 = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 ) . A polynomialin K [ 𝑛, 𝑘 ] is called integer-linear over K if all of its irreducible factors are integer-linear. A rational function in K ( 𝑛, 𝑘 ) is called integer-linear over K if its denomi-nator and numerator are both integer-linear. Definition 5.4.
A hypergeometric term 𝑇 over K ( 𝑛, 𝑘 ) is called a factorial term ifthe shift-quotients 𝜎 𝑛 ( 𝑇 ) /𝑇 and 𝜎 𝑘 ( 𝑇 ) /𝑇 are integer-linear over K . A proper term over K ( 𝑛, 𝑘 ) is the product of a factorial term and a polynomial in K [ 𝑛, 𝑘 ] . We have the following existence criterion for telescopers according to [67, 4].
Theorem 5.5 (Existence criterion) . Let 𝑇 be a hypergeometric term over F ( 𝑘 ) andlet 𝐾 = 𝑢/𝑣 with 𝑢, 𝑣 ∈ F [ 𝑘 ] , gcd( 𝑢, 𝑣 ) = 1 be a kernel of 𝑇 w.r.t. 𝑘 and 𝑆 acorresponding shell of 𝑇 . Assume that applying Algorithm 3.17, i.e., the modifiedAbramov-Petkovšek reduction w.r.t. 𝑘 to 𝑇 yields 𝑇 = Δ 𝑘 ( 𝑔𝐻 ) + (︁ 𝑎𝑏 + 𝑞𝑣 )︁ 𝐻, (5.1) where 𝑔 ∈ F ( 𝑘 ) , 𝐻 = 𝑇 /𝑆 , and 𝑎/𝑏 + 𝑞/𝑣 is a residual form of 𝑆 w.r.t. 𝐾 , thatis, 𝑎, 𝑏 ∈ F [ 𝑘 ] with deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ) , 𝑏 is shift-free and strongly coprime with 𝐾 w.r.t. 𝑘 , and 𝑞 ∈ W 𝐾 . Then 𝑇 has a telescoper w.r.t. 𝑘 if and only if 𝑏 is integer-linear over K .Proof. Since the kernel 𝐾 = 𝜎 𝑘 ( 𝐻 ) /𝐻 is shift-reduced w.r.t. 𝑘 , it follows from [8,Theorem 8] that 𝐻 is a factorial term over F ( 𝑘 ). Thus 𝐾 is integer-linear over K ,and then so are the numerator 𝑢 and the denominator 𝑣 .We first show the sufficiency. Since 𝑏 is integer-linear over K , the term 𝐻/ ( 𝑏𝑣 ) isagain a factorial term. Hence (︁ 𝑎𝑏 + 𝑞𝑣 )︁ 𝐻 = ( 𝑎𝑣 + 𝑏𝑞 ) 𝐻𝑏𝑣 is a proper term, whose telescopers exist according to the fundamental lemma in [67].By (5.1), 𝑇 has a telescoper w.r.t. 𝑘 .To show the necessity, assume that 𝑇 has a telescoper w.r.t. 𝑘 . Then the term ( 𝑎/𝑏 + 𝑞/𝑣 ) 𝐻 is proper by [4, Theorem 10]. Thus 𝐻/ ( 𝑏𝑣 ) is a factorial term. Note that 𝜎 𝑘 ( 𝐻/ ( 𝑏𝑣 )) 𝐻/ ( 𝑏𝑣 ) = 𝑢𝜎 𝑘 ( 𝑣 ) 𝑏𝜎 𝑘 ( 𝑏 ) . Hence, 𝑏/𝜎 𝑘 ( 𝑏 ) is integer-linear over K as 𝑢, 𝑣 are integer-linear. Because 𝑏 is shift-free w.r.t. 𝑘 , so gcd( 𝑏, 𝜎 𝑘 ( 𝑏 )) ∈ F . The assertion follows by noticing that all elementsin F are integer-linear.2 Chapter 5. Creative Telescoping for Hypergeometric Terms
Let 𝑇 be a hypergeometric term over F ( 𝑘 ). If there exists a telescoper for 𝑇 w.r.t. 𝑘 by Theorem 5.5, then all telescopers for 𝑇 w.r.t. 𝑘 together with the zero operatorform a left ideal of the principal ideal ring F ⟨ 𝑆 𝑛 ⟩ . We refer to a generator of thisideal as a minimal telescoper for 𝑇 w.r.t. 𝑘 . Roughly speaking, a minimal telescoperis a telescoper of the minimal order.Since 1990, various algorithms [69, 70, 71, 44, 6] have been designed to computea minimal telescoper for a given hypergeometric term, typically the classical Zeil-berger’s algorithm [69]. When telescopers exist, Zeilberger’s algorithm constructs atelescoper for a given hypergeometric term 𝑇 by iteratively using Gosper’s algorithmto detect the summability of 𝐿 ( 𝑇 ) for an ansatz 𝐿 = 𝜌 ∑︁ 𝑖 =0 ℓ 𝑖 𝑆 𝑖𝑛 ∈ F ⟨ 𝑆 𝑛 ⟩ , where ℓ 𝑖 are indeterminates. In order to get a telescoper, one needs to solve a linearsystem with unknowns ℓ 𝑖 and also unknowns from the certificate. Any nontrivialsolution gives rise to a telescoper and a corresponding certificate simultaneously.There is no obvious way to avoid the computation of certificates in Zeilberger’salgorithm.In order to separate the computations of telescopers and certificates, we followthe ideas in the continuous case [14, 18, 16, 15], and use the modified Abramov-Petkovšek reduction to develop a creative telescoping algorithm. The algorithm isoutlined below. Algorithm 5.6 (Reduction-based creative telescoping) .Input : A hypergeometric term 𝑇 over F ( 𝑘 ). Output : A minimal telescoper for 𝑇 w.r.t. 𝑘 and a corresponding certificate iftelescopers exist; “No telescoper exists!”, otherwise. Find a kernel 𝐾 and shell 𝑆 of 𝑇 w.r.t. 𝑘 such that 𝑇 = 𝑆𝐻 with 𝐾 = 𝜎 𝑘 ( 𝐻 ) /𝐻 . Apply the modified Abramov-Petkovšek reduction to 𝑇 to get 𝑇 = Δ 𝑘 ( 𝑔 𝐻 ) + 𝑟 𝐻. (5.2)If 𝑟 = 0, then return (1 , 𝑔 𝐻 ). If the denominator of 𝑟 is not integer-linear, return “No telescoper exists!”. Set 𝑁 = 𝜎 𝑛 ( 𝐻 ) /𝐻 and 𝑅 = ℓ 𝑟 , where ℓ is an indeterminate.For 𝑖 = 1 , , . . . do View 𝜎 𝑛 ( 𝑟 𝑖 − ) 𝑁 𝐻 as a hypergeometric term with kernel 𝐾 andshell 𝜎 𝑛 ( 𝑟 𝑖 − ) 𝑁 . Using Algorithm 3.5 and Algorithm 3.16 w.r.t. 𝐾 ,find 𝑔 ′ 𝑖 ∈ F and a residual form ˜ 𝑟 𝑖 w.r.t. 𝐾 such that 𝜎 𝑛 ( 𝑟 𝑖 − ) 𝑁 𝐻 = Δ 𝑘 ( 𝑔 ′ 𝑖 𝐻 ) + ˜ 𝑟 𝑖 𝐻. .2. Telescoping via reductions Set ˜ 𝑔 𝑖 = 𝜎 𝑛 ( 𝑔 𝑖 − ) 𝑁 + 𝑔 ′ 𝑖 , so that 𝜎 𝑖𝑛 ( 𝑇 ) = Δ 𝑘 (˜ 𝑔 𝑖 𝐻 ) + ˜ 𝑟 𝑖 𝐻. (5.3) Apply Algorithm 4.20 to ˜ 𝑟 𝑖 w.r.t. 𝐾 and 𝑅 , to find 𝑔 𝑖 , 𝑟 𝑖 ∈ F ( 𝑘 )such that 𝑟 𝑖 ≡ 𝑘 ˜ 𝑟 𝑖 mod V 𝐾 , 𝜎 𝑖𝑛 ( 𝑇 ) = Δ 𝑘 ( 𝑔 𝑖 𝐻 ) + 𝑟 𝑖 𝐻, (5.4)and 𝑅 + ℓ 𝑖 𝑟 𝑖 is a residual form w.r.t. 𝐾 , where ℓ 𝑖 is an indeterminate. Update 𝑅 to 𝑅 + ℓ 𝑖 𝑟 𝑖 .Find ℓ 𝑗 ∈ F such that 𝑅 = 0 by solving a linear system in ℓ , . . . , ℓ 𝑖 over F . If there is a nontrivial solution, return ⎛⎝ 𝑖 ∑︁ 𝑗 =0 ℓ 𝑗 𝑆 𝑗𝑛 , 𝑖 ∑︁ 𝑗 =0 ℓ 𝑗 𝑔 𝑗 𝐻 ⎞⎠ . Theorem 5.7.
Let 𝑇 be a hypergeometric term over F ( 𝑘 ) . If 𝑇 has a telescoper, thenAlgorithm 5.6 terminates and returns a telescoper of minimal order for 𝑇 w.r.t. 𝑘 .Proof. By Theorem 3.18, 𝑟 = 0 in step 2 implies that 𝑇 is summable, and thus 1 isa minimal telescoper for 𝑇 w.r.t. 𝑘 . Now let 𝑟 obtained from step 2 be of the form 𝑟 = 𝑎 /𝑏 + 𝑞 /𝑣 , where 𝑎 , 𝑏 , 𝑣 ∈ F [ 𝑘 ], deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), 𝑏 is strongly coprimewith 𝐾 , 𝑞 ∈ W 𝐾 , and 𝑣 is the denominator of 𝐾 . According to [8, Theorem 8], 𝐾 isinteger-linear and so is 𝑣 . It follows that 𝑏 is integer-linear if and only if 𝑏 𝑣 is. ByTheorem 5.5, 𝑇 has a telescoper if and only if the denominator of 𝑟 is integer-linear.Thus, steps 2 and 3 are correct.It follows from (5.2) and 𝜎 𝑛 ( 𝑟 𝐻 ) = 𝜎 𝑛 ( 𝑟 ) 𝑁 𝐻 that (5.3) holds for 𝑖 = 1. ByAlgorithm 4.20, there exists a rational function 𝑢 ∈ F ( 𝑘 ) and a residual form 𝑟 w.r.t. 𝐾 such that˜ 𝑟 = 𝐾𝜎 𝑘 ( 𝑢 ) − 𝑢 + 𝑟 , i.e., 𝑟 ≡ 𝑘 ˜ 𝑟 mod V 𝐾 , and 𝑅 + ℓ 𝑟 is again a residual form for all ℓ , ℓ ∈ F . Setting 𝑔 = ˜ 𝑔 + 𝑢 , wesee that (5.4) holds for 𝑖 = 1. By a direct induction on 𝑖 , (5.4) holds in the loop ofstep 4.Assume that 𝐿 = ∑︀ 𝜌𝑖 =0 𝑐 𝑖 𝑆 𝑖𝑛 is a minimal telescoper for 𝑇 with 𝜌 ∈ N , 𝑐 𝑖 ∈ F and 𝑐 𝜌 ̸ = 0. Then 𝐿 ( 𝑇 ) is summable w.r.t. 𝑘 . By Theorem 3.18, ∑︀ 𝜌𝑖 =0 𝑐 𝑖 𝑟 𝑖 is equal tozero. Thus, the linear homogeneous system (over F ) obtained by equating ∑︀ 𝜌𝑖 =0 ℓ 𝑖 𝑟 𝑖 to zero has a nontrivial solution, which yields a minimal telescoper. Remark 5.8.
Algorithm 5.6 indeed separates the computation of minimal telescop-ers from that of certificates. In applications where certificates are irrelevant, we candrop step 4 .
2, and in step 4 . 𝑔 𝑖 and 𝑟 𝑖 with 𝑟 𝑖 ≡ 𝑘 ˜ 𝑟 𝑖 mod V 𝐾 , 𝜎 𝑖𝑛 ( 𝑟 𝑖 − ) 𝑁 𝐻 = Δ 𝑘 ( 𝑔 𝑖 𝐻 ) + 𝑟 𝑖 𝐻 Chapter 5. Creative Telescoping for Hypergeometric Terms and 𝑅 + ℓ 𝑖 𝑟 𝑖 is a residual form w.r.t. 𝐾 , where ℓ 𝑖 is an indeterminate. Moreover, therational function 𝑔 𝑖 can be discarded, and we do not need to calculate ∑︀ 𝑖𝑗 =0 ℓ 𝑗 𝑔 𝑗 𝐻 in the end. Remark 5.9.
Instead of applying the modified reduction to 𝜎 𝑛 ( 𝑟 𝑖 − ) 𝑁 𝐻 in step 4 . 𝜎 𝑖𝑛 ( 𝑇 ) directly, but our experimentssuggest that this variant takes considerably more time. This observation agrees withthe advices given in [6, Example 6].Since Algorithm 5.6 performs the same function as Zeilberger’s algorithm, it isalso applicable to the examples and applications indicated in [54]. In other words, itcan be used to evaluate definite sums and prove identities of hypergeometric termsefficiently. Example 5.10.
Consider the hypergeometric term 𝑇 = (︂ 𝑛𝑘 )︂ . Then the respectiveshift-quotients of 𝑇 with respect to 𝑛 and 𝑘 are 𝑓 = 𝜎 𝑛 ( 𝑇 ) 𝑇 = ( 𝑛 + 1) ( 𝑛 + 1 − 𝑘 ) and 𝑔 = 𝜎 𝑘 ( 𝑇 ) 𝑇 = ( 𝑛 − 𝑘 ) ( 𝑘 + 1) . Since 𝑔 is shift-reduced w.r.t. 𝑘 , its kernel is equal to 𝑔 itself, and the correspondingshell is 1, implying that 𝐻 = 𝑇 in step 1 of Algorithm 5.6. In step 4, applying themodified Abramov-Petkovšek reduction to 𝑇, 𝜎 𝑛 ( 𝑇 ) , 𝜎 𝑛 ( 𝑇 ), incrementally, yields 𝜎 𝑖𝑘 ( 𝑇 ) = Δ 𝑘 ( 𝑔 𝑖 𝐻 ) + 𝑞 𝑖 𝑣 𝐻, where 𝑖 = 0 , , 𝑣 = ( 𝑘 + 1) , 𝑞 = 12 ( 𝑛 + 1)( 𝑛 − 𝑛 + 3 𝑘 ( 𝑘 − 𝑛 + 1) + 1) , 𝑞 = ( 𝑛 + 1) ,𝑞 = ( 𝑛 + 1) ( 𝑛 + 2) (︁ 𝑛 − 𝑛𝑘 + 17 𝑛 + 20 + 12 𝑘 + 12 𝑘 )︁ , and 𝑔 , 𝑔 , 𝑔 ∈ F ( 𝑘 ) which are too complicated to be reproduced here. By findingan F -linear dependency among 𝑞 , 𝑞 , 𝑞 , we see that 𝐿 = ( 𝑛 + 2) 𝑆 𝑛 − (7 𝑛 + 21 𝑛 + 16) 𝑆 𝑛 − 𝑛 + 1) is a minimal telescoper for 𝑇 w.r.t. 𝑘 . For a corresponding certificate 𝐺 , one canchoose to leave it as an unnormalized term 𝐺 = ( 𝑛 + 2) 𝑔 − (7 𝑛 + 21 𝑛 + 16) 𝑔 − 𝑛 + 1) 𝑔 , or normalize it as one rational function according to the specific requirements. .3. Implementation and timings We have implemented Algorithm 5.6 in
Maple 18 . The procedure is named as
ReductionCT in the Maple package
ShiftReductionCT . See Appendix A for moredetails.In this section, we compare the runtime of the new procedure to the performanceof Zeilberger’s algorithm. All timings are measured in seconds on a Linux computerwith 388Gb RAM and twelve 2.80GHz Dual core processors. No parallelism wasused in this experiment. In addition, we also compare the memory requirements ofall procedures, which is shown in Appendix B. For brevity, we denote• Z : the procedure SumTools[Hypergeometric] [ Zeilberger ], which is based onZeilberger’s algorithm;•
RCT 𝑡𝑐 : the procedure ReductionCT in ShiftReductionCT , which computes aminimal telescoper and a corresponding normalized certificate;•
RCT 𝑡 : the procedure ReductionCT in ShiftReductionCT , which computes aminimal telescoper without constructing a certificate.• order : the order of the resulting minimal telescoper.
Example 5.11.
Consider bivariate hypergeometric terms of the form 𝑇 = 𝑓 ( 𝑛, 𝑘 ) 𝑔 ( 𝑛 + 𝑘 ) 𝑔 (2 𝑛 + 𝑘 ) Γ(2 𝛼𝑛 + 𝑘 )Γ( 𝑛 + 𝛼𝑘 )where 𝑓 ∈ Z [ 𝑛, 𝑘 ] of degree 𝑑 , and for 𝑖 = 1 , 𝑔 𝑖 = 𝑝 𝑖 𝜎 𝜆𝑧 ( 𝑝 𝑖 ) 𝜎 𝜇𝑧 ( 𝑝 𝑖 ) with 𝑝 𝑖 ∈ Z [ 𝑧 ] ofdegree 𝑑 and 𝛼, 𝜆, 𝜇 ∈ N . For different choices of 𝑑 , 𝑑 , 𝛼, 𝜇, 𝜆 , Table 5.1 comparesthe timings of the four procedures. Remark 5.12.
The difference between
RCT 𝑡𝑐 and RCT 𝑡 mainly comes from thetime needed to bring the rational function 𝑔 in the certificate 𝑔𝐻 on a commondenominator. When it is acceptable to keep the certificate as an unnormalized linearcombination of rational functions, their timings are virtually the same.6 Chapter 5. Creative Telescoping for Hypergeometric Terms ( 𝑑 , 𝑑 , 𝛼, 𝜆, 𝜇 ) Z RCT 𝑡𝑐 RCT 𝑡 order (1 , , , ,
5) 17.12 5.00 1.80 4(1 , , , ,
5) 74.91 26.18 5.87 6(1 , , , ,
5) 445.41 92.74 17.34 7(1 , , , ,
5) 649.57 120.88 23.59 7(2 , , , ,
10) 354.46 58.01 4.93 4(2 , , , ,
10) 576.31 363.25 53.15 6(2 , , , ,
10) 2989.18 1076.50 197.75 7(2 , , , ,
10) 3074.08 1119.26 223.41 7(2 , , , ,
15) 2148.10 245.07 11.22 4(2 , , , ,
15) 2036.96 1153.38 153.21 6(2 , , , ,
15) 11240.90 3932.26 881.12 7(2 , , , ,
15) 10163.30 3954.47 990.60 7(3 , , , ,
10) 18946.80 407.06 43.01 6(3 , , , ,
10) 46681.30 2040.21 465.88 8(3 , , , ,
10) 172939.00 5970.10 1949.71 9
Table 5.1:
Timing comparison of Zeilberger’s algorithm to reduction-based creativetelescoping with and without construction of a certificate (in seconds) hapter 6
Order Bounds for
Minimal Telescopers In the previous chapter, we have presented a reduction-based creative telescopingalgorithm for bivariate hypergeometric terms, namely Algorithm 5.6. Roughly speak-ing, its basic idea is as follows. Using the modified Abramov-Petkovšek reductionfrom Chapter 3, we first reduce a given hypergeometric term and its shifts to somerequired “standard forms” (called remainders in the sequel), such that the differ-ence between the original function and its remainder is summable. Then computinga telescoper amounts to finding a linear dependence among these remainders. In or-der to show that this algorithm terminates, we show that for every summable term,its remainder is zero. This ensures that the algorithm terminates by the existencecriterion given in Theorem 5.5, and in fact it will find the smallest possible tele-scoper, but it does not provide a bound on its order. Another possible approach isto show that the vector space spanned by the remainders has a finite dimension.Then, as soon as the number of remainders exceeds this dimension, we can be surethat a telescoper will be found. This approach was taken in [15, 16, 23]. As a nice sideresult, this approach provides an independent proof of the existence of telescopers,and even a bound on the order of minimal telescopers.In this chapter, we show that the approach for the differential case also worksfor the shift case, i.e., the remainders in the shift case also form a finite-dimensionalvector space, so as to eliminate the discrepancy. As a result, we obtain a new argu-ment for the termination of Algorithm 5.6, and also get new bounds for the orderof minimal telescopers for hypergeometric terms. We do not find exactly the samebounds that are already in the literature [49, 6]. Comparing our bounds to the knownbounds in the literature, it appears that for “generic” input (see Subsection 6.4.1for a definition), the values often agree (of course, because the known bounds arealready generically sharp). However, there are some special examples in which ourbounds are better than the known bounds. On the other hand, our bounds are neverworse than the known ones. In addition, we give a variant of Algorithm 5.6 basedon the new bounds. An experimental comparison is presented in the final section. The main results in this chapter are published in [38]. Chapter 6. Order Bounds for Minimal Telescopers
In this section, we generalize the notion of shift-equivalence in Chapter 4 to thebivariate case, and then derive a useful decomposition for an integer-linear polyno-mial.Using the same notations as the previous chapter, K is a field of characteristiczero, and K ( 𝑛, 𝑘 ) is the field of rational functions in 𝑛 and 𝑘 over K . Let 𝜎 𝑛 and 𝜎 𝑘 be the shift operators w.r.t. 𝑛 and 𝑘 , respectively. Definition 6.1.
Two polynomials 𝑝, 𝑞 ∈ K [ 𝑛, 𝑘 ] are called shift-equivalent w.r.t. 𝑛 and 𝑘 if there exist integers ℓ, 𝑚 such that 𝑞 = 𝜎 ℓ𝑛 𝜎 𝑚𝑘 ( 𝑝 ) . We denote it by 𝑝 ∼ 𝑛,𝑘 𝑞 . Clearly ∼ 𝑛,𝑘 is an equivalence relation. In particular, when ℓ = 0 or 𝑚 = 0,the above definition degenerates to Definition 4.1. Thus ∼ 𝑛 or ∼ 𝑘 implies ∼ 𝑛,𝑘 .Choosing the pure lexicographic order 𝑛 ≺ 𝑘 , we say a polynomial is monic if itshighest term has coefficient 1. A rational function is said to be shift-homogeneous ifall non-constant monic irreducible factors of its denominator and numerator belongto the same shift-equivalence class.By grouping together the factors in the same shift-equivalence class, every ratio-nal function 𝑟 ∈ K ( 𝑛, 𝑘 ) can be decomposed into the form 𝑟 = 𝑐 𝑟 . . . 𝑟 𝑠 , (6.1)where 𝑐 ∈ K , 𝑠 ∈ N , each 𝑟 𝑖 is a shift-homogeneous rational function, and any twonon-constant monic irreducible factors of 𝑟 𝑖 and 𝑟 𝑗 are pairwise shift-inequivalentwhenever 𝑖 ̸ = 𝑗 . We call each 𝑟 𝑖 a shift-homogeneous component of 𝑟 and (6.1)a shift-homogeneous decomposition of 𝑟 . Noticing that the field K ( 𝑛, 𝑘 ) is a uniquefactorization domain, one can easily show that the shift-homogeneous decompositionis unique up to the order of the factors and multiplication by nonzero constants.Let 𝑝 ∈ K [ 𝑛, 𝑘 ] be an irreducible integer-linear polynomial. Then it is of theform 𝑝 = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 ) for some 𝑃 ∈ K [ 𝑧 ] and 𝜆, 𝜇 ∈ Z , not both zero. W.l.o.g.,we further assume that 𝜇 ≥ 𝜆, 𝜇 ) = 1. Under this assumption, makingansatz and comparing coefficients yield the uniqueness of 𝑃 since Z is a uniquefactorization domain. In view of this, we call the pair ( 𝑃, { 𝜆, 𝜇 } ) the univariaterepresentation of the integer-linear polynomial 𝑝 . By Bézout’s relation, there existunique integers 𝛼, 𝛽 with | 𝛼 | < | 𝜇 | and | 𝛽 | < | 𝜆 | such that 𝛼𝜆 + 𝛽𝜇 = 1. Define 𝛿 ( 𝜆,𝜇 ) to be 𝜎 𝛼𝑛 𝜎 𝛽𝑘 . For brevity, we just write 𝛿 if ( 𝜆, 𝜇 ) is clear from the context. Notethat 𝛿 ( 𝑃 ( 𝑧 )) = 𝑃 ( 𝑧 + 1) with 𝑧 = 𝜆𝑛 + 𝜇𝑘 , which allows us to treat integer-linearpolynomials as univariate ones. For a Laurent polynomial 𝜉 = ∑︀ 𝜌𝑖 = ℓ 𝑚 𝑖 𝛿 𝑖 in Z [ 𝛿, 𝛿 − ]with ℓ, 𝜌, 𝑚 𝑖 ∈ Z and ℓ ≤ 𝜌 , define 𝑝 𝜉 = 𝛿 ℓ ( 𝑝 𝑚 ℓ ) 𝛿 ℓ +1 ( 𝑝 𝑚 ℓ +1 ) · · · 𝛿 𝜌 ( 𝑝 𝑚 𝜌 ) . Let 𝑝, 𝑞 ∈ K [ 𝑛, 𝑘 ] be two irreducible integer-linear polynomials of the forms 𝑝 = 𝑃 ( 𝜆 𝑛 + 𝜇 𝑘 ) and 𝑞 = 𝑄 ( 𝜆 𝑛 + 𝜇 𝑘 ) , where ( 𝑃, { 𝜆 , 𝜇 } ) and ( 𝑄, { 𝜆 , 𝜇 } ) are the univariate representations of 𝑝 and 𝑞 ,respectively. Namely, 𝑃, 𝑄 ∈ K [ 𝑧 ], 𝜆 , 𝜇 , 𝜆 , 𝜇 ∈ Z , 𝜇 , 𝜇 ≥ 𝜆 , 𝜇 ) = .2. Shift-relation of residual forms 𝜆 , 𝜇 ) = 1. It is readily seen that 𝑝 ∼ 𝑛,𝑘 𝑞 if and only if 𝜆 = 𝜆 , 𝜇 = 𝜇 and 𝑞 = 𝑝 𝛿 ℓ for some integer ℓ , in which 𝛿 = 𝛿 ( 𝜆 ,𝜇 ) = 𝛿 ( 𝜆 ,𝜇 ) .Given a shift-homogeneous and integer-linear rational function 𝑟 ∈ K ( 𝑛, 𝑘 ), let ℎ be a monic, irreducible and integer-linear polynomial in K [ 𝑛, 𝑘 ] with the propertythat all monic irreducible factors of the numerator and denominator of 𝑟 are equal tosome shift of ℎ w.r.t. 𝑛 and 𝑘 . Assume that the univariate representation of ℎ is thepair ( 𝑃 ℎ , { 𝜆 ℎ , 𝜇 ℎ } ). Then 𝑟 can be written as 𝑐 ℎ 𝜉 ℎ for some 𝑐 ∈ K and 𝜉 ℎ ∈ Z [ 𝛿 − , 𝛿 ]with 𝛿 = 𝛿 ( 𝜆 ℎ ,𝜇 ℎ ) . We call ( 𝑃 ℎ , { 𝜆 ℎ , 𝜇 ℎ } , 𝜉 ℎ ) a univariate representation of 𝑟 . Assumethat ( 𝑃 𝑔 , { 𝜆 𝑔 , 𝜇 𝑔 } , 𝜉 𝑔 ) is another univariate representation of 𝑟 with 𝑔 ∈ K [ 𝑛, 𝑘 ]. Bythe conclusion made in the preceding paragraph, we find that 𝑔 = ℎ 𝛿 ℓ for some ℓ ∈ Z ,or, equivalently, 𝑃 𝑔 ( 𝑧 ) = 𝑃 ℎ ( 𝑧 + ℓ ). Moreover, ( 𝜆 𝑔 , 𝜇 𝑔 ) = ( 𝜆 ℎ , 𝜇 ℎ ). It follows that 𝜉 𝑔 = 𝛿 ℓ 𝜉 ℎ . In particular, deg 𝑧 ( 𝑃 ℎ ) is equal to deg 𝑧 ( 𝑃 𝑔 ) and the nonzero coefficients of 𝜉 ℎ are exactly the same as those of 𝜉 𝑔 . When the choice of ℎ and 𝑔 is insignificant, wesay that a tuple ( 𝑃, { 𝜆, 𝜇 } , 𝜉 ) is a univariate representation of 𝑟 if the polynomial 𝑃 ∈ K [ 𝑧 ] is irreducible and 𝑟 ( 𝑛, 𝑘 ) = 𝑐𝑃 ( 𝜆𝑛 + 𝜇𝑘 ) 𝜉 for some 𝑐 ∈ K . Note that thecoefficients of 𝜉 are all nonnegative if 𝑟 is a polynomial.Let 𝑟 ∈ K ( 𝑛, 𝑘 ) be integer-linear with the shift-homogeneous decomposition 𝑟 = 𝑐 𝑟 · · · 𝑟 𝑠 . For 𝑖 = 1 , . . . , 𝑠 , assume that 𝑈 𝑖 = ( 𝑃 𝑖 , ( 𝜆 𝑖 , 𝜇 𝑖 ) , 𝜉 𝑖 ) is a univariate representationof 𝑟 𝑖 . Then we call the tuple ( 𝑐, ( 𝑈 , . . . , 𝑈 𝑠 ))a univariate representation of 𝑟 .To avoid unnecessary duplication, we make a notational convention. Convention 6.2.
Let 𝑇 be a hypergeometric term over K ( 𝑛, 𝑘 ) with a multiplicativedecomposition 𝑆𝐻 , where 𝑆 ∈ K ( 𝑛, 𝑘 ) and 𝐻 is a hypergeometric term whose shift-quotient 𝐾 w.r.t. 𝑘 is shift-reduced w.r.t. 𝑘 . By [8, Theorem 8], we know 𝐾 isinteger-linear over K . Write 𝐾 = 𝑢/𝑣 where 𝑢, 𝑣 ∈ K ( 𝑛 )[ 𝑘 ] and gcd( 𝑢, 𝑣 ) = 1 . In this section, we describe a relation among residual forms of a given hypergeometricterm and its shifts. This relation enables us to derive a shift-free common multipleof significant denominators of those residual forms, provided that telescopers exist.The existence of this common multiple implies that the residual forms span a finite-dimensional vector space over K ( 𝑛 ), and then lead to order bounds for the minimaltelescopers presented in the next section. Lemma 6.3.
With Convention 6.2, let 𝑟 be a residual form of 𝑆 w.r.t. 𝐾 . Then 𝜎 𝑛 ( 𝐾 ) and 𝜎 𝑛 ( 𝑆 ) are a kernel and a corresponding shell of 𝜎 𝑛 ( 𝑇 ) w.r.t. 𝑘 , respectively.Moreover, 𝜎 𝑛 ( 𝑟 ) is a residual form of 𝜎 𝑛 ( 𝑆 ) w.r.t. 𝜎 𝑛 ( 𝐾 ) .Proof. By Convention 6.2, 𝜎 𝑛 ( 𝑇 ) = 𝜎 𝑛 ( 𝑆 ) 𝜎 𝑛 ( 𝐻 ) and 𝜎 𝑛 ( 𝐾 ) is the shift-quotientof 𝜎 𝑛 ( 𝐻 ) w.r.t. 𝑘 . To prove the first assertion, one needs to show that 𝜎 𝑛 ( 𝐾 ) is0 Chapter 6. Order Bounds for Minimal Telescopers shift-reduced w.r.t. 𝑘 . This can be proven by observing that, for any two polynomials 𝑝 , 𝑝 ∈ K ( 𝑛 )[ 𝑘 ], we have gcd( 𝜎 𝑛 ( 𝑝 ) , 𝜎 𝑛 ( 𝑝 )) = 1 if and only if gcd( 𝑝 , 𝑝 ) = 1.For the second assertion, since 𝑟 is a residual form w.r.t. 𝐾 , we write 𝑟 = 𝑎𝑏 + 𝑞𝑣 , where 𝑎, 𝑏, 𝑞 ∈ K ( 𝑛 )[ 𝑘 ], deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ), gcd( 𝑎, 𝑏 ) = 1, 𝑏 is shift-free and stronglycoprime with 𝐾 , and 𝑞 ∈ W 𝐾 . It is clear that deg 𝑘 ( 𝜎 𝑛 ( 𝑎 )) < deg 𝑘 ( 𝜎 𝑛 ( 𝑏 )) andgcd( 𝜎 𝑛 ( 𝑎 ) , 𝜎 𝑛 ( 𝑏 )) = 1. By the above observation, 𝜎 𝑛 ( 𝑏 ) is shift-free and stronglycoprime with 𝜎 𝑛 ( 𝐾 ).Note that 𝜎 𝑛 ∘ deg 𝑘 = deg 𝑘 ∘ 𝜎 𝑛 and 𝜎 𝑛 ∘ lc 𝑘 = lc 𝑘 ∘ 𝜎 𝑛 , where lc 𝑘 ( 𝑝 ) is the leadingcoefficient of 𝑝 ∈ K ( 𝑛 )[ 𝑘 ] w.r.t. 𝑘 . So the standard complements W 𝐾 and W 𝜎 𝑛 ( 𝐾 ) for polynomial reduction have the same echelon basis according to the case study inSubsection 3.2.1. It follows from 𝑞 ∈ W 𝐾 that 𝜎 𝑛 ( 𝑞 ) ∈ W 𝜎 𝑛 ( 𝐾 ) . Accordingly, 𝜎 𝑛 ( 𝑟 )is a residual form of 𝜎 𝑛 ( 𝑆 ) w.r.t. 𝜎 𝑛 ( 𝐾 ). Theorem 6.4.
With Convention 6.2, for every nonnegative integer 𝑖 assume 𝜎 𝑖𝑛 ( 𝑇 ) = Δ 𝑘 ( 𝑔 𝑖 𝐻 ) + (︂ 𝑎 𝑖 𝑏 𝑖 + 𝑞 𝑖 𝑣 )︂ 𝐻, (6.2) where 𝑔 𝑖 ∈ K ( 𝑛, 𝑘 ) , 𝑎 𝑖 , 𝑏 𝑖 ∈ K ( 𝑛 )[ 𝑘 ] with deg 𝑘 ( 𝑎 𝑖 ) < deg 𝑘 ( 𝑏 𝑖 ) , gcd( 𝑎 𝑖 , 𝑏 𝑖 ) = 1 , 𝑏 𝑖 isshift-free w.r.t. 𝑘 and strongly coprime with 𝐾 , and 𝑞 𝑖 belongs to W 𝐾 . Then 𝑏 𝑖 isshift-related to 𝜎 𝑖𝑛 ( 𝑏 ) , i.e., 𝑏 𝑖 ≈ 𝑘 𝜎 𝑖𝑛 ( 𝑏 ) .Proof. We proceed by induction on 𝑖 . For 𝑖 = 0, the reflexivity of the relation ≈ 𝑘 implies that 𝑏 ≈ 𝑘 𝑏 .Assume that 𝑏 𝑖 − ≈ 𝑘 𝜎 𝑖 − 𝑛 ( 𝑏 ) for 𝑖 ≥
1. Note that 𝐾 is also a kernel of 𝜎 𝑖 − 𝑛 ( 𝑇 )and 𝜎 𝑖𝑛 ( 𝑇 ) w.r.t. 𝑘 . Let 𝑆 𝑖 − and 𝑆 𝑖 be the corresponding shells, respectively. Con-sider the equality 𝜎 𝑖 − 𝑛 ( 𝑇 ) = Δ 𝑘 ( 𝑔 𝑖 − 𝐻 ) + (︂ 𝑎 𝑖 − 𝑏 𝑖 − + 𝑞 𝑖 − 𝑣 )︂ 𝐻, where 𝑔 𝑖 − ∈ K ( 𝑛, 𝑘 ) and 𝑎 𝑖 − /𝑏 𝑖 − + 𝑞 𝑖 − /𝑣 is a residual form of 𝑆 𝑖 − w.r.t. 𝐾 .Applying 𝜎 𝑛 to both sides yields 𝜎 𝑖𝑛 ( 𝑇 ) = 𝜎 𝑛 (Δ 𝑘 ( 𝑔 𝑖 − 𝐻 )) + 𝜎 𝑛 (︂ 𝑎 𝑖 − 𝑏 𝑖 − + 𝑞 𝑖 − 𝑣 )︂ 𝜎 𝑛 ( 𝐻 )= Δ 𝑘 ( 𝜎 𝑛 ( 𝑔 𝑖 − 𝐻 )) + (︂ 𝜎 𝑛 ( 𝑎 𝑖 − ) 𝜎 𝑛 ( 𝑏 𝑖 − ) + 𝜎 𝑛 ( 𝑞 𝑖 − ) 𝜎 𝑛 ( 𝑣 ) )︂ 𝜎 𝑛 ( 𝐻 )It follows from Lemma 6.3 that 𝜎 𝑛 ( 𝐾 ) and 𝜎 𝑛 ( 𝑆 𝑖 − ) are a kernel and the corre-sponding shell 𝜎 𝑖𝑛 ( 𝑇 ) w.r.t. 𝑘 , and 𝜎 𝑛 ( 𝑎 𝑖 − ) /𝜎 𝑛 ( 𝑏 𝑖 − ) + 𝜎 𝑛 ( 𝑞 𝑖 − ) /𝜎 𝑛 ( 𝑣 ) is a residualform of 𝑆 𝑖 w.r.t. 𝜎 𝑛 ( 𝐾 ). By (6.2) with 𝑖 = 1, we know that 𝑎 𝑖 /𝑏 𝑖 + 𝑞 𝑖 /𝑣 is a residualform of 𝑆 𝑖 w.r.t. 𝐾 . By Theorem 4.10, 𝑏 𝑖 ≈ 𝑘 𝜎 𝑛 ( 𝑏 𝑖 − ). Thus 𝑏 𝑖 ≈ 𝑘 𝜎 𝑖𝑛 ( 𝑏 ) by theinduction hypothesis. .2. Shift-relation of residual forms 𝑓 ∈ K ( 𝑛 )[ 𝑘 ], there alwaysexists 𝑔 ∈ K ( 𝑛 )[ 𝑘 ] such that 𝑓 ≈ 𝑘 𝑔 and 𝑔 is strongly coprime with 𝐾 . Lemma 6.5.
With Convention 6.2, assume that 𝑝 is an irreducible polynomialin K ( 𝑛 )[ 𝑘 ] . Then there exists an integer 𝑚 such that 𝜎 𝑚𝑘 ( 𝑝 ) is strongly coprimewith 𝐾 .Proof. According to the definition of strong coprimeness, there is one and only oneof the following three cases true.
Case 1. 𝑝 is strongly coprime with 𝐾 . Then the lemma follows by letting 𝑚 = 0. Case 2.
There exists an integer 𝑘 ≥ 𝜎 𝑘𝑘 ( 𝑝 ) | 𝑢 . Then for every integer ℓ ,we have gcd( 𝜎 ℓ𝑘 ( 𝑝 ) , 𝑣 ) = 1, since 𝐾 is shift-reduced w.r.t. 𝑘 . Let 𝑚 = max { 𝑖 ∈ N | 𝜎 𝑖𝑘 ( 𝑝 ) | 𝑢 } + 1 . One can see that 𝜎 𝑚𝑘 ( 𝑝 ) is strongly coprime with 𝐾 . Case 3.
There exists an integer 𝑘 ≤ 𝜎 𝑘𝑘 ( 𝑝 ) | 𝑣 . Then for every integer ℓ ,we have gcd( 𝜎 ℓ𝑘 ( 𝑝 ) , 𝑢 ) = 1, since 𝐾 is shift-reduced w.r.t. 𝑘 . Letting 𝑚 = min { 𝑖 ∈ N | 𝜎 𝑖𝑘 ( 𝑝 ) | 𝑣 } − 𝜎 𝑚𝑘 ( 𝑝 ) is strongly coprime with 𝐾 .The next lemma shows that for any integer-linear polynomial in K [ 𝑛, 𝑘 ], thenumber of shift-equivalence classes w.r.t. 𝑘 produced by shifting the polynomial asa univariate one is finite. Lemma 6.6.
Let 𝑞 ∈ K [ 𝑛, 𝑘 ] be integer-linear, and then 𝑞 = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 ) for 𝑃 ∈ K [ 𝑧 ] and 𝜆, 𝜇 ∈ Z not both zero. Then any shift of 𝑞 w.r.t. 𝑛 or 𝑧 = 𝜆𝑛 + 𝜇𝑘 is shift-equivalent to 𝛿 𝑗 ( 𝑞 ) w.r.t. 𝑘 for 𝛿 = 𝛿 ( 𝜆,𝜇 ) and ≤ 𝑗 ≤ 𝜇 − . More precisely, let 𝑆 = { 𝛿 𝑗 ( 𝑞 ) | 𝑗 = 0 , . . . , 𝜇 − } , 𝑆 = { 𝜎 𝑖𝑛 ( 𝑞 ) | 𝑖 ∈ N } and 𝑆 = { 𝛿 𝑗 ( 𝑞 ) | 𝑗 ∈ N } . Then for any element 𝑓 in 𝑆 ∪ 𝑆 , there exists 𝑔 ∈ 𝑆 such that 𝑓 ∼ 𝑘 𝑔 .Proof. Assume that 𝑓 ∈ 𝑆 ∪ 𝑆 . Since 𝜎 𝑛 = 𝛿 𝜆 , there exists a nonnegative integer 𝑖 such that 𝑓 = 𝛿 𝑖 ( 𝑞 ) = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑖 ) . By Euclidean division, there exist unique integers 𝑗, ℓ with 0 ≤ 𝑗 ≤ 𝜇 −
1, suchthat 𝑖 = ℓ𝜇 + 𝑗 . It follows that 𝑓 = 𝜎 ℓ𝑘 ( 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 )) = 𝜎 ℓ𝑘 ( 𝛿 𝑗 ( 𝑞 )) . Letting 𝑔 = 𝛿 𝑗 ( 𝑞 ) completes the proof.Now we are ready to compute a common multiple as mentioned before.2 Chapter 6. Order Bounds for Minimal Telescopers
Theorem 6.7.
With Convention 6.2, assume that 𝑇 = Δ 𝑘 ( 𝑔𝐻 ) + (︁ 𝑎𝑏 + 𝑞𝑣 )︁ 𝐻, (6.3) where 𝑔 ∈ K ( 𝑛, 𝑘 ) , 𝑎, 𝑏, 𝑞 ∈ K ( 𝑛 )[ 𝑘 ] , deg 𝑘 ( 𝑎 ) < deg 𝑘 ( 𝑏 ) , gcd( 𝑎, 𝑏 ) = 1 , 𝑏 is shift-free w.r.t. 𝑘 and strongly coprime with 𝐾 , and 𝑞 ∈ W 𝐾 . Further assume that 𝑏 isinteger-linear and has a univariate representation ( 𝑐, ( 𝑈 , . . . , 𝑈 𝑠 )) , where 𝑈 𝑗 = ( 𝑃 𝑗 , ( 𝜆 𝑗 , 𝜇 𝑗 ) , 𝜉 𝑗 ) , 𝑗 = 1 , . . . , 𝑠. Then there exists 𝐵 ∈ K ( 𝑛 )[ 𝑘 ] such that 𝑏 | 𝐵 and for all 𝑖 ∈ N , 𝜎 𝑖𝑛 ( 𝑇 ) = Δ 𝑘 ( 𝑔 𝑖 𝐻 ) + (︁ 𝑎 𝑖 𝐵 + 𝑞 𝑖 𝑣 )︁ 𝐻 (6.4) for some 𝑔 𝑖 ∈ K ( 𝑛, 𝑘 ) , 𝑎 𝑖 ∈ K ( 𝑛 )[ 𝑘 ] with deg 𝑘 ( 𝑎 𝑖 ) < deg 𝑘 ( 𝐵 ) , and 𝑞 𝑖 ∈ W 𝐾 . More-over,(i) 𝐵 is shift-free w.r.t. 𝑘 and strongly coprime with 𝐾 ;(ii) deg 𝑘 ( 𝐵 ) = ∑︀ 𝑠𝑗 =1 𝜇 𝑗 𝑚 𝑗 deg 𝑘 ( 𝑃 𝑗 ) , where 𝑚 𝑗 is the maximum of the coefficientsof 𝜉 𝑗 .Proof. Since the shift-homogeneous components of 𝑏 are coprime to each other, itsuffices to consider the case when 𝑏 is shift-homogeneous. W.l.o.g., assume that 𝑏 isshift-homogeneous and has a univariate representation ( 𝑃, { 𝜆, 𝜇 } , 𝜉 ) such that 𝑏 = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 ) 𝜉 . Write 𝜉 = ∑︀ 𝑑𝑖 =0 𝑚 ′ 𝑖 𝛿 𝑖 where 𝑑 ∈ N , 𝑚 ′ 𝑖 ∈ Z and 𝛿 = 𝛿 ( 𝜆,𝜇 ) .If 𝜇 = 0 then 𝑏 ∈ K ( 𝑛 ). By the modified Abramov-Petkovšek reduction we canassume that (6.2) holds for every 𝑖 > 𝑏 𝑖 ∈ K ( 𝑛 ) by Theorem 6.4. Theassertion follows by letting 𝐵 = 1.Otherwise we have 𝜇 >
0. By Lemma 6.6, for every 𝑖 ∈ N there are uniqueintegers 𝑗 and ℓ 𝑗 with 0 ≤ 𝑗 ≤ 𝜇 − 𝑃 ( 𝜆𝑛 + 𝜇𝑘 ) 𝛿 𝑖 = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 ) 𝜎 ℓ𝑗𝑘 , which is equivalent to 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑖 ) = 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝜇ℓ 𝑗 + 𝑗 ) . Since 𝑃 is irreducible, we have 𝑖 = 𝜇ℓ 𝑗 + 𝑗 . Let 𝑚 ′′ 𝑗 = 𝑚 ′ 𝜇ℓ 𝑗 + 𝑗 . Since 𝑏 is shift-freew.r.t. 𝑘 , 𝑏 = 𝜇 − ∏︁ 𝑗 =0 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 ) 𝑚 ′′ 𝑗 𝜎 ℓ𝑗𝑘 . For each 𝑗 , if 𝑚 ′′ 𝑗 ̸ = 0 then set 𝑚 𝑗 = ℓ 𝑗 ; otherwise by Lemma 6.5, let 𝑚 𝑗 be an integerso that 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 ) 𝜎 𝑚𝑗𝑘 is strongly coprime with 𝐾 . Let 𝑚 = max ≤ 𝑗 ≤ 𝜇 − { 𝑚 ′′ 𝑗 } and 𝐵 = 𝜇 − ∏︁ 𝑗 =0 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 ) 𝑚 𝜎 𝑚𝑗𝑘 . (6.5) .2. Shift-relation of residual forms 𝑘 ( 𝐵 ) = 𝜇𝑚 deg 𝑘 ( 𝑃 ). Since 𝑚 𝑗 = ℓ 𝑗 when 𝑚 ′′ 𝑗 ̸ = 0, every irreducible factorof 𝑏 divides 𝐵 and thus 𝑏 | 𝐵 by the maximum of 𝑚 . Because 0 ≤ 𝑗 ≤ 𝜇 −
1, so 𝐵 is shift-free w.r.t. 𝑘 . Moreover, 𝐵 is strongly coprime with 𝐾 by the choice of 𝑚 𝑗 .It remains to show that (6.4) holds for every nonnegative integer 𝑖 . To prove this,we first show 𝜎 𝑛 ( 𝐵 ) ≈ 𝑘 𝐵 . By (6.5), we have 𝐵 ≈ 𝑘 𝜇 − ∏︁ 𝑗 =0 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 ) 𝑚 , which yields 𝜎 𝑛 ( 𝐵 ) ≈ 𝑘 𝜇 − ∏︁ 𝑗 =0 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 + 𝜆 ) 𝑚 . By Lemma 6.6, there exists a unique integer ℓ with 0 ≤ ℓ ≤ 𝜇 − 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + 𝑗 + 𝜆 ) ∼ 𝑘 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + ℓ ) . Conversely, for any 0 ≤ ℓ ≤ 𝜇 −
1, there exists a unique integer 0 ≤ 𝑗 ≤ 𝜇 − 𝜎 𝑛 ( 𝐵 ) ≈ 𝑘 𝜇 − ∏︁ ℓ =0 𝑃 ( 𝜆𝑛 + 𝜇𝑘 + ℓ ) 𝑚 ≈ 𝑘 𝐵. For 𝑖 = 0, letting 𝑔 = 𝑔 , 𝑎 = 𝑎𝐵/𝑏 and 𝑞 = 𝑞 gives (6.4). Since 𝜎 𝑛 ( 𝐵 ) ≈ 𝑘 𝐵 ,we have 𝜎 𝑖𝑛 ( 𝐵 ) ≈ 𝑘 𝜎 𝑖 − 𝑛 ( 𝐵 ) for every positive integer 𝑖 , and then 𝜎 𝑖𝑛 ( 𝐵 ) ≈ 𝑘 𝐵 .On the other hand, by the modified Abramov-Petkovšek reduction (6.2) holdsfor every 𝑖 ≥
0, in which 𝑏 = 𝑏 . According to Theorem 6.4, 𝑏 𝑖 ≈ 𝑘 𝜎 𝑖𝑛 ( 𝑏 ). It followsfrom 𝑏 | 𝐵 that 𝜎 𝑖𝑛 ( 𝑏 ) | 𝜎 𝑖𝑛 ( 𝐵 ). Consequently, we have 𝑏 𝑖 ≈ 𝑘 𝜎 𝑖𝑛 ( 𝑏 ) | 𝜎 𝑖𝑛 ( 𝐵 ) ≈ 𝑘 𝐵. Thus there is ˜ 𝑏 𝑖 ∈ K ( 𝑛 )[ 𝑘 ] dividing 𝐵 so that ˜ 𝑏 𝑖 ≈ 𝑘 𝑏 𝑖 . Moreover, ˜ 𝑏 𝑖 is strongly co-prime with 𝐾 as 𝐵 is. By the shifting property of significant denominators (i.e.,Lemma 4.17 and Remark 4.18), there exist ˜ 𝑔 𝑖 ∈ K ( 𝑛, 𝑘 ), ˜ 𝑎 𝑖 , ˜ 𝑞 𝑖 ∈ K ( 𝑛 )[ 𝑘 ] withdeg 𝑘 (˜ 𝑎 𝑖 ) < deg 𝑘 (˜ 𝑏 𝑖 ), and ˜ 𝑞 𝑖 ∈ W 𝐾 such that 𝜎 𝑖𝑛 ( 𝑇 ) = Δ 𝑘 ( ˜ 𝑔 𝑖 𝐻 ) + (˜ 𝑎 𝑖 / ˜ 𝑏 𝑖 + ˜ 𝑞 𝑖 /𝑣 ) 𝐻 .The assertion follows by noticing 𝜎 𝑖𝑛 ( 𝑇 ) = Δ 𝑘 ( ˜ 𝑔 𝑖 𝐻 ) + (︂ ˜ 𝑎 𝑖 𝐵/ ˜ 𝑏 𝑖 𝐵 + ˜ 𝑞 𝑖 𝑣 )︂ 𝐻. Under the assumptions of Theorem 6.7, applying Algorithm 3.17 to 𝑇 w.r.t. 𝑘 yields 𝑇 = Δ 𝑘 ( 𝑔𝐻 ) + 𝑟𝐻, where 𝑔 ∈ K ( 𝑛, 𝑘 ) and 𝑟 is a residual form w.r.t. 𝐾 . ByTheorems 4.6 and 4.10, 𝑏 and the significant denominator 𝑟 𝑑 of 𝑟 are shift-relatedw.r.t. 𝑘 , and thus so are the respective shift-homogeneous components. W.l.o.g.,assume that 𝑏 is shift-homogeneous (then so is 𝑟 𝑑 ). Let ( 𝑃 𝑏 , { 𝜆 𝑏 , 𝜇 𝑏 } , 𝜉 𝑏 ) be a uni-variate representation of 𝑏 and ( 𝑃 𝑟 𝑑 , { 𝜆 𝑟 𝑑 , 𝜇 𝑟 𝑑 } , 𝜉 𝑟 𝑑 ) be one of 𝑟 𝑑 . Definition 4.5 yields4 Chapter 6. Order Bounds for Minimal Telescopers that ( 𝜆 𝑏 , 𝜇 𝑏 ) = ( 𝜆 𝑟 𝑑 , 𝜇 𝑟 𝑑 ) and for each integer 𝑖 , there exists a unique integer 𝑗 andanother integer ℓ 𝑖𝑗 such that 𝑃 𝑏 ( 𝑧 ) 𝛿 𝑖 = 𝜎 ℓ 𝑖𝑗 𝑘 (︁ 𝑃 𝑟 𝑑 ( 𝑧 ) 𝛿 𝑗 )︁ = 𝑃 𝑟 𝑑 ( 𝑧 + 𝜇 𝑟 𝑑 ℓ 𝑖𝑗 ) 𝛿 𝑗 with 𝛿 = 𝛿 ( 𝜆 𝑏 ,𝜇 𝑏 ) . Moreover, the nonzero coefficients of 𝜉 𝑏 are exactly the same as those of 𝜉 𝑟 𝑑 . Insummary, we have the following remark. Remark 6.8.
Although the form of 𝐵 in Theorem 6.7 depends on the choice of 𝑏 ,the shift-equivalence classes w.r.t. ∼ 𝑛,𝑘 as well as the degree of 𝐵 w.r.t. 𝑘 dependonly on the hypergeometric term 𝑇 . In this section, we show that Theorem 6.7 implies that some residual forms { 𝑎 𝑖 /𝑏 𝑖 + 𝑞 𝑖 /𝑣 } 𝑖 ≥ satisfying (6.2) form a finite-dimensional vector space over K ( 𝑛 ), and thenderive an upper bound for the order of minimal telescopers. Theorem 6.9.
With the assumptions of Theorem 6.7, we have that the order of aminimal telescoper for 𝑇 w.r.t. 𝑘 is no more than max { deg 𝑘 ( 𝑢 ) , deg 𝑘 ( 𝑣 ) } − (cid:74) deg 𝑘 ( 𝑣 − 𝑢 ) ≤ deg 𝑘 ( 𝑢 ) − (cid:75) + 𝑠 ∑︁ 𝑗 =1 𝜇 𝑗 𝑚 𝑗 deg 𝑘 ( 𝑃 𝑗 ) , where (cid:74) 𝜙 (cid:75) equals if 𝜙 is true, otherwise it is .Proof. Let 𝐿 = ∑︀ 𝜌𝑖 =0 𝑒 𝑖 𝑆 𝑖𝑛 be a minimal telescoper for 𝑇 w.r.t. 𝑘 , where 𝜌 ∈ N and 𝑒 , . . . , 𝑒 𝜌 ∈ K ( 𝑛 ) not all zero. By Theorem 6.7, there exists 𝐵 ∈ K ( 𝑛 )[ 𝑘 ] suchthat (6.4) holds for every nonnegative integer 𝑖 . Then by Theorem 5.7, the residualforms { 𝑎 𝑖 /𝐵 + 𝑞 𝑖 /𝑣 } 𝜌𝑖 =0 are linearly dependent over K ( 𝑛 ); equivalently, the followinglinear system with unknowns 𝑒 , . . . , 𝑒 𝜌 ⎧⎨⎩ 𝐴 𝜌 = 𝑒 𝑎 + 𝑒 𝑎 + · · · + 𝑒 𝜌 𝑎 𝜌 = 0 𝑄 𝜌 = 𝑒 𝑞 + 𝑒 𝑞 + · · · + 𝑒 𝜌 𝑞 𝜌 = 0 (6.6)has a nontrivial solution in K ( 𝑛 ) 𝜌 +1 . Since deg 𝑘 ( 𝑎 𝑖 ) < deg 𝑘 ( 𝐵 ),deg 𝑘 ( 𝐴 𝜌 ) < deg 𝑘 ( 𝐵 ) = 𝑠 ∑︁ 𝑗 =1 𝜇 𝑗 𝑚 𝑗 deg 𝑘 ( 𝑃 𝑗 ) . (6.7)Note that W 𝐾 is a vector space, so 𝑄 𝜌 ∈ W 𝐾 . By Proposition 3.15, the number ofnonzero terms w.r.t. 𝑘 in 𝑄 𝜌 is no more than the dimension dim K ( 𝑛 ) ( W 𝐾 ), which isbounded by max { deg 𝑘 ( 𝑢 ) , deg 𝑘 ( 𝑣 ) } − (cid:74) deg 𝑘 ( 𝑣 − 𝑢 ) ≤ deg 𝑘 ( 𝑢 ) − (cid:75) . (6.8)Comparing coefficients of like powers of 𝑘 of the linear system (6.6) yields at mostdeg 𝑘 ( 𝐴 𝜌 ) + dim K ( 𝑛 ) ( W 𝐾 ) + 1 (6.9) .3. Upper and lower order bounds 𝜌 exceedsdeg 𝑘 ( 𝐴 𝜌 ) + dim K ( 𝑛 ) ( W 𝐾 ). It implies that the order of a minimal telescoper for 𝑇 w.r.t. 𝑘 is no more than the number (6.9). Therefore, the theorem follows by (6.7)and (6.8).In addition, we can further obtain a lower order bound for telescopers for 𝑇 . Theorem 6.10.
With the assumptions of Theorem 6.7, further assume that 𝑇 isnot summable w.r.t. 𝑘 . Then the order of a telescoper for 𝑇 w.r.t. 𝑘 is at least max 𝑝 | 𝑏, deg 𝑘 ( 𝑝 ) > multiplicity 𝛼 monic & irred. min {︁ 𝜌 ∈ N ∖ { } : 𝜎 ℓ𝑘 ( 𝑝 ) 𝛼 | 𝜎 𝜌𝑛 ( 𝑏 ) for some ℓ ∈ Z }︁ . Proof.
Let 𝐿 = ∑︀ 𝜌𝑖 =0 𝑒 𝑖 𝑆 𝑖𝑛 be a minimal telescoper for 𝑇 w.r.t. 𝑘 , where 𝜌 ∈ N and 𝑒 , . . . , 𝑒 𝜌 ∈ K ( 𝑛 ) not all zero. Since 𝑇 is not summable w.r.t. 𝑘 , we have 𝜌 ≥ ≤ 𝑖 ≤ 𝜌 . Since 𝐿 isa minimal telescoper, 𝑒 ̸ = 0 and by Theorem 5.7, 𝑒 𝑎𝑏 + 𝑒 𝑎 𝑏 + · · · + 𝑒 𝜌 𝑎 𝜌 𝑏 𝜌 = 0 . By partial fraction decomposition, for any monic irreducible factor 𝑝 of 𝑏 withdeg 𝑘 ( 𝑝 ) > 𝛼 >
0, there exists an integer 𝑖 with 1 ≤ 𝑖 ≤ 𝜌 sothat 𝑝 𝛼 is also a factor of 𝑏 𝑖 . By Theorem 6.4, 𝑏 𝑖 ≈ 𝑘 𝜎 𝑖𝑛 ( 𝑏 ). Thus there is a factor 𝑝 ′ of 𝜎 𝑖𝑛 ( 𝑏 ) with multiplicity at least 𝛼 such that 𝑝 ′ ∼ 𝑘 𝑝 . Let 𝑖 𝑝 be the minimal onewith this property. Then the assertion follows by the fact that for each factor 𝑝 of 𝑏 there exists no telescoper for 𝑇 of order less than 𝑖 𝑝 .Together with the bounds given above, we can further develop a variant of Al-gorithm 5.6 by omitting step 4 . 𝑖 reaches andexceeds the lower bound. Algorithm 6.11 (Bound and Reduction-based creative telescoping) .Input : A hypergeometric term 𝑇 over F ( 𝑘 ). Output : A minimal telescoper for 𝑇 w.r.t. 𝑘 and a corresponding certificate iftelescopers exist; “No telescoper exists!”, otherwise. Similar to steps 1 – 3 of Algorithm 5.6. Compute the upper bound 𝑏 𝑢 ∈ N and lower bound 𝑏 𝑙 ∈ N for the orderof minimal telescopers for 𝑇 w.r.t. 𝑘 , respectively. Set 𝑁 = 𝜎 𝑛 ( 𝐻 ) /𝐻 and 𝑅 = ℓ 𝑟 , where ℓ is an indeterminate.For 𝑖 = 1 , , . . . , 𝑏 𝑢 do Similar to steps 4 . . 𝑔 𝑖 , 𝑟 𝑖 ∈ K ( 𝑛, 𝑘 )such that (5.4) holds, and 𝑅 + ℓ 𝑖 𝑟 𝑖 is a residual form w.r.t. 𝐾 ,where ℓ 𝑖 is an indeterminate. Update 𝑅 to 𝑅 + ℓ 𝑖 𝑟 𝑖 . If 𝑖 > 𝑏 𝑙 then find ℓ 𝑗 ∈ F such that 𝑅 = 0by solving a linear system in ℓ , . . . , ℓ 𝑖 over F .If there is a nontrivial solution, return (︁∑︀ 𝑖𝑗 =0 ℓ 𝑗 𝑆 𝑗𝑛 , ∑︀ 𝑖𝑗 =0 ℓ 𝑗 𝑔 𝑗 𝐻 )︁ . Chapter 6. Order Bounds for Minimal Telescopers
In 2005, upper and lower bounds for the order of telescopers for hypergeometricterms have been studied in [49] and [6], respectively. In this section, we are going toreview these known bounds and also compare them to our bounds.
Let 𝑇 be a proper hypergeometric term over K ( 𝑛, 𝑘 ), i.e., it can be written in theform 𝑇 = 𝑝𝑤 𝑛 𝑧 𝑘 𝑚 ∏︁ 𝑖 =1 ( 𝛼 𝑖 𝑛 + 𝛼 ′ 𝑖 𝑘 + 𝛼 ′′ 𝑖 − 𝛽 𝑖 𝑛 − 𝛽 ′ 𝑖 𝑘 + 𝛽 ′′ 𝑖 − 𝜇 𝑖 𝑛 + 𝜇 ′ 𝑖 𝑘 + 𝜇 ′′ 𝑖 − 𝜈 𝑖 𝑛 − 𝜈 ′ 𝑖 𝑘 + 𝜈 ′′ 𝑖 − , (6.10)where 𝑝 ∈ K [ 𝑛, 𝑘 ], 𝑤, 𝑧 ∈ K , 𝑚 ∈ N is fixed, 𝛼 𝑖 , 𝛼 ′ 𝑖 , 𝛽 𝑖 , 𝛽 ′ 𝑖 , 𝜇 𝑖 , 𝜇 ′ 𝑖 , 𝜈 𝑖 , 𝜈 ′ 𝑖 are nonnegativeintegers and 𝛼 ′′ 𝑖 , 𝛽 ′′ 𝑖 , 𝜇 ′′ 𝑖 , 𝜈 ′′ 𝑖 ∈ K . Further assume that there exist no integers 𝑖 and 𝑗 with 1 ≤ 𝑖, 𝑗 ≤ 𝑚 such that (︀ 𝛼 𝑖 = 𝜇 𝑗 and 𝛼 ′ 𝑖 = 𝜇 ′ 𝑗 and 𝛼 ′′ 𝑖 − 𝜇 ′′ 𝑗 ∈ N )︀ or (︀ 𝛽 𝑖 = 𝜈 𝑗 and 𝛽 ′ 𝑖 = 𝜈 ′ 𝑗 and 𝛽 ′′ 𝑖 − 𝜈 ′′ 𝑗 ∈ N )︀ . We refer to this as the generic situation. Then Apagodu and Zeilberger [49] statedthat the order of a minimal telescoper for 𝑇 w.r.t. 𝑘 is bounded by 𝐵 𝐴𝑍 = max {︃ 𝑚 ∑︁ 𝑖 =1 ( 𝛼 ′ 𝑖 + 𝜈 ′ 𝑖 ) , 𝑚 ∑︁ 𝑖 =1 ( 𝛽 ′ 𝑖 + 𝜇 ′ 𝑖 ) }︃ , and this bound is generically sharp.We now show that 𝐵 𝐴𝑍 is at least the order bound given in Theorem 6.9. Re-ordering the factorial terms in (6.10) if necessary, let 𝒮 be the maximal set of integers 𝑖 with 1 ≤ 𝑖 ≤ 𝑚 satisfying (︀ 𝛼 𝑖 = 𝜇 𝑖 and 𝛼 ′ 𝑖 = 𝜇 ′ 𝑖 and 𝜇 ′′ 𝑖 − 𝛼 ′′ 𝑖 ∈ N )︀ or (︀ 𝛽 𝑖 = 𝜈 𝑖 and 𝛽 ′ 𝑖 = 𝜈 ′ 𝑖 and 𝜈 ′′ 𝑖 − 𝛽 ′′ 𝑖 ∈ N )︀ . Rewrite 𝑇 as 𝑟𝑤 𝑛 𝑧 𝑘 𝑚 ∏︁ 𝑖 =1 , 𝑖/ ∈𝒮 ( 𝛼 𝑖 𝑛 + 𝛼 ′ 𝑖 𝑘 + 𝛼 ′′ 𝑖 − 𝛽 𝑖 𝑛 − 𝛽 ′ 𝑖 𝑘 + 𝛽 ′′ 𝑖 − 𝜇 𝑖 𝑛 + 𝜇 ′ 𝑖 𝑘 + 𝜇 ′′ 𝑖 − 𝜈 𝑖 𝑛 − 𝜈 ′ 𝑖 𝑘 + 𝜈 ′′ 𝑖 − , where 𝑟 ∈ K ( 𝑛, 𝑘 ). For 𝑞 ∈ K [ 𝑛, 𝑘 ] and 𝑚 ∈ N , let 𝑞 𝑚 = 𝑞 ( 𝑞 + 1)( 𝑞 + 2) · · · ( 𝑞 + 𝑚 − 𝑞 = 1. By an easy calculation, 𝐾 = 𝑧 ∏︁ 𝑖 ( 𝛼 𝑖 𝑛 + 𝛼 ′ 𝑖 𝑘 + 𝛼 ′′ 𝑖 ) 𝛼 ′ 𝑖 ( 𝜈 𝑖 𝑛 − 𝜈 ′ 𝑖 𝑘 + 𝜈 ′′ 𝑖 − 𝜇 ′ 𝑖 ) 𝜈 ′ 𝑖 ( 𝜇 𝑖 𝑛 + 𝜇 ′ 𝑖 𝑘 + 𝜇 ′′ 𝑖 ) 𝜇 ′ 𝑖 ( 𝛽 𝑖 𝑛 − 𝛽 ′ 𝑖 𝑘 + 𝛽 ′′ 𝑖 − 𝛽 ′ 𝑖 ) 𝛽 ′ 𝑖 (6.11) .4. Comparison of bounds 𝑖 from 1 to 𝑚 such that 𝑖 / ∈ 𝒮 , 𝛼 ′ 𝑖 , 𝛽 ′ 𝑖 > 𝜇 ′ 𝑖 , 𝜈 ′ 𝑖 >
0, is a kernel of 𝑇 and 𝑆 = 𝑟 is a corresponding shell. Let 𝐾 = 𝑢/𝑣 with 𝑢, 𝑣 ∈ K ( 𝑛 )[ 𝑘 ]and gcd( 𝑢, 𝑣 ) = 1. Note that the right-hand side of (6.11) already has the reducedform, then a straightforward calculation yieldsdeg 𝑘 ( 𝑢 ) = 𝑚 ∑︁ 𝑖 =1 ,𝑖/ ∈𝒮 ( 𝛼 ′ 𝑖 + 𝜈 ′ 𝑖 ) and deg 𝑘 ( 𝑣 ) = 𝑚 ∑︁ 𝑖 =1 ,𝑖/ ∈𝒮 ( 𝛽 ′ 𝑖 + 𝜇 ′ 𝑖 ) . Applying the modified Abramov-Petkovšek reduction to 𝑇 w.r.t. 𝑘 yields (6.3),in which 𝑏 is integer-linear. Since 𝑏 only comes from the shift-free part of the denom-inator of 𝑟 , it factors into shift-inequivalent integer-linear polynomials of degree onewhich are separately shift-equivalent to either ( 𝜇 𝑖 𝑛 + 𝜇 ′ 𝑖 𝑘 + 𝜇 ′′ 𝑖 ) or ( 𝛽 𝑖 𝑛 − 𝛽 ′ 𝑖 𝑘 + 𝛽 ′′ 𝑖 )w.r.t. 𝑛, 𝑘 for some 𝑖 ∈ 𝒮 . Note that each 𝑖 in 𝒮 corresponds to at most one integer-linear factor of 𝑏 , and increases the multiplicity of the corresponding factor in 𝑏 byat most 1. Hence, the bound given in Theorem 6.9 is no more thanmax { deg 𝑘 ( 𝑢 ) , deg 𝑘 ( 𝑣 ) } − (cid:74) deg 𝑘 ( 𝑣 − 𝑢 ) ≤ deg 𝑘 ( 𝑢 ) − (cid:75) + 𝑚 ∑︁ 𝑖 =1 ,𝑖 ∈𝒮 ( 𝛽 ′ 𝑖 + 𝜇 ′ 𝑖 ) , which is exactly equal to 𝐵 𝐴𝑍 − (cid:74) deg 𝑘 ( 𝑣 − 𝑢 ) ≤ deg 𝑘 ( 𝑢 ) − (cid:75) , since ∑︀ 𝑚𝑖 =1 ,𝑖 ∈𝒮 ( 𝛼 ′ 𝑖 + 𝜈 ′ 𝑖 ) = ∑︀ 𝑚𝑖 =1 ,𝑖 ∈𝒮 ( 𝛽 ′ 𝑖 + 𝜇 ′ 𝑖 ).In general, i.e., in the generic situation, the order bound in Theorem 6.9 isalmost the same as 𝐵 𝐴𝑍 , which is not suprising since 𝐵 𝐴𝑍 is already genericallysharp. However, our bound can be much better in some special examples. Example 6.12.
Consider a rational function 𝑇 = 𝛼 𝑘 + 𝛼 𝑘 − 𝛼𝛽𝑘 + 2 𝛼𝑛𝑘 + 𝑛 ( 𝑛 + 𝛼𝑘 + 𝛼 )( 𝑛 + 𝛼𝑘 )( 𝑛 + 𝛽𝑘 ) , where 𝛼, 𝛽 are positive integers and 𝛼 ̸ = 𝛽 . Rewriting 𝑇 into the proper form (6.10)yields 𝐵 𝐴𝑍 = 𝛼 + 𝛽 . On the other hand, 1 is the only kernel of 𝑇 since 𝑇 is a rationalfunction. By the modified Abramov-Petkovšek reduction, 𝑏 = 𝑛 + 𝛽𝑘 in (6.3). ByTheorem 6.9, a minimal telescoper for 𝑇 w.r.t. 𝑘 has order no more than 𝛽 , whichis in fact the real order of minimal telescopers for 𝑇 w.r.t. 𝑘 . Remark 6.13.
Together with [4, Theorem 10], the upper order bound 𝐵 𝐴𝑍 on min-imal telescopers derived in [49] can be also applied to non-proper hypergeometricterms. On the other hand, Theorem 6.9 can be directly applied to any hypergeomet-ric term provided that its telescopers exist. With Convention 6.2, further assume that 𝑇 has the initial reduction (6.3), in which 𝑏 is integer-linear. Let 𝐻 ′ = 𝐻/𝑣 . A direct calculation leads to 𝜎 𝑘 ( 𝐻 ′ ) 𝐻 ′ = 𝑢𝜎 𝑘 ( 𝑣 ) , Chapter 6. Order Bounds for Minimal Telescopers which can be easily checked to be shift-reduced w.r.t. 𝑘 . Let 𝑑 ′ ∈ K ( 𝑛 )[ 𝑘 ] be thedenominator of 𝜎 𝑛 ( 𝐻 ′ ) /𝐻 ′ . Then the algorithm LowerBound in [6] asserts that theorder of telescopers for 𝑇 w.r.t. 𝑘 is at least 𝐵 𝐴𝐿 = max 𝑝 | 𝑏 irred. & monicdeg 𝑘 ( 𝑝 ) > min ⎧⎪⎪⎨⎪⎪⎩ 𝜌 ∈ N ∖ { } : 𝜎 ℓ𝑘 ( 𝑝 ) | 𝜎 𝜌𝑛 ( 𝑏 )or 𝜎 ℓ𝑘 ( 𝑝 ) | 𝜎 𝜌 − 𝑛 ( 𝑑 ′ ) for some ℓ ∈ Z ⎫⎪⎪⎬⎪⎪⎭ Comparing to 𝐵 𝐴𝐿 from above, one easily sees that the lower bound given in Theo-rem 6.10 can be better but never worse than 𝐵 𝐴𝐿 . Example 6.14.
Consider a hypergeometric term 𝑇 = 1( 𝑛 − 𝛼𝑘 − 𝛼 )( 𝑛 − 𝛼𝑘 − , where 𝛼 ∈ N and 𝛼 >
1. By the algorithm
LowerBound , a telescoper for 𝑇 w.r.t. 𝑘 has order at least 2. On the other hand, a telescoper for 𝑇 w.r.t. 𝑘 has order atleast 𝛼 by Theorem 6.10. In fact, 𝛼 is exactly the order of minimal telescopers for 𝑇 w.r.t. 𝑘 . In Maple 18 , we have implemented Algorithm 6.11 and embedded it into the pack-age
ShiftReductionCT , under the name of
BoundReductionCT . For a detailedexplanation, one may refer to Appendix A.In this section, we focus on the two procedures –
BoundReductionCT and
Re-ductionCT in the package
ShiftReductionCT , and their runtime is compared. Alltimings are measured in seconds on a Linux computer with 388Gb RAM and twelve2.80GHz Dual core processors. No parallelism was used in this experiment. More-over, a comparison of the memory requirements is given in Appendix B. For brevity,we denote•
RCT 𝑡𝑐 : the procedure ReductionCT in ShiftReductionCT , which computes aminimal telescoper and a corresponding normalized certificate;•
RCT 𝑡 : the procedure ReductionCT in ShiftReductionCT , which computes aminimal telescoper without constructing a certificate.•
BRCT 𝑡𝑐 : the procedure BoundReductionCT in ShiftReductionCT , which com-putes a minimal telescoper and a corresponding normalized certificate;•
BRCT 𝑡 : the procedure BoundReductionCT in ShiftReductionCT , which com-putes a minimal telescoper without constructing a certificate.• LB : the lower bound for telescopers given in Theorem 6.10.• order : the order of the resulting minimal telescoper. .5. Implementation and timings Example 6.15.
Consider the same hypergeometric term as in Example 6.14, i.e., 𝑇 = 1( 𝑛 − 𝛼𝑘 − 𝛼 )( 𝑛 − 𝛼𝑘 − , where 𝛼 is an integer greater than 1. For different choices of 𝛼 , Table 6.1 showsthe timings of the above procedures. Note that since the term 𝑇 in this example isvery simple, there is little difference in the timings for the two procedures with andwithout construction of a certificate. 𝛼 RCT 𝑡 RCT 𝑡𝑐 BRCT 𝑡 BRCT 𝑡𝑐 LB order
20 2.00 2.02 1.07 1.13 20 2030 7.01 7.19 2.86 2.96 30 3040 20.08 20.13 7.06 7.18 40 4050 42.15 42.68 14.96 15.05 50 5060 104.07 106.31 25.54 25.93 60 6070 225.67 229.04 45.76 45.97 70 70
Table 6.1:
Timing comparison of two reduction-basedcreative telescoping with andwithout construction of a certificate for Example 6.15 (in seconds)
Example 6.16 (Example 6 in [6]) . Consider the hypergeometric term 𝑇 = Δ 𝑘 ( 𝑇 ) + 𝑇 , where 𝑇 = 1( 𝑛𝑘 − 𝑛 − 𝛼𝑘 − 𝑚 (2 𝑛 + 𝑘 + 3)! and 𝑇 = 1( 𝑛 − 𝛼𝑘 − 𝑛 + 𝑘 + 3)!for 𝛼, 𝑚 positive integers. For different choices of 𝛼 and 𝑚 , we compare the timingsof the procedures from above. Table 6.2 shows the final experimental results.( 𝑚, 𝛼 ) RCT 𝑡 RCT 𝑡𝑐 BRCT 𝑡 BRCT 𝑡𝑐 LB order (1,1) 0.20 0.24 0.20 0.23 1 2(1,10) 5.25 9.56 4.60 8.74 10 11(1,15) 57.06 76.01 37.73 58.69 15 16(1,20) 538.59 656.99 264.04 324.09 20 21(2,10) 5.29 9.11 4.43 8.36 10 11(2,15) 79.34 96.48 40.26 54.85 15 16(2,20) 574.00 658.20 282.54 377.84 20 21
Table 6.2:
Timing comparison of two reduction-basedcreative telescoping with andwithout construction of a certificate for Example 6.16 (in seconds) Chapter 6. Order Bounds for Minimal Telescopers
Remark 6.17.
Compared to linear dependence, determining linear independencetakes much less time because with high probability, independence can be recognizedby a computation in a homomorphic image. For this reason, the procedure
Bound-ReductionCT makes no big difference from the procedure
ReductionCT if the lowerbound is far away from the real order of minimal telescopers. In fact, their performalmost the same in this case. art II
Limits ofP-recursive sequences hapter 7D-finite Functions and
P-recursive Sequences
In this chapter, we recall [34, 41] basic notions related to the class of D-finite func-tions and P-recursive sequences, and also present some useful properties.
Recall [41] that a formal power series is an infinite series of the form 𝑓 ( 𝑧 ) = ∞ ∑︁ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 , where 𝑧 is a formal indeterminate. It generalizes the notions of polynomials andpower series in some sense. A formal power series differs from a polynomial in thatit allows an infinite number of terms, and it differs from power series by assuminga formal variable and ignoring analytic properties. One way to view a formal powerseries 𝑓 ( 𝑧 ) is to take it as an infinite sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 , where the powers indicate theorder of terms. We will also call a formal power series 𝑓 ( 𝑥 ) the generating function ofits coefficient sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 . Note that these three notions – formal power series,sequences, generating functions – all refer to the same object.For a ring 𝑅 , we denote by 𝑅 [[ 𝑧 ]] the ring of formal power series endowed withtermwise addition (+) and Cauchy product ( · ): (︃ ∞ ∑︁ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 )︃ + (︃ ∞ ∑︁ 𝑛 =0 𝑏 𝑛 𝑧 𝑛 )︃ = ∞ ∑︁ 𝑛 =0 ( 𝑎 𝑛 + 𝑏 𝑛 ) 𝑧 𝑛 , (︃ ∞ ∑︁ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 )︃ · (︃ ∞ ∑︁ 𝑛 =0 𝑏 𝑛 𝑧 𝑛 )︃ = ∞ ∑︁ 𝑛 =0 (︃ 𝑛 ∑︁ 𝑘 =0 𝑎 𝑘 𝑏 𝑛 − 𝑘 )︃ 𝑧 𝑛 , and by 𝑅 N the ring of infinite sequences endowed with termwise addition (+) and Hadamard product ( ⊙ ): ( 𝑎 𝑛 ) ∞ 𝑛 =0 + ( 𝑏 𝑛 ) ∞ 𝑛 =0 = ( 𝑎 𝑛 + 𝑏 𝑛 ) ∞ 𝑛 =0 , ( 𝑎 𝑛 ) ∞ 𝑛 =0 ⊙ ( 𝑏 𝑛 ) ∞ 𝑛 =0 = ( 𝑎 𝑛 𝑏 𝑛 ) ∞ 𝑛 =0 . Chapter 7. D-finite Functions and P-recursive Sequences
Also recall [34] that a complex function 𝑓 ( 𝑧 ) is called analytic at a point 𝜁 ∈ C iffor any 𝑧 in a neighborhood of 𝜁 , it can be represented by a convergent power seriesover C , 𝑓 ( 𝑧 ) = ∞ ∑︁ 𝑛 =0 𝑎 𝑛 ( 𝑧 − 𝜁 ) 𝑛 , where 𝑎 𝑛 ∈ C for all 𝑛 ∈ N . A function is analytic in an open set if it is analytic at every point of the set.Throughout the chapter, let 𝑅 be a subring of C and F be a subfield of C .We consider linear operators that act on sequences or power series and analyticfunctions. Recall from the previous chapters that we write 𝜎 𝑛 for the shift operatorw.r.t. 𝑛 which maps a sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 to ( 𝑎 𝑛 +1 ) ∞ 𝑛 =0 . Also we denote by F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ the ring of linear recurrence operators of the form 𝐿 := 𝑝 + 𝑝 𝑆 𝑛 + · · · + 𝑝 𝜌 𝑆 𝜌𝑛 ,with 𝑝 , . . . , 𝑝 𝜌 ∈ F [ 𝑛 ], where 𝑆 𝑛 𝑟 = 𝜎 𝑛 ( 𝑟 ) 𝑆 𝑛 for all 𝑟 ∈ F [ 𝑛 ]. This ring forms an Orealgebra. Analogously, we write 𝐷 𝑧 for the derivation operator w.r.t. 𝑧 which maps apower series or function 𝑓 ( 𝑧 ) to its derivative 𝑓 ′ ( 𝑧 ) = 𝑑𝑑𝑧 𝑓 ( 𝑧 ). Also the set of linearoperators of the form 𝐿 := 𝑝 + 𝑝 𝐷 𝑧 + · · · + 𝑝 𝜌 𝐷 𝜌𝑧 , with 𝑝 , . . . , 𝑝 𝜌 ∈ F [ 𝑧 ], formsan Ore algebra; we denote it by F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ . For an introduction to Ore algebras andtheir actions, please refer to [17]. When 𝑝 𝜌 ̸ = 0, we call 𝜌 the order of the operatorand lc( 𝐿 ) := 𝑝 𝜌 its leading coefficient . Definition 7.1.
1. A sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ 𝑅 N is called P-recursive or D-finite over F if there existsa nonzero operator 𝐿 = ∑︀ 𝜌𝑗 =0 𝑝 𝑗 ( 𝑛 ) 𝑆 𝑗𝑛 ∈ F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ such that 𝐿 · 𝑎 𝑛 = 𝑝 𝜌 ( 𝑛 ) 𝑎 𝑛 + 𝜌 + · · · + 𝑝 ( 𝑛 ) 𝑎 𝑛 +1 + 𝑝 ( 𝑛 ) 𝑎 𝑛 = 0 for all 𝑛 ∈ N .2. A formal power series 𝑓 ( 𝑧 ) ∈ 𝑅 [[ 𝑧 ]] is called D-finite over F if there exists anonzero operator 𝐿 = ∑︀ 𝜌𝑗 =0 𝑝 𝑗 ( 𝑧 ) 𝐷 𝑗𝑧 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ such that 𝐿 · 𝑓 ( 𝑧 ) = 𝑝 𝜌 ( 𝑧 ) 𝐷 𝜌𝑧 𝑓 ( 𝑧 ) + · · · + 𝑝 ( 𝑧 ) 𝐷 𝑧 𝑓 ( 𝑧 ) + 𝑝 ( 𝑧 ) 𝑓 ( 𝑧 ) = 0 .
3. A formal power series 𝑓 ( 𝑧 ) ∈ F [[ 𝑧 ]] is called algebraic over F if there exists anonzero bivariate polynomial 𝑃 ( 𝑧, 𝑦 ) ∈ F [ 𝑧, 𝑦 ] such that 𝑃 ( 𝑧, 𝑓 ( 𝑧 )) = 0 . In general, D-finite power series are called D-finite functions instead. A formalpower series is D-finite if and only if its coefficient sequence is P-recursive. Manyelementary functions like rational functions, exponentials, logarithms, sine, algebraicfunctions, etc., as well as many special functions, like hypergeometric series, theerror function, Bessel functions, etc., are D-finite. Hence their respective coefficientsequences are P-recursive.
The class of D-finite functions (resp. P-recursive sequences) is closed under certainoperations: addition, multiplication, derivative (resp. forward shift) and integration(resp. summation). In particular, the set of D-finite functions (resp. P-recursive .2. Useful properties F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ -module (resp. a left- F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ -module). Also, if 𝑓 is a D-finite function and 𝑔 is an algebraic function, then the composition 𝑓 ∘ 𝑔 isD-finite. These and further closure properties are easily proved by linear algebraarguments, whose proofs can be found for instance in [59, 57, 41]. We will make freeuse of these facts.We will be considering singularities of D-finite functions. Recall from the clas-sical theory of linear differential equations [40] that a linear differential equation 𝑝 ( 𝑧 ) 𝑓 ( 𝑧 ) + · · · + 𝑝 𝜌 ( 𝑧 ) 𝑓 ( 𝜌 ) ( 𝑧 ) = 0 with polynomial coefficients 𝑝 , . . . , 𝑝 𝜌 ∈ F [ 𝑧 ]and 𝑝 𝜌 ̸ = 0 has a basis of analytic solutions in a neighborhood of every point 𝜁 ∈ C ,except possibly at roots of 𝑝 𝜌 . The roots of 𝑝 𝜌 are therefore called the singularities of the equation (or the corresponding linear operator). If 𝜁 ∈ C is a singularity ofthe equation but the equation nevertheless admits a basis of analytic solutions atthis point, we call it an apparent singularity . It is well-known [40, 25] that for anygiven linear differential equation with some apparent and some non-apparent sin-gularities, we can always construct another linear differential equation (typically ofhigher order) whose solution space contains the solution space of the first equationand whose only singularities are the non-apparent singularities of the first equation.This process is known as desingularization.For later use, we will give a proof of the composition closure property for D-finitefunctions which pays attention to the singularities. Theorem 7.2.
Let 𝑃 ( 𝑧, 𝑦 ) ∈ F [ 𝑧, 𝑦 ] be a square-free polynomial of degree 𝑑 , andlet 𝐿 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ be nonzero. Let 𝜁 ∈ C be such that 𝑃 defines 𝑑 distinct analyticalgebraic functions 𝑔 ( 𝑧 ) with 𝑃 ( 𝑧, 𝑔 ( 𝑧 )) = 0 in a neighborhood of 𝜁 , and assume thatfor none of these functions, the value 𝑔 ( 𝜁 ) ∈ C is a singularity of 𝐿 . Fix a solution 𝑔 of 𝑃 and an analytic solution 𝑓 of 𝐿 defined in a neighborhood of 𝑔 ( 𝜁 ) . Then thereexists a nonzero operator 𝑀 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ with 𝑀 · ( 𝑓 ∘ 𝑔 ) = 0 which does not have 𝜁 among its singularities.Proof. (borrowed from [42]) Consider the operator ˜ 𝐿 = 𝐿 ( 𝑔, ( 𝑔 ′ ) − 𝐷 𝑧 ) ∈ F ( 𝑧 ) ⟨ 𝐷 𝑧 ⟩ .It is easy to check that 𝐿 · 𝑓 = 0 if and only if ˜ 𝐿 · ( 𝑓 ∘ 𝑔 ) = 0 for every solution 𝑔 of 𝑃 near 𝜁 . Therefore, if 𝑓 , . . . , 𝑓 𝜌 is a basis of the solution space of 𝐿 near 𝑔 ( 𝜁 ),then 𝑓 ∘ 𝑔, . . . , 𝑓 𝜌 ∘ 𝑔 is a basis of the solution space of ˜ 𝐿 near 𝜁 .Let 𝑔 , . . . , 𝑔 𝑑 be all the solutions of 𝑃 near 𝜁 , and let 𝑀 be the least commonleft multiple of all the operators 𝐿 ( 𝑔 𝑗 , ( 𝑔 ′ 𝑗 ) − 𝐷 𝑧 ). Then the solution space of 𝑀 near 𝜁 is generated by all the functions 𝑓 𝑖 ∘ 𝑔 𝑗 . Since the coefficients of 𝑀 aresymmetric w.r.t. the conjugates 𝑔 , . . . , 𝑔 𝑑 , they belong to the ground field F ( 𝑧 ),and after clearing denominators (from the left) if necessary, we may assume that 𝑀 is an operator in F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ whose solution space is generated by functions that areanalytic at 𝜁 . Therefore, by the remarks made about desingularization, it is possibleto replace 𝑀 by an operator (possibly of higher order) which does not have 𝜁 amongits singularities.By a similar argument, we see that algebraic extensions of the coefficient field ofthe recurrence operators are useless. Moreover, it is also not useful to make F biggerthan the quotient field of 𝑅 .6 Chapter 7. D-finite Functions and P-recursive Sequences
Lemma 7.3.
1. If E is an algebraic extension field of F and ( 𝑎 𝑛 ) ∞ 𝑛 =0 is P-recursive over E , thenit is also P-recursive over F .2. If 𝑅 ⊆ F and ( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ 𝑅 N is P-recursive over F , then it is also P-recursiveover Quot( 𝑅 ) , the quotient field of 𝑅 .3. If F is closed under complex conjugation and ( 𝑎 𝑛 ) ∞ 𝑛 =0 is P-recursive over F ,then so are (¯ 𝑎 𝑛 ) ∞ 𝑛 =0 , (Re( 𝑎 𝑛 )) ∞ 𝑛 =0 , and (Im( 𝑎 𝑛 )) ∞ 𝑛 =0 .Proof.
1. Let 𝐿 ∈ E [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ be an annihilating operator of ( 𝑎 𝑛 ) ∞ 𝑛 =0 . Then, since 𝐿 has only finitely many coefficients, 𝐿 ∈ F ( 𝜃 )[ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ for some 𝜃 ∈ E . Let 𝑀 be the least common left multiple of all the conjugates of 𝐿 . Then 𝑀 is anannihilating operator of ( 𝑎 𝑛 ) ∞ 𝑛 =0 which belongs to F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ . The claim follows.2. Let us write K = Quot( 𝑅 ). Let 𝐿 ∈ F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ be a nonzero annihilating opera-tor of ( 𝑎 𝑛 ) ∞ 𝑛 =0 . Since F is an extension field of K , it is a vector space over K .Write 𝐿 = 𝜌 ∑︁ 𝑚 =0 𝑑 𝑚 ∑︁ 𝑗 =0 𝑝 𝑚𝑗 𝑛 𝑗 𝑆 𝑚𝑛 , where 𝑟, 𝑑 𝑚 ∈ N and 𝑝 𝑚𝑗 ∈ F not all zero. Then the set of the coefficients 𝑝 𝑖𝑗 belongs to a finite dimensional subspace of F . Let { 𝛼 , . . . , 𝛼 𝑠 } be a basis ofthis subspace over K . Then for each pair ( 𝑚, 𝑗 ), there exists 𝑐 𝑚𝑗ℓ ∈ K suchthat 𝑝 𝑚𝑗 = ∑︀ 𝑠ℓ =1 𝑐 𝑚𝑗ℓ 𝛼 ℓ , which gives0 = 𝐿 · 𝑎 𝑛 = 𝑠 ∑︁ ℓ =1 𝛼 ℓ ⎛⎝ 𝜌 ∑︁ 𝑚 =0 𝑑 𝑚 ∑︁ 𝑗 =0 𝑐 𝑚𝑗ℓ 𝑛 𝑗 𝑎 𝑛 + 𝑚 ⎞⎠⏟ ⏞ =: 𝑏 𝑛 ∈ K . For all 𝑛 ∈ N , it follows from the linear independence of { 𝛼 , . . . , 𝛼 𝑠 } over K that 𝑏 𝑛 = 0. Therefore 𝜌 ∑︁ 𝑚 =0 ⎛⎝ 𝑑 𝑚 ∑︁ 𝑗 =0 𝑐 𝑚𝑗ℓ 𝑛 𝑗 ⎞⎠⏟ ⏞ ∈ K [ 𝑛 ] 𝑆 𝑚𝑛 · 𝑎 𝑛 = 0 for all 𝑛 ∈ N and ℓ = 1 , . . . , 𝑠. Thus ( 𝑎 𝑛 ) ∞ 𝑛 =0 has a nonzero annihilating operator with coefficients in K [ 𝑛 ].3. Since ( 𝑎 𝑛 ) ∞ 𝑛 =0 is P-recursive over F , there exists a nonzero operator 𝐿 in F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ such that 𝐿 · 𝑎 𝑛 = 0. Hence ¯ 𝐿 · ¯ 𝑎 𝑛 = 0 where ¯ 𝐿 is the operator obtained from 𝐿 by taking the complex conjugate of each coefficient. Since F is closed undercomplex conjugation by assumption, we see that ¯ 𝐿 belongs to F [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ , andhence (¯ 𝑎 𝑛 ) ∞ 𝑛 =0 is P-recursive over F .Because of Re( 𝑎 𝑛 ) = ( 𝑎 𝑛 + ¯ 𝑎 𝑛 ) and Im( 𝑎 𝑛 ) = 𝑖 ( 𝑎 𝑛 − ¯ 𝑎 𝑛 ) with 𝑖 the imaginaryunit, the other two assertions follow by closure properties. .2. Useful properties Maple implementation, namely the
NumGfun pack-age, for computing such evaluations. These algorithms perform arbitrary-precisionevaluations with full error control. hapter 8D-finite Numbers As mentioned in the introduction, the class of algebraic numbers and the class ofalgebraic functions are naturally connected to each other. For instance, evaluatingan algebraic function over Q at an algebraic point gives an algebraic number. Alsothe values of compositional inverses of algebraic functions at algebraic points arealgebraic. In particular, roots of an algebraic function over Q are all algebraic num-bers. Moreover, we will see below that every algebraic number can appear as a limitof the coefficient sequence of an algebraic function. However, the class of algebraicnumbers is quite small. Almost all real and complex numbers are not algebraic,including many important numbers like 𝜋 and Euler’s number e.Motivated by the above relation, we aim to establish a similar correspondencebetween numbers and the class of D-finite functions. To this end, we introduce thefollowing class of numbers. Definition 8.1.
Let 𝑅 be a subring of C and let F be a subfield of C .1. A number 𝜉 ∈ C is called D-finite (with respect to 𝑅 and F ) if there exists aconvergent sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 in 𝑅 N with lim 𝑛 →∞ 𝑎 𝑛 = 𝜉 and some polynomials 𝑝 , . . . , 𝑝 𝜌 ∈ F [ 𝑛 ] , 𝑝 𝜌 ̸ = 0 , not all zero, such that 𝑝 ( 𝑛 ) 𝑎 𝑛 + 𝑝 ( 𝑛 ) 𝑎 𝑛 +1 + · · · + 𝑝 𝜌 ( 𝑛 ) 𝑎 𝑛 + 𝜌 = 0 for all 𝑛 ∈ N .2. The set of all D-finite numbers with respect to 𝑅 and F is denoted by 𝒟 𝑅, F . If 𝑅 = F , we also write 𝒟 F := 𝒟 F , F for short. It turns out that the class of D-finite numbers is closely related to the classof (regular or singular) holonomic constants [35], i.e., the set of all finite values ofD-finite functions at (regular or singular) algebraic points.In this chapter, we show that D-finite numbers are in fact holonomic constants,and conversely, the regular holonomic constants, i.e., the values D-finite functionscan assume at non-singular algebraic number arguments, are essentially D-finitenumbers over the Gaussian rational field. Together with the work on arbitrary-precision evaluation of D-finite functions [26, 62, 63, 64, 47, 48], it follows that The main results in this chapter are joint work with M. Kauers [39]. Chapter 8. D-finite Numbers
D-finite numbers are computable in the sense that for every D-finite number 𝜉 thereexists an algorithm which for any given 𝑛 ∈ N computes a numeric approximation of 𝜉 with a guaranteed precision of 10 − 𝑛 . Consequently, all non-computable numbershave no chance to be D-finite. Besides these artificial examples, we do not know ofany explicit real numbers which are not in 𝒟 Q , and we believe that it may be verydifficult to find some.We see from Definition 8.1 that the class 𝒟 𝑅, F depends on two subrings of C : thering 𝑅 where the sequence lives, and the field F over which the difference equationis defined. Obviously, different choices of subrings may or may not lead to differentclasses of D-finite numbers. One goal for this chapter is to investigate what kindof choices of 𝑅 and F can be made without changing the resulting class of D-finitenumbers. Throughout the chapter, 𝑅 is a subring of C and F is a subfield of C , as in Defini-tion 8.1 above. Thanks to many mathematicians’ work, we can easily recognize formany constants that they in fact belong to 𝒟 Q . Example 8.2.
1. Archimedes’ constant 𝜋 . Let 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =0 𝑘 (︂ 𝑘 + 1 − 𝑘 + 4 − 𝑘 + 5 − 𝑘 + 6 )︂ . It is clear that ( 𝑓 𝑛 ) ∞ 𝑛 =0 is a P-recursive sequence in Q . According to the Bailey-Borwein-Plouffe formula [13], lim 𝑛 →∞ 𝑓 𝑛 = 𝜋 .2. Euler’s number e. By the Taylor series of the exponential function, we havelim 𝑛 →∞ 𝑓 𝑛 = e where 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =0 𝑘 ! . It is clear that the terms 𝑓 𝑛 form a P-recursive sequence over Q .3. Logarithmic value log 2. By the Taylor series of the natural logarithm, we finda P-recursive sequence ( 𝑓 𝑛 ) ∞ 𝑛 =0 ∈ Q N with 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =1 ( − 𝑘 +1 𝑘 , such that lim 𝑛 →∞ 𝑓 𝑛 = log(2).4. Pythagoras’ constant √
2. One easily finds a P-recursive sequence ( 𝑓 𝑛 ) ∞ 𝑛 =0 over Q with 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =0 (︂ 𝑘 )︂ , and we have lim 𝑛 →∞ 𝑓 𝑛 = √ .2. Algebraic numbers 𝜁 (3). By the definition, we see thatlim 𝑛 →∞ 𝑓 𝑛 = 𝜁 (3) with 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =1 𝑘 . It is readily seen that ( 𝑓 𝑛 ) ∞ 𝑛 =0 ∈ Q N is D-finite.6. The number 1 /𝜋 . Thanks to Ramanujan, we know that the terms 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =0 (︂ 𝑘𝑘 )︂ 𝑘 + 52 𝑘 +4 , tend to 1 /𝜋 as 𝑛 → ∞ and form a P-recursive sequence over Q .7. Euler’s constant 𝛾 . A desired P-recursive sequence is found by Fischler andRivoal at their work [31]. They showed thatlim 𝑛 →∞ 𝑓 𝑛 = 𝛾 with 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =1 ( − 𝑘 (︂ 𝑛𝑘 )︂ 𝑘 (︂ − 𝑘 ! )︂ .
8. Any value of the Gamma function to a rational number Γ( 𝛼 ) with 𝛼 < Q .Again, Fischler and Rivoal [31] proved thatlim 𝑛 →∞ 𝑓 𝑛 = Γ( 𝛼 ) with 𝑓 𝑛 = 𝑛 ∑︁ 𝑘 =0 (︂ 𝑛 + 𝛼𝑘 + 𝛼 )︂ ( − 𝑘 𝑘 !( 𝑘 + 𝛼 ) . Before turning to general D-finite numbers, let us consider the subclass of alge-braic functions. We will show that in this case, the possible limits are preciselythe algebraic numbers. For the purpose of this chapter, let us say that a sequence( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ F N is algebraic over F if the corresponding power series ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 ∈ F [[ 𝑧 ]]is algebraic in the sense of Definition 7.1. Since algebraic functions are D-finite(Abel’s theorem), it is clear that algebraic sequences are P-recursive. We will write 𝒜 F for the set of all numbers 𝜉 ∈ C which are limits of convergent algebraic sequencesover F .Recall [34] that two sequences ( 𝑎 𝑛 ) ∞ 𝑛 =0 , ( 𝑏 𝑛 ) ∞ 𝑛 =0 are called asymptotically equiv-alent , written 𝑎 𝑛 ∼ 𝑏 𝑛 ( 𝑛 → ∞ ), if the quotient 𝑎 𝑛 /𝑏 𝑛 converges to 1 as 𝑛 → ∞ .Similarly, two complex functions 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) are called asymptotically equivalent ata point 𝜁 ∈ C , written 𝑓 ( 𝑧 ) ∼ 𝑔 ( 𝑧 ) ( 𝑧 → 𝜁 ), if the quotient 𝑓 ( 𝑧 ) /𝑔 ( 𝑧 ) converges to 1as 𝑧 approaches 𝜁 . These notions are connected by the following classical theorem. Theorem 8.3.
1. (Transfer theorem [33, 34]) For every 𝛼 ∈ C ∖ Z ≤ we have [ 𝑧 𝑛 ] 1(1 − 𝑧 ) 𝛼 ∼ 𝑛 𝛼 − Γ( 𝛼 ) ( 𝑛 → ∞ ) , where Γ( 𝑧 ) stands for the Gamma function and the notation [ 𝑧 𝑛 ] 𝑓 ( 𝑧 ) refers tothe coefficient of 𝑧 𝑛 in the power series 𝑓 ( 𝑧 ) ∈ F [[ 𝑧 ]] . Chapter 8. D-finite Numbers
2. (Basic Abelian theorem [32]) Let ( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ F N be a sequence that satisfies theasymptotic estimate 𝑎 𝑛 ∼ 𝑛 𝛼 ( 𝑛 → ∞ ) , where 𝛼 ≥ . Then the generating function 𝑓 ( 𝑧 ) = ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 satisfies theasymptotic estimate 𝑓 ( 𝑧 ) ∼ Γ( 𝛼 + 1)(1 − 𝑧 ) 𝛼 +1 ( 𝑧 → − ) . This estimate remains valid when 𝑧 tends to in any sector with vertex at symmetric about the horizontal axis, and with opening angle less than 𝜋 . To show that 𝒜 F is in fact a field, we need the following lemma. It indicates thatdepending on whether F is a real field or not, every real algebraic number or everyalgebraic number can appear as a limit. Lemma 8.4.
Let 𝑝 ( 𝑧 ) ∈ F [ 𝑧 ] be an irreducible polynomial of degree 𝑑 . Then there isa square-free polynomial 𝑃 ( 𝑧, 𝑦 ) ∈ F [ 𝑧, 𝑦 ] of degree 𝑑 in 𝑦 and admitting 𝑑 distinctanalytic algebraic functions 𝑓 ( 𝑧 ) ∈ F [[ 𝑧 ]] with 𝑃 ( 𝑧, 𝑓 ( 𝑧 )) = 0 in a neighborhood of such that is the only dominant singularity of each 𝑓 and1. if F ⊆ R , then for each root 𝜉 ∈ ¯ F ∩ R of 𝑝 ( 𝑧 ) there exists a solution 𝑓 ( 𝑧 ) of 𝑃 ( 𝑧, 𝑦 ) with lim 𝑛 →∞ [ 𝑧 𝑛 ] 𝑓 ( 𝑧 ) = 𝜉 ;2. if F ∖ R ̸ = ∅ , then for each root 𝜉 ∈ ¯ F of 𝑝 ( 𝑧 ) there exists a solution 𝑓 ( 𝑧 ) of 𝑃 ( 𝑧, 𝑦 ) with lim 𝑛 →∞ [ 𝑧 𝑛 ] 𝑓 ( 𝑧 ) = 𝜉 .Proof. The two assertions can be proved simultaneously as follows.Let 𝜀 > 𝑝 have a distance of morethan 𝜀 to each other. Such an 𝜀 exists because 𝑝 is a polynomial, and polynomialshave only finitely many roots. The roots of a polynomial depend continuously onits coefficients. Therefore there exists a real number 𝛿 > 𝛿 won’t perturb the roots by more than 𝜀/
2. Any positivesmaller number than 𝛿 will have the same property. By the choice of 𝜀 , any suchperturbation of the polynomial will have exactly one (real or complex) root in eachof the balls of radius 𝜀/ 𝑝 .Let 𝜉 be a root of 𝑝 . If 𝜉 = 0, then 𝑝 ( 𝑧 ) = 𝑧 . Letting 𝑃 ( 𝑧, 𝑦 ) = 𝑦 yields theassertions. Now assume that 𝜉 ̸ = 0. Let 𝑚 ∈ F be the maximal modulus of coefficientsof 𝑝 . Then 𝑚 ̸ = 0 since 𝑝 is irreducible. Therefore, we always can find a number 𝑎 ∈ F such that | 𝑎 − 𝜉 | < 𝛿/𝑚 , with the 𝛿 from above. Indeed, we have the following casedistinction.For part 1 where F ⊆ R , we only consider 𝜉 ∈ ¯ F ∩ R . In this case, F is dense in R since F ⊇ Q . Hence such 𝑎 ∈ F ⊆ R exists.For part 2 where F ∖ R ̸ = ∅ , there exists a non-real complex number 𝛼 in F . Therefore, Q ( 𝛼 ) is dense in C . Since Q ( 𝛼 ) ⊆ F , such 𝑎 ∈ F is guaranteed by the density of F in C .After finding 𝑎 ∈ F with | 𝑎 − 𝜉 | < 𝛿/𝑚 , for both cases, we have | 𝑝 ( 𝑎 ) | = | 𝑝 ( 𝑎 ) − 𝑝 ( 𝜉 ) | ≤ 𝑚 | 𝑎 − 𝜉 | < 𝛿. .2. Algebraic numbers 𝛿 by | 𝑝 ( 𝑎 ) | for such a choice of 𝑎 . The remaining argument works forboth cases.Consider the perturbation ˜ 𝑝 ( 𝑦 ) = 𝑝 ( 𝑦 ) − 𝑝 ( 𝑎 )(1 − 𝑧 ). For any 𝑧 ∈ [0 , |− 𝑝 ( 𝑎 )(1 − 𝑧 ) | < | 𝑝 ( 𝑎 ) | = 𝛿. Therefore, as 𝑧 moves from 0 to 1, each root of 𝑝 ( 𝑦 ) − 𝑝 ( 𝑎 ) moves to the corre-sponding root of 𝑝 ( 𝑦 ), which belongs to the same ball. In particular, the root 𝑎 of˜ 𝑝 | 𝑧 =0 will move to the root 𝜉 of ˜ 𝑝 | 𝑧 =1 . Define 𝑃 ( 𝑧, 𝑦 ) = 𝑝 ((1 − 𝑧 ) 𝑦 ) − 𝑝 ( 𝑎 )(1 − 𝑧 ) ∈ F [ 𝑧, 𝑦 ] . We claim that 𝑃 ( 𝑧, 𝑦 ) determines an analytic algebraic function 𝑓 ( 𝑧 ) in F [[ 𝑧 ]]with the dominant singularity 1 and whose coefficient sequence converges to 𝜉 . Toprove this, we make an ansatz 𝑓 ( 𝑧 ) = ∞ ∑︁ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 , where the 𝑎 is from above and ( 𝑎 𝑛 ) ∞ 𝑛 =1 are to determined. Observe that for any 𝑐 ( 𝑧 ) ∈ F [ 𝑧 ], 𝑐 ( 𝑧 ) / (1 − 𝑧 ) is a root of 𝑃 ( 𝑧, 𝑦 ) if and only if 𝑐 ( 𝑧 ) is a root of ˜ 𝑝 ( 𝑦 ), so 𝑓 ( 𝑧 ) admits the following Laurent expansion at 𝑧 = 1, 𝑓 ( 𝑧 ) = 𝜉 − 𝑧 + ∞ ∑︁ 𝑛 =0 𝑏 𝑛 (1 − 𝑧 ) 𝑛 for 𝑏 𝑛 ∈ C . Hence 𝑧 = 1 is a singularity of 𝑓 ( 𝑧 ) as 𝜉 ̸ = 0.The above argument also implies that 𝑧 = 1 is the only dominant singularityof 𝑓 ( 𝑧 ). Indeed, note that 𝑧 = 1 is the only root of the leading coefficient of 𝑃 ( 𝑧, 𝑦 )w.r.t. 𝑦 , so the other singularities of 𝑓 ( 𝑧 ) could only be branch points, i.e., roots ofdiscriminant of 𝑃 ( 𝑧, 𝑦 ) w.r.t. 𝑦 . However, the choices of 𝜀 and 𝛿 make it impossiblefor 𝑓 ( 𝑧 ) to have branch points in the disk | 𝑧 | ≤
1, because in order to have a branchpoint, two roots of the polynomial 𝑃 ( 𝑧, 𝑦 ) w.r.t. 𝑦 would need to touch each other,and we have ensured that they are always separated by more than 𝜀 . Consequently, 𝑧 = 1 is the dominant singularity of 𝑓 ( 𝑧 ), which gives 𝑎 𝑛 ∼ 𝜉 as 𝑛 → ∞ by part 1of Theorem 8.3. Therefore lim 𝑛 →∞ 𝑎 𝑛 = 𝜉 since 𝜉 ̸ = 0.To complete the proof, it remains to show that the coefficients of 𝑓 ( 𝑧 ) are indeedin F . This is observed by plugging the ansatz of 𝑓 ( 𝑧 ) into 𝑃 ( 𝑧, 𝑦 ) and comparing thecoefficients of like powers of 𝑧 to zero. Since 𝑝 ( 𝑧 ) is irreducible and 𝜉 is arbitrary,one sees that 𝑃 ( 𝑧, 𝑦 ) admits 𝑑 distinct analytic solutions in F [[ 𝑧 ]] in a neighborhoodof 0.The following theorem clarifies the converse direction for algebraic sequences.It turns out that every element in 𝒜 F is algebraic over F . Theorem 8.5.
Let F be a subfield of C .1. If F ⊆ R , then 𝒜 F = ¯ F ∩ R .2. If F ∖ R ̸ = ∅ , then 𝒜 F = ¯ F . Chapter 8. D-finite Numbers
Proof.
1. Let 𝜉 ∈ ¯ F ∩ R . Then there is an irreducible polynomial 𝑝 ( 𝑧 ) ∈ F [ 𝑧 ] suchthat 𝑝 ( 𝜉 ) = 0. By part 1 of Lemma 8.4, 𝜉 is in fact a limit of an algebraicsequence in F N , which implies 𝜉 ∈ 𝒜 F .To show the converse inclusion, we let 𝜉 ∈ 𝒜 F . When 𝜉 = 0, there is nothingto show. Assume that 𝜉 ̸ = 0. Then there is an algebraic sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ F N such that lim 𝑛 →∞ 𝑎 𝑛 = 𝜉 . Since 𝜉 ̸ = 0, 𝑎 𝑛 ∼ 𝜉 ( 𝑛 → ∞ ).Let 𝑓 ( 𝑧 ) = ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 . Clearly 𝑓 ( 𝑧 ) is an algebraic function over F . By part 2of Theorem 8.3, 𝑓 ( 𝑧 ) ∼ 𝜉/ (1 − 𝑧 ) ( 𝑧 → − ), implying that 𝑧 = 1 is a simplepole of 𝑓 ( 𝑧 ) and 𝑓 ( 𝑧 ) = 𝜉 − 𝑧 + ∞ ∑︁ 𝑛 =0 𝑏 𝑛 (1 − 𝑧 ) 𝑛 for ( 𝑏 𝑛 ) ∞ 𝑛 =0 ∈ C N . Setting 𝑔 ( 𝑧 ) = 𝑓 ( 𝑧 )(1 − 𝑧 ) establishes that 𝑔 ( 𝑧 ) = 𝜉 + ∑︀ ∞ 𝑛 =0 𝑏 𝑛 (1 − 𝑧 ) 𝑛 +1 , andthen 𝑔 ( 𝑧 ) is analytic at 1. Sending 𝑧 to 1 gives 𝑔 (1) = 𝜉 . By closure properties, 𝑔 ( 𝑧 ) is again an algebraic function over F . Thus 𝜉 = 𝑔 (1) ∈ ¯ F ∩ R as F ⊆ R .2. By part 2 of Lemma 8.4 and a similar argument as above, we have 𝒜 F = ¯ F .If we were to consider the class 𝒞 F of limits of convergent sequences in F satisfyinglinear difference equations with constant coefficients over F , sometimes called C-finitesequences, then an argument analogous to the above proof would imply that 𝒞 F ⊆ F ,because the power series corresponding to such sequences are rational functions, andthe values of rational functions over F at points in F evidently gives values in F . Theconverse direction F ⊆ 𝒞 F is trivial, so 𝒞 F = F . Corollary 8.6. If F ⊆ R , then ¯ F = 𝒜 F ( 𝑖 ) = 𝒜 F [ 𝑖 ] = 𝒜 F + 𝑖 𝒜 F , where 𝑖 is theimaginary unit.Proof. Since 𝒜 F is a ring and 𝑖 = − ∈ F ⊆ 𝒜 F , we have 𝒜 F [ 𝑖 ] = 𝒜 F + 𝑖 𝒜 F .Since 𝑖 ∈ ¯ F and F ⊆ R , ¯ F is closed under complex conjugation and then¯ F = (¯ F ∩ R ) + 𝑖 (¯ F ∩ R ) = 𝒜 F + 𝑖 𝒜 F , by part 1 of Theorem 8.5. It follows from part 2 of Theorem 8.5 that 𝒜 F ( 𝑖 ) = F ( 𝑖 ).Since 𝒜 F ⊆ 𝒜 F ( 𝑖 ) and 𝑖 ∈ 𝒜 F ( 𝑖 ) , we have¯ F = 𝒜 F + 𝑖 𝒜 F ⊆ 𝒜 F ( 𝑖 ) = F ( 𝑖 ) = ¯ F . The assertion holds.The following lemma says that every element in ¯ F can be represented as thevalue at 1 of an analytic algebraic function vanishing at zero, provided that F isdense in C . This will be used in the next section to extend the evaluation domain. Lemma 8.7.
Let F be a subfield of C with F ∖ R ̸ = ∅ . Let 𝑝 ( 𝑧 ) ∈ F [ 𝑧 ] be anirreducible polynomial of degree 𝑑 . Assume that 𝜉 , . . . , 𝜉 𝑑 are all the (distinct) rootsof 𝑝 in ¯ F . Then there is a square-free polynomial 𝑃 ( 𝑧, 𝑦 ) ∈ F [ 𝑧, 𝑦 ] of degree 𝑑 in 𝑦 andadmitting 𝑑 distinct analytic algebraic functions 𝑔 ( 𝑧 ) , . . . , 𝑔 𝑑 ( 𝑧 ) with 𝑃 ( 𝑧, 𝑔 𝑗 ( 𝑧 )) = 0 in a neighborhood of such that all 𝑔 𝑗 ’s are analytic in the disk | 𝑧 | ≤ with 𝑔 𝑗 (0) = 0 and, after reordering (if necessary), 𝑔 𝑗 (1) = 𝜉 𝑗 . .3. Rings of D-finite numbers Proof.
By part 2 of Lemma 8.4, there exists a bivariate square-free polynomial˜ 𝑃 ( 𝑧, 𝑦 ) ∈ F [ 𝑧, 𝑦 ] of degree 𝑑 in 𝑦 and admitting 𝑑 distinct analytic algebraic func-tions 𝑓 ( 𝑧 ) , . . . , 𝑓 𝑑 ( 𝑧 ) with 𝑃 ( 𝑧, 𝑓 𝑗 ( 𝑧 )) = 0 in a neighborhood of 0 such that 1 is theonly dominant singularity of each 𝑓 𝑗 ( 𝑧 ) and, after reordering (if necessary),lim 𝑛 →∞ [ 𝑧 𝑛 ] 𝑓 𝑗 ( 𝑧 ) = 𝜉 𝑗 , 𝑗 = 1 , . . . 𝑑. If 𝜉 𝑗 = 0 for some 𝑗 then 𝑝 ( 𝑧 ) = 𝑧 . Letting 𝑃 ( 𝑧, 𝑦 ) = 𝑦 yields the assertion.Otherwise all roots 𝜉 , . . . , 𝜉 𝑑 are nonzero, and thus [ 𝑧 𝑛 ] 𝑓 𝑗 ( 𝑧 ) ∼ 𝜉 𝑗 ( 𝑛 → ∞ ) foreach 𝑗 . By part 2 of Theorem 8.3, 𝑓 𝑗 ( 𝑧 ) ∼ 𝜉 𝑗 − 𝑧 ( 𝑧 → − ) , which implies that 𝑧 = 1 is a simple pole of each 𝑓 𝑗 . Let 𝑔 𝑗 ( 𝑧 ) = 𝑓 𝑗 ( 𝑧 ) 𝑧 (1 − 𝑧 ). Then 𝑔 ( 𝑧 ) , . . . , 𝑔 𝑑 ( 𝑧 ) are distinct and each 𝑔 𝑗 ( 𝑧 ) ∈ F [[ 𝑧 ]] is analytic for any 𝑧 in the disk | 𝑧 | ≤
1. Moreover, 𝑔 𝑗 (0) = 0 and 𝑔 𝑗 (1) = 𝜉 𝑗 . By closure properties, 𝑔 𝑗 ( 𝑧 ) is againalgebraic over F . Define a square-free polynomial 𝑃 ( 𝑧, 𝑦 ) = 𝑑 ∏︁ 𝑗 =1 ( 𝑦 − 𝑔 𝑗 ( 𝑧 )) = 𝑑 ∏︁ 𝑗 =1 (︀ 𝑦 − 𝑓 𝑗 ( 𝑧 ) 𝑧 (1 − 𝑧 ) )︀ ∈ F ( 𝑧 )[ 𝑦 ] . Then 𝑃 ∈ F [ 𝑧, 𝑦 ] since 𝑃 is symmetric in 𝑓 , . . . , 𝑓 𝑑 . The lemma follows. Let us now return to the study of D-finite numbers. Let 𝑅 be a subring of C and F be a subfield of C . Recall that by Definition 8.1, the elements of 𝒟 𝑅, F are exactlylimits of convergent sequences in 𝑅 N which are P-recursive over F . Some facts aboutP-recursive sequences translate directly into facts about 𝒟 𝑅, F . Proposition 8.8. 𝑅 ⊆ 𝒟 𝑅, F and 𝒜 F ⊆ 𝒟 F .2. If 𝑅 ⊆ 𝑅 then 𝒟 𝑅 , F ⊆ 𝒟 𝑅 , F , and if F ⊆ E then 𝒟 𝑅, F ⊆ 𝒟 𝑅, E .3. 𝒟 𝑅, F is a subring of C . Moreover, if 𝑅 is an F -algebra then so is 𝒟 𝑅, F .4. If E is an algebraic extension field of F , then 𝒟 𝑅, F = 𝒟 𝑅, E .5. If 𝑅 ⊆ F , then 𝒟 𝑅, F = 𝒟 𝑅, Quot( 𝑅 ) .6. If 𝑅 and F are closed under complex conjugation, then so is 𝒟 𝑅, F .In this case, we have 𝒟 𝑅, F ∩ R = 𝒟 𝑅 ∩ R , F .Moreover, if the imaginary unit 𝑖 ∈ 𝒟 𝑅, F then 𝒟 𝑅, F = 𝒟 𝑅 ∩ R , F + 𝑖 𝒟 𝑅 ∩ R , F .Proof.
1. The first inclusion is clear because every element of 𝑅 is the limit ofa constant sequence, and every constant sequence is P-recursive. The secondinclusion follows from the fact that algebraic functions are D-finite, and thecoefficient sequences of D-finite functions are P-recursive.6 Chapter 8. D-finite Numbers
2. Clear.3. Follows directly from the corresponding closure properties for P-recursive se-quences.4. Follows directly from part 1 of Lemma 7.3.5. Follows directly from part 2 of Lemma 7.3.6. For any convergent sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ 𝑅 N , we haveRe (︁ lim 𝑛 →∞ 𝑎 𝑛 )︁ = lim 𝑛 →∞ Re( 𝑎 𝑛 ) , Im (︁ lim 𝑛 →∞ 𝑎 𝑛 )︁ = lim 𝑛 →∞ Im( 𝑎 𝑛 ) , and thus lim 𝑛 →∞ 𝑎 𝑛 = lim 𝑛 →∞ ¯ 𝑎 𝑛 . Hence the first assertion follows by (¯ 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ 𝑅 N and part 3 of Lemma 7.3.Since 𝑅 is closed under complex conjugation, (Re( 𝑎 𝑛 )) ∞ 𝑛 =0 ∈ ( 𝑅 ∩ R ) N . Thenthe inclusion 𝒟 𝑅, F ∩ R ⊆ 𝒟 𝑅 ∩ R , F can be shown similarly as the first assertion.The converse direction holds by part 2. Thus 𝒟 𝑅, F ∩ R = 𝒟 𝑅 ∩ R , F .If 𝑖 ∈ 𝒟 𝑅, F , then 𝒟 𝑅 ∩ R , F + 𝑖 𝒟 𝑅 ∩ R , F ⊆ 𝒟 𝑅, F since 𝒟 𝑅 ∩ R , F ⊆ 𝒟 𝑅, F . To showthe converse inclusion, let 𝜉 ∈ 𝒟 𝑅, F . Then 𝜉 ∈ 𝒟 𝑅, F by the first assertion.Since 𝑖 ∈ 𝒟 𝑅, F and 𝑅 is closed under complex conjugation, Re( 𝜉 ) , Im( 𝜉 ) bothbelong to 𝒟 𝑅, F ∩ R = 𝒟 𝑅 ∩ R , F by the second assertion. Therefore we have 𝜉 = Re( 𝜉 ) + 𝑖 Im( 𝜉 ) ∈ 𝒟 𝑅 ∩ R , F + 𝑖 𝒟 𝑅 ∩ R , F . Example 8.9.
1. We have 𝒟 Q ( √ , Q ( 𝜋, √ = 𝒟 Q ( √ , Q ( √ = 𝒟 Q ( √ , Q . The first identity holds bypart 5, the second by part 4 of the proposition.2. We have 𝒟 ¯ Q , Q = 𝒟 ¯ Q , R . The inclusion “ ⊆ ” is clear by part 2. For the inclu-sion “ ⊇ ”, let 𝜉 ∈ 𝒟 ¯ Q , R . Then 𝜉 = 𝑎 + 𝑖𝑏 for some 𝑎, 𝑏 ∈ R , and there exists asequence ( 𝑎 𝑛 + 𝑖𝑏 𝑛 ) ∞ 𝑛 =0 in ¯ Q N and a nonzero operator 𝐿 ∈ R [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ such that 𝐿 · ( 𝑎 𝑛 + 𝑖𝑏 𝑛 ) = 0 and lim 𝑛 →∞ ( 𝑎 𝑛 + 𝑖𝑏 𝑛 ) = 𝑎 + 𝑖𝑏 . Since the coefficients of 𝐿 are real, we then have 𝐿 · 𝑎 𝑛 = 0 and 𝐿 · 𝑏 𝑛 = 0. Furthermore, we see thatlim 𝑛 →∞ 𝑎 𝑛 = 𝑎 and lim 𝑛 →∞ 𝑏 𝑛 = 𝑏 . Therefore, 𝑎, 𝑏 ∈ 𝒟 ¯ Q ∩ R , R part 5 = 𝒟 ¯ Q ∩ R , ¯ Q ∩ R part 4 = 𝒟 ¯ Q ∩ R , Q , which implies 𝑎 + 𝑖𝑏 ∈ 𝒟 ¯ Q ∩ R , Q + 𝑖 𝒟 ¯ Q ∩ R , Q part 6 = 𝒟 ¯ Q , Q , as claimed.Lemma 8.7 motivates the following theorem, which says that every D-finite num-ber is essentially the value at 1 of an analytic D-finite function, and thus a holonomicconstant. Theorem 8.10.
Let 𝑅 be a subring of C and F be a subfield of C . Then for any 𝜉 ∈𝒟 𝑅, F , there exists 𝑔 ( 𝑧 ) ∈ 𝑅 [[ 𝑧 ]] D-finite over F and analytic at such that 𝜉 = 𝑔 (1) . .3. Rings of D-finite numbers Proof.
The statement is clear when 𝜉 = 0. Assume that 𝜉 is nonzero. Then thereexists a sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 ∈ 𝑅 N , P-recursive over F , such that lim 𝑛 →∞ 𝑎 𝑛 = 𝜉 . Since 𝜉 is nonzero, we have 𝑎 𝑛 ∼ 𝜉 ( 𝑛 → ∞ ). Let 𝑓 ( 𝑧 ) = ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 . Then by Theorem 8.3,we see that 𝑓 ( 𝑧 ) ∼ 𝜉 − 𝑧 ( 𝑧 → − ) , which implies that 𝑧 = 1 is a simple pole of 𝑓 ( 𝑧 ). Let 𝑔 ( 𝑧 ) = 𝑓 ( 𝑧 )(1 − 𝑧 ). Then 𝑔 ( 𝑧 )belongs to 𝑅 [[ 𝑧 ]] and is analytic at 𝑧 = 1. Write 𝑓 ( 𝑧 ) = 𝜉 − 𝑧 + ∞ ∑︁ 𝑛 =0 𝑏 𝑛 (1 − 𝑧 ) 𝑛 with 𝑏 𝑛 ∈ C . It follows that 𝑔 ( 𝑧 ) = 𝑓 ( 𝑧 )(1 − 𝑧 ) = 𝜉 + ∑︀ ∞ 𝑛 =0 𝑏 𝑛 (1 − 𝑧 ) 𝑛 +1 , which gives 𝜉 = 𝑔 (1). Theassertion follows by noticing that 𝑔 ( 𝑧 ) is D-finite over F due to closure properties. Example 8.11.
We have 𝜁 (3) = ∑︀ ∞ 𝑛 =1 1 𝑛 = Li (1), where Li ( 𝑧 ) = ∑︀ ∞ 𝑛 =1 1 𝑛 𝑧 𝑛 isthe polylogarithm function, D-finite over Q and analytic at 1.Note that the above theorem implies that D-finite numbers are computable whenthe ring 𝑅 and the field F consist of computable numbers. This allows the construc-tion of (artificial) numbers that are not D-finite.Some kind of converse of Theorem 8.10 can be proved for the case when F is nota subfield of R , namely F ∖ R ̸ = ∅ . Note that this condition is equivalent to sayingthat F is dense in C . To this end, we first need to develop several lemmas.The following lemma says that the value of a D-finite function at any non-singularpoint in ¯ F can be represented by the value at 1 of another D-finite function. Lemma 8.12.
Let F be a subfield of C with F ∖ R ̸ = ∅ and 𝑅 be a subring of C containing F . Assume that 𝑓 ( 𝑧 ) ∈ 𝒟 𝑅, F [[ 𝑧 ]] is analytic and annihilated by a nonzerooperator 𝐿 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ with zero an ordinary point. Then for any non-singular point 𝜁 ∈ ¯ F of 𝐿 , there exists an analytic function ℎ ( 𝑧 ) ∈ 𝒟 𝑅, F [[ 𝑧 ]] and a nonzero operator 𝑀 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ with and ordinary points such that 𝑀 · ℎ ( 𝑧 ) = 0 and 𝑓 ( 𝜁 ) = ℎ (1) .Proof. Let 𝜁 ∈ ¯ F be a non-singular point of 𝐿 . Then there exists an irreduciblepolynomial 𝑝 ( 𝑧 ) ∈ F [ 𝑧 ] such that 𝑝 ( 𝜁 ) = 0. Let 𝜁 = 𝜁, . . . , 𝜁 𝑑 be all the roots of 𝑝 in ¯ F . By Lemma 8.7, there exists a square-free polynomial 𝑃 ( 𝑧, 𝑦 ) ∈ F [ 𝑧, 𝑦 ] of degree 𝑑 in 𝑦 and admitting 𝑑 distinct analytic algebraic functions 𝑔 ( 𝑧 ) , . . . , 𝑔 𝑑 ( 𝑧 ) with 𝑃 ( 𝑧, 𝑔 𝑗 ( 𝑧 )) = 0 in a neighborhood of 0. Moreover, 𝑔 ( 𝑧 ) , . . . , 𝑔 𝑑 ( 𝑧 ) are all analytic inthe disk | 𝑧 | ≤ 𝑔 𝑗 (1) = 𝜁 𝑗 and 𝑔 𝑗 (0) = 0.Since 𝑔 (1) = 𝜁 is not a singularity of 𝐿 by assumption, none of 𝑔 𝑗 (1) = 𝜁 𝑗 is asingularity of 𝐿 . Suppose otherwise that for some 2 ≤ ℓ ≤ 𝑑 , the point 𝑔 ℓ (1) = 𝜁 ℓ isa root of lc( 𝐿 ). Since lc( 𝐿 ) ∈ F [ 𝑧 ] and 𝑝 is the minimal polynomial of 𝜁 ℓ over F , weknow that 𝑝 divides lc( 𝐿 ) over F . Thus 𝜁 is also a root of lc( 𝐿 ), a contradiction.Note that 𝑔 ( 𝑧 ) , . . . , 𝑔 𝑑 ( 𝑧 ) are analytic in the disk | 𝑧 | ≤ 𝑔 𝑗 (0) = 0. ByTheorem 7.2, there exists a nonzero operator 𝑀 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ with 𝑀 · ( 𝑓 ∘ 𝑔 ) = 0which does not have 0 or 1 among its singularities. By part 1 of Proposition 8.8, F ⊆ 𝑅 ⊆ 𝒟 𝑅, F . Since 𝑓 ( 𝑧 ) ∈ 𝒟 𝑅, F [[ 𝑧 ]] and 𝑔 ( 𝑧 ) ∈ F [[ 𝑧 ]] with 𝑔 (0) = 0, wehave 𝑓 ( 𝑔 ( 𝑧 )) ∈ 𝒟 𝑅, F [[ 𝑧 ]]. Setting ℎ ( 𝑧 ) = 𝑓 ( 𝑔 ( 𝑧 )) completes the proof.8 Chapter 8. D-finite Numbers
With the above lemma, it suffices to consider the case when the evaluation pointis in 𝑅 ∩ F . This is exactly what the next two lemmas are concerned about. Lemma 8.13.
Assume that 𝑓 ( 𝑧 ) = ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 ∈ 𝑅 [[ 𝑧 ]] is D-finite over F andconvergent in some neighborhood of . Let 𝜁 ∈ 𝑅 ∩ F be in the disk of convergence.Then 𝑓 ( 𝑘 ) ( 𝜁 ) ∈ 𝒟 𝑅, F for all 𝑘 ∈ N .Proof. For 𝑘 ∈ N , it is well-known that 𝑓 ( 𝑘 ) ( 𝑧 ) ∈ 𝑅 [[ 𝑧 ]] is also D-finite and has thesame radius of convergence at zero as 𝑓 ( 𝑧 ). Note that since 𝑓 ( 𝑧 ) is D-finite over F ,so is 𝑓 ( 𝑘 ) ( 𝑧 ). Thus to prove the lemma, it suffices to show the case when 𝑘 = 0, i.e., 𝑓 ( 𝜁 ) ∈ 𝒟 𝑅, F .Since 𝑓 ( 𝑧 ) is D-finite over F , the coefficient sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 is P-recursiveover F . Note that 𝜁 ∈ 𝑅 ∩ F is in the disk of convergence of 𝑓 ( 𝑧 ) at zero, so 𝑓 ( 𝜁 ) = ∞ ∑︁ 𝑛 =0 𝑎 𝑛 𝜁 𝑛 = lim 𝑛 →∞ 𝑛 ∑︁ ℓ =0 𝑎 ℓ 𝜁 ℓ . Since ( 𝜁 𝑛 ) ∞ 𝑛 =0 is P-recursive over F , the assertion follows by noticing that the se-quence ( ∑︀ 𝑛ℓ =0 𝑎 ℓ 𝜁 ℓ ) ∞ 𝑛 =0 ∈ 𝑅 N is P-recursive over F due to closure properties. Example 8.14.
Since exp( 𝑧 ) = ∑︀ ∞ 𝑛 =0 1 𝑛 ! 𝑧 𝑛 ∈ Q [[ 𝑧 ]] is D-finite over Q , and convergeseverywhere, we get from the lemma that the numbers e , / e , √ e belong to 𝒟 Q , Q . Moreprecisely, since we are currently only considering non-real fields F , we could say thatthe function exp( 𝑧 ) is D-finite over ¯ Q , therefore e , / e , √ e all belong to 𝒟 Q , ¯ Q , butby Proposition 8.8, 𝒟 Q , ¯ Q = 𝒟 Q , Q . Lemma 8.15.
Let 𝑅 be a subring of C containing F . Let 𝑓 ( 𝑧 ) = ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 in 𝒟 𝑅, F [[ 𝑧 ]] be an analytic function. Assume that there exists a nonzero operator 𝐿 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ with zero an ordinary point such that 𝐿 · 𝑓 ( 𝑧 ) = 0 . Let 𝑟 > be thesmallest modulus of roots of lc( 𝐿 ) and let 𝜁 ∈ F with | 𝜁 | < 𝑟 . Then 𝑓 ( 𝑘 ) ( 𝜁 ) ∈ 𝒟 𝑅, F for all 𝑘 ∈ N .Proof. Let 𝜌 be the order of 𝐿 . Since zero is an ordinary point of 𝐿 , there existP-recursive sequences ( 𝑐 (0) 𝑛 ) ∞ 𝑛 =0 , . . . , ( 𝑐 ( 𝜌 − 𝑛 ) ∞ 𝑛 =0 in F N ⊆ 𝑅 N with 𝑐 ( 𝑚 ) 𝑗 equal to theKronecker delta 𝛿 𝑚𝑗 for 𝑚, 𝑗 = 0 , . . . , 𝜌 −
1, so that the set { ∑︀ ∞ 𝑛 =0 𝑐 ( 𝑚 ) 𝑛 𝑧 𝑛 } 𝜌 − 𝑚 =0 formsa basis of the solution space of 𝐿 near zero. Note that the singularities of solutionsof 𝐿 can only be roots of lc( 𝐿 ). Hence the power series 𝑓 ( 𝑧 ) = ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 as wellas ∑︀ ∞ 𝑛 =0 𝑐 ( 𝑚 ) 𝑛 𝑧 𝑛 for 𝑚 = 0 , . . . , 𝜌 − | 𝑧 | < 𝑟 . It followsfrom | 𝜁 | < 𝑟 and Lemma 8.13 that the set { ∑︀ ∞ 𝑛 =0 𝑐 ( 𝑚 ) 𝑛 𝜁 𝑛 } 𝜌 − 𝑚 =0 belongs to 𝒟 𝑅, F . Since 𝑎 , . . . , 𝑎 𝜌 − ∈ 𝒟 𝑅, F , 𝑓 ( 𝜁 ) = ∞ ∑︁ 𝑛 =0 𝑎 𝑛 𝜁 𝑛 = 𝑎 ∞ ∑︁ 𝑛 =0 𝑐 (0) 𝑛 𝜁 𝑛 + · · · + 𝑎 𝜌 − ∞ ∑︁ 𝑛 =0 𝑐 ( 𝜌 − 𝑛 𝜁 𝑛 is D-finite by closure properties. In the same vein, we find that for 𝑘 >
0, thederivative 𝑓 ( 𝑘 ) ( 𝜁 ) also belongs to 𝒟 𝑅, F . .3. Rings of D-finite numbers Example 8.16.
1. We know from Proposition 8.8 that √ ∈ 𝒟 Q . The series( 𝑧 + 1) √ = 1 + √ 𝑧 + (1 − √ 𝑧 + · · · ∈ Q ( √ 𝑧 ]] ⊆ 𝒟 Q [[ 𝑧 ]]is D-finite over Q , an annihilating operator is ( 𝑧 + 1) 𝐷 𝑧 + ( 𝑧 + 1) 𝐷 𝑧 −
2. Herewe have the radius 𝑟 = 1. Taking 𝜁 = √ −
1, the lemma implies that √ √ belongs to 𝒟 Q .2. Observe that the lemma refers to the singularities of the operator rather thanto the singularities of the particular solution at hand. For example, it does notimply that 𝐽 (1) ∈ 𝒟 Q , Q , where 𝐽 ( 𝑧 ) is the first Bessel function, because itsannihilating operator is 𝑧 𝐷 𝑧 + 𝑧𝐷 𝑧 + ( 𝑧 − 𝐽 ( 𝑧 ) ∈ Q [[ 𝑧 ]] is analytic at 0.Of course, in this particular example we see from the series representation 𝐽 (1) = ∑︀ ∞ 𝑛 =0 ( − / 𝑛 ( 𝑛 +1) 𝑛 ! that the value belongs to 𝒟 Q , Q .3. The hypergeometric function 𝑓 ( 𝑧 ) := 𝐹 ( , , , 𝑧 + ) can be viewed as anelement of 𝒟 Q , Q [[ 𝑧 ]]: 𝑓 ( 𝑧 ) = √ ∞ ∑︁ 𝑛 =0 (1 / 𝑛 (1 / 𝑛 𝑛 ! ( − 𝑛 ⏟ ⏞ ∈𝒟 Q + √ ∞ ∑︁ 𝑛 =0 (1 / 𝑛 (4 / 𝑛 (2) 𝑛 𝑛 ! ( − 𝑛 ⏟ ⏞ ∈𝒟 Q 𝑧 + 2 √ ∞ ∑︁ 𝑛 =0 (1 / 𝑛 (7 / 𝑛 (3) 𝑛 𝑛 ! ( − 𝑛 ⏟ ⏞ ∈𝒟 Q 𝑧 + · · · . The function 𝑓 is annihilated by the operator 𝐿 = 3(2 𝑧 − 𝑧 + 1) 𝐷 𝑧 + (22 𝑧 − 𝐷 𝑧 + 2 . This operator has a singularity at 𝑧 = 1 /
2, and there is no annihilating oper-ator of 𝑓 which does not have a singularity there. Although 𝑓 (1 /
2) = Γ(1 / / / Theorem 8.17.
Let F be a subfield of C with F ∖ R ̸ = ∅ and let 𝑅 be a subring of C containing F . Assume that 𝑓 ( 𝑧 ) ∈ 𝒟 𝑅, F [[ 𝑧 ]] is analytic and there exists a nonzerooperator 𝐿 ∈ F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ with zero an ordinary point such that 𝐿 · 𝑓 ( 𝑧 ) = 0 . Furtherassume that 𝜁 ∈ ¯ F is not a singularity of 𝐿 . Then 𝑓 ( 𝑘 ) ( 𝜁 ) belongs to 𝒟 𝑅, F for all 𝑘 ∈ N . Chapter 8. D-finite Numbers 𝛽 = 0 𝛽 𝛽 𝛽 𝛽 𝑠 − 𝛽 𝑠 − 𝛽 𝑠 = 𝜁 𝒫 𝑟 𝑟 𝑟 𝑟 𝑠 − 𝑟 𝑠 Figure 8.1: a simple path 𝒫 with a finite cover ⋃︀ 𝑠𝑗 =0 ℬ 𝑟 𝑗 ( 𝛽 𝑗 ) ( stands for the rootsof lc( 𝐿 )) Proof.
By Lemma 8.12, it suffices to show the assertion holds for 𝜁 = 1 (or moregenerally 𝜁 ∈ F ). Now assume that 𝜁 ∈ F . We apply the method of analytic contin-uation.Let 𝒫 be a simple path with a finite cover ⋃︀ 𝑠𝑗 =0 ℬ 𝑟 𝑗 ( 𝛽 𝑗 ), where 𝑠 ∈ N , 𝛽 = 0, 𝛽 𝑠 = 𝜁 , 𝛽 𝑗 ∈ F , 𝑟 𝑗 > 𝛽 𝑗 and the zero set of lc( 𝐿 ), and ℬ 𝑟 𝑗 ( 𝛽 𝑗 )is the open circle centered at 𝛽 𝑗 and with radius 𝑟 𝑗 . Moreover, 𝛽 𝑗 +1 is inside ℬ 𝑟 𝑗 ( 𝛽 𝑗 )for each 𝑗 (as illustrated by Figure 8.1). Such a path exists because F is dense in C and the zero set of lc( 𝐿 ) is finite. Since the path 𝒫 avoids all roots of lc( 𝐿 ), thefunction 𝑓 ( 𝑧 ) is analytic along 𝒫 . We next use induction on the index 𝑗 to showthat 𝑓 ( 𝑘 ) ( 𝛽 𝑗 ) ∈ 𝒟 𝑅, F for all 𝑘 ∈ N .It is trivial when 𝑗 = 0 as 𝑓 ( 𝑘 ) ( 𝛽 ) = 𝑓 ( 𝑘 ) (0) ∈ 𝒟 𝑅, F for 𝑘 ∈ N by assumption.Assume now that 0 < 𝑗 ≤ 𝑠 and 𝑓 ( 𝑘 ) ( 𝛽 𝑗 − ) ∈ 𝒟 𝑅, F for all 𝑘 ∈ N . We consider 𝑓 ( 𝛽 𝑗 )and its derivatives.Recall that 𝑟 𝑗 − > 𝛽 𝑗 − and the zero set of lc( 𝐿 ).Since 𝑓 ( 𝑧 ) is analytic at 𝛽 𝑗 − , it is representable by a convergent power series ex-pansion 𝑓 ( 𝑧 ) = ∞ ∑︁ 𝑛 =0 𝑓 ( 𝑛 ) ( 𝛽 𝑗 − ) 𝑛 ! ( 𝑧 − 𝛽 𝑗 − ) 𝑛 for all | 𝑧 − 𝛽 𝑗 − | < 𝑟 𝑗 − . By the induction hypothesis, 𝑓 ( 𝑛 ) ( 𝛽 𝑗 − ) /𝑛 ! ∈ 𝒟 𝑅, F for all 𝑛 ∈ N and thus 𝑓 ( 𝑧 )belongs to 𝒟 𝑅, F [[ 𝑧 − 𝛽 𝑗 − ]].Let 𝑍 = 𝑧 − 𝛽 𝑗 − , i.e., 𝑧 = 𝑍 + 𝛽 𝑗 − . Define 𝑔 ( 𝑍 ) = 𝑓 ( 𝑍 + 𝛽 𝑗 − ) and ˜ 𝐿 tobe the operator obtained by replacing 𝑧 in 𝐿 by 𝑍 + 𝛽 𝑗 . Since 𝛽 𝑗 − ∈ F ⊆ 𝒟 𝑅, F and 𝐷 𝑧 = 𝐷 𝑍 , we have 𝑔 ( 𝑍 ) ∈ 𝒟 𝑅, F [[ 𝑍 ]] and ˜ 𝐿 ∈ F [ 𝑍 ] ⟨ 𝐷 𝑍 ⟩ . Note that 𝐿 · 𝑓 ( 𝑧 ) = 0and 𝛽 𝑗 − is an ordinary point of 𝐿 as 𝑟 𝑗 − >
0. It follows that ˜ 𝐿 · 𝑔 ( 𝑍 ) = 0 and zerois an ordinary point of ˜ 𝐿 . Moreover, we see that 𝑟 𝑗 − is now the smallest modulusof roots of lc( ˜ 𝐿 ). Since | 𝛽 𝑗 − 𝛽 𝑗 − | < 𝑟 𝑗 − , applying Lemma 8.15 to 𝑔 ( 𝑍 ) yields 𝑓 ( 𝑘 ) ( 𝛽 𝑗 ) = 𝑔 ( 𝑘 ) ( 𝛽 𝑗 − 𝛽 𝑗 − ) ∈ 𝒟 𝑅, F for 𝑘 ∈ N . .4. Open questions 𝑗 = 𝑠 . The theorem follows. Example 8.18.
By the above theorem, exp( √
2) and log(1+ √
3) both belong to 𝒟 Q .We also have e 𝜋 ∈ 𝒟 Q . This is because e 𝜋 = ( − − 𝑖 with 𝑖 the imaginary unit, isequal to the value of the D-finite function ( 𝑧 +1) − 𝑖 ∈ Q ( 𝑖 )[[ 𝑧 ]] at 𝑧 = − 𝜋 ∈ 𝒟 Q ( 𝑖 ) ∩ R = 𝒟 Q . Furthermore, as remarked in theintroduction, the numbers obtained by evaluating a G-function at algebraic numberswhich avoid the singularities of its annihilating operator are in 𝒟 Q ( 𝑖 ) , because G-functions are D-finite. We have introduced the notion of D-finite numbers and made some first steps towardsunderstanding their nature. We believe that, similarly as for D-finite functions, theclass is interesting because it has good mathematical and computational propertiesand because it contains many special numbers that are of independent interest. Weconclude this chapter with some possible directions of future research.
Evaluation at singularities.
While every singularity of a D-finite function mustalso be a singularity of its annihilating operator, the converse is in general nottrue. We have seen above that evaluating a D-finite function at a point which isnot a singularity of its annihilating operator yields a D-finite number. It wouldbe natural to wonder about the values of a D-finite function at singularities of itsannihilating operator, including those at which the given function is not analyticbut its evaluation is finite. Also, we always consider zero as an ordinary point of theannihilating operator. If this is not the case, the method used in this chapter fails,as pointed out by part 2 of Example 8.16.
Quotients of D-finite numbers.
The set of algebraic numbers forms a field, butwe do not have a similar result for D-finite numbers. It is known that the set ofD-finite functions does not form a field. Instead, Harris and Sibuya [37] showed thata D-finite function 𝑓 admits a D-finite multiplicative inverse if and only if 𝑓 ′ /𝑓 isalgebraic. This explains for example why both e and 1 / e are D-finite, but it doesnot explain why both 𝜋 and 1 /𝜋 are D-finite. It would be interesting to know moreprecisely under which circumstances the multiplicative inverse of a D-finite number isD-finite. Is 1 / log(2) a D-finite number? Are there choices of 𝑅 and F for which 𝒟 𝑅, F is a field? Roots of D-finite functions.
A similar pending analogy concerns compositionalinverses. We know that if 𝑓 is an algebraic function, then so is its compositionalinverse 𝑓 − . The analogous statement for D-finite functions is not true. Nevertheless,it could still be true that the values of compositional inverses of D-finite functions areD-finite numbers, although this seems somewhat unlikely. A particularly interestingspecial case is the question whether (or under which circumstances) the roots of aD-finite function are D-finite numbers.2 Chapter 8. D-finite Numbers
Evaluation at D-finite number arguments.
We see that the class 𝒞 F of limits ofconvergent C-finite sequences is the same as the values of rational functions at pointsin F , namely the field F . Similarly, the class 𝒜 F of limits of convergent algebraicsequences essentially consists of the values of algebraic functions at points in ¯ F .Continuing this pattern, is the value of a D-finite function at a D-finite numberagain a D-finite number? If so, this would imply that also numbers like e e ee areD-finite. Since 1 / (1 − 𝑧 ) is a D-finite function, it would also imply that D-finitenumbers form a field. ppendices ppendix A The
ShiftReductionCT
Package
In order to be able to experiment with the algorithms proposed in the first partof this thesis, we have implemented all of them and encapsulated the proceduresas a Maple package, namely the
ShiftReductionCT package. This package wasdeveloped for
Maple 18 and it is available upon request from the author. Here is adescription of the package. > eval(ShiftReductionCT); module ( ) option package; export ReductionCT , BoundReductionCT , ModifiedAbramovPetkovsekReduction , ShiftMAPReduction , IsSummable , ShellReduction , KernelReductionCT , PolynomialReduction , TranslateDRF , VerifyMAPReduction , VerifyRCT ; description "Creative Telescoping for Bivariate Hypergeometric Terms viathe Modified Abramov-Petkovsek Reduction"; end module This appendix is intended to give a detailed instruction for the package. All exportcommands will be discussed in the order of their first appearance in the thesis,but only some of them will be emphasized particularly. By applying them to someconcrete examples, we show the usage of the package as well as its applications.These examples are chosen to take virtually no computation time.The appendix contains a whole Maple session. The inputs are given exactly inthe way how the commands need to be used in Maple and displayed in the typeof Maple notation, while the outputs are displayed in 2-D math notation. To startwith, we load the package in Maple. > read(ShiftReductionCT): > with(ShiftReductionCT): Appendix A. The
ShiftReductionCT
Package
Commands related to Chapter 3
We first consider univariate hypergeometric terms. Let 𝑇 be the hypergeometricterm in Example 3.7 (or Example 3.19). > T:=k^2*k!/(k+1); 𝑇 := 𝑘 𝑘 ! 𝑘 + 1By commands from the built-in Maple package SumTools[Hypergeometric] , wefind a kernel 𝐾 = 𝑘 + 1 and its corresponding shell 𝑆 = 𝑘 / ( 𝑘 + 1) of 𝑇 .The command ShellReduction performs Algorithm 3.5 and returns a decomposi-tion of the form (3.3) for the shell 𝑆 with respect to its kernel 𝐾 . > res:=ShellReduction(numer(K),denom(K),numer(S),denom(S),k); 𝑟𝑒𝑠 := [︂[︂ − 𝑘 + 1 ]︂ , − , 𝑘 + 2 , 𝑘 ]︂ Using the notations in (3.3), we check the correctness by > S1:=add(res[1][i],i=1..nops(res[1])): > a:=res[2]: b:=res[3]: p:=res[4]: > normal(K*subs(y=y+1,S1)-S1+(a/b+p/denom(K))-S); PolynomialReduction , namely Algorithm 3.16, projects a polyno-mial onto the image space of the map for polynomial reduction with respect to ashift-reduced rational function, and the standard complement of the image space. > res:=PolynomialReduction(p,numer(K),denom(K),k); 𝑟𝑒𝑠 := [1] , > f:=add(res[1][i],i=1..nops(res[1])): > q:=res[2]: normal(numer(K)*subs(k=k+1,f)-denom(K)*f+q-p); SumDecomposition , which is in the package
Sum-Tools[Hypergeometric] , is implemented based on the Abramov-Petkovšek reduc-tion. It computes a minimal additive decomposition described in Section 3.1 for agiven hypergeometric term. > SumTools[Hypergeometric][SumDecomposition](T,k); ⎡⎢⎢⎢⎢⎢⎣ 𝑘 𝑘 − ∏︁ _ 𝑘 =1 (_ 𝑘 + 1) 𝑘 + 1 , − 𝑘 − ∏︁ _ 𝑘 =1 (_ 𝑘 + 1) 𝑘 + 2 ⎤⎥⎥⎥⎥⎥⎦ ppendix A. The ShiftReductionCT
Package
ModifiedAbramovPetkovsekReduction . This command canbe used in the following (default) way. > res:=ModifiedAbramovPetkovsekReduction(T,k); 𝑟𝑒𝑠 := [︂[︂ 𝑘𝑘 + 1 , − 𝑘 + 2 ]︂ , 𝑘 ! ]︂ Using the notations in Algorithm 3.17, we have > f:=res[1][1]: r:=res[1][2]: H:=res[2]: The package also provides the command
VerifyMAPReduction to verify the reduction.This command is used according to the presented form of the result. In the defaultcase, we say > VerifyMAPReduction(res,T,k); true
Moreover, we can change the outputs of
ModifiedAbramovPetkovsekReduction byspecifying the third argument. For example, we would like to display the resultin terms of hypergeometric terms, > res:=ModifiedAbramovPetkovsekReduction(T,k,output= > hypergeometric); > VerifyMAPReduction(res,T,k,output=hypergeometric); 𝑟𝑒𝑠 := [︂ 𝑘𝑘 ! 𝑘 + 1 , − 𝑘 ! 𝑘 + 2 ]︂ true or we can also perform it as a list of functions, which specifies the standard form ofthe residual forms. > res:=ModifiedAbramovPetkovsekReduction(T,k,output=list); > VerifyMAPReduction(res,T,k,output=list); 𝑟𝑒𝑠 := [︂[︂[︂ − 𝑘 + 1 , , ]︂ , [ − , 𝑘 + 2 , ]︂ , 𝑘 ! ]︂ true As mentioned in Section 3.3, we also implement a procedure based on the modi-fied Abramov-Petkovšek reduction, which is only used to determine hypergeometricsummability and performs in a similar way as Gosper’s algorithm, namely the com-mand
IsSummable . > IsSummable(T,k); false
The built-in Maple command for Gosper’s algorithm is
Gosper in the package
SumTools[Hypergeometric] .8 Appendix A. The
ShiftReductionCT
Package > SumTools[Hypergeometric][Gosper](T,k);
Error, (in SumTools:-Hypergeometric:-Gosper) no solution found
Commands related to Chapter 4
In Chapter 4, we showed that the sum of two residual forms is congruent toa residual form (see Theorem 4.19), which plays an important role in developingthe reduction-based creative telescoping algorithm for hypergeometric terms (i.e.,Algorithm 5.6).To prove Theorem 4.19, we introduced two congruences in Lemma 4.15. Thesetwo congruences stand for two types of kernel reduction in the shift case, that is,denominator type and numerator type, respectively. We implemented them by thecommand
KernelReduction . To call it in Maple, using the notations from Lemma 4.15,one just says > KernelReduction(p1,numer(K),denom(K),m,k,type=denominator); or > KernelReduction(p2,numer(K),denom(K),m,k,type=numerator);
The key idea of Algorithm 4.20 is to move the significant denominator of aresidual form to a required form according to a given residual form, so that the re-sulting sum is again a residual form. This process was implemented as the command
TranslateDRF . We also provide a command named
SignificantDenom to extract thesignificant denominator of a residual form.Now let’s consider Example 4.11. For 𝐾 = 1 /𝑘 shift-reduced, we have two resid-uals form w.r.t. 𝐾 : 𝑟 = 1 / (2 𝑘 + 1) and 𝑠 = 1 / (2 𝑘 + 3). > K:=1/k: r:=1/(2*k+1): s:=1/(2*k+3):
One can compute a residual form of 𝑟 + 𝑠 in terms of the significant denominatorof 𝑟 by > res:=TranslateDRF(s, SignificantDenom(r,K,k), K, k); > S1:=res[1]: a:=res[2][1]: b:=res[2][2]: q:=res[2][3]: > new:=r+normal(a/b)+q/denom(K); > normal(K*subs(k=k+1, S1)-S1+new-r-s); 𝑏 := 𝑘 + 12 𝑟𝑒𝑠 := [︂ − 𝑘 + 1) , [︂ − , 𝑘 + 12 , ]︂]︂ 𝑛𝑒𝑤 := − 𝑘 + 1) + 12 𝑘 𝑟 + 𝑠 in terms of the significant denominator of 𝑠 , ppendix A. The ShiftReductionCT
Package > b:=SignificantDenom(s,K,k); > res:=TranslateDRF(r, b, K, k); > new:=s+normal(res[2][1]/b)+res[2][2]/denom(K); > normal(K*subs(k=k+1, res[1])-res[1]+new-r-s); 𝑟𝑒𝑠 := [︂ − 𝑘 + 1 , [︂ − , 𝑘 + 32 , ]︂]︂ 𝑛𝑒𝑤 := 13(2 𝑘 + 3) + 13 𝑘 Commands related to Chapter 5
Now let’s turn our attention to bivariate hypergeometric terms. Consider thefollowing hypergeometric term from Example 5.10. > T:=binomial(n,k)^3; 𝑇 := binomial( 𝑛, 𝑘 ) Based on the modified Abramov-Petkovšek reduction, Algorithm 5.6 is imple-mented in the command
ReductionCT , which (by default) returns the (monic) min-imal telescoper for a given hypergeometric term. > ReductionCT(T,n,k,Sn); − 𝑛 + 2 𝑛 + 1) 𝑛 + 4 𝑛 + 4 − (7 𝑛 + 21 𝑛 + 16) 𝑆𝑛𝑛 + 4 𝑛 + 4 + 𝑆𝑛 As illustrated by the following commands, if a fifth argument is specified then thecommand also returns a corresponding certificate, whose form depends on the speci-fication. More precisely, we get a certificate as a list of a normalized rational functionand a hypergeometric term by saying > res:=ReductionCT(T,n,k,Sn,output=normalized); 𝑟𝑒𝑠 := [︃ − 𝑛 + 2 𝑛 + 1) 𝑛 + 4 𝑛 + 4 − (7 𝑛 + 21 𝑛 + 16) 𝑆𝑛𝑛 + 4 𝑛 + 4 + 𝑆𝑛 , [︃ − 𝑛 − 𝑘 ) ( 𝑛 + 4 𝑛 + 4)( − 𝑛 − 𝑘 ) (︁ 𝑘 (4 𝑘 𝑛 − 𝑘 𝑛 + 27 𝑘𝑛 − 𝑛 + 8 𝑘 𝑛 − 𝑘 𝑛 + 147 𝑘𝑛 − 𝑛 + 4 𝑘 − 𝑘 𝑛 + 291 𝑘𝑛 − 𝑛 − 𝑘 + 249 𝑘𝑛 − 𝑛 + 78 𝑘 − 𝑛 − )︃ , binomial( 𝑛, 𝑘 ) ]︃]︃ or get one as a list of a linear combination of several simple rational functions anda hypergeometric term by0 Appendix A. The
ShiftReductionCT
Package > res:=ReductionCT(T,n,k,Sn,output=unnormalized); 𝑟𝑒𝑠 := [︃ − 𝑛 + 2 𝑛 + 1) 𝑛 + 4 𝑛 + 4 − (7 𝑛 + 21 𝑛 + 16) 𝑆𝑛𝑛 + 4 𝑛 + 4 + 𝑆𝑛 , [︃ 𝑛 + 2 𝑛 + 1) 𝑛 + 4 𝑛 + 4 − (7 𝑛 + 21 𝑛 + 16)( 𝑛 + 3 𝑛 + 3 𝑛 + 1)( 𝑛 + 4 𝑛 + 4)( − 𝑛 − 𝑘 ) − ( 𝑛 + 1) (6 𝑘 + 3 𝑘𝑛 + 𝑛 + 6 𝑘 + 4 𝑛 + 4)( 𝑛 + 4 𝑛 + 4)( − 𝑛 − 𝑘 ) + 1( 𝑛 + 4 𝑛 + 4)( − 𝑛 − 𝑘 ) (12 𝑘 𝑛 − 𝑘𝑛 + 11 𝑛 + 36 𝑘 𝑛 − 𝑘𝑛 + 62 𝑛 + 36 𝑘 𝑛 − 𝑘𝑛 + 140 𝑛 + 12 𝑘 − 𝑘𝑛 + 158 𝑛 − 𝑘 + 89 𝑛 + 20) − (( 𝑛 + 1) + 3( 𝑛 + 1) + 3 𝑛 + 4)( 𝑛 + 1) ( − 𝑛 − − 𝑘 ) ( − 𝑛 − 𝑘 ) , binomial( 𝑛, 𝑘 ) ]︃]︃ The result returned by the command
ReductionCT can be verified by the com-mand
VerifyRCT . > VerifyRCT(res,T,n,k,Sn); true
Maple’s implementation for Zeilberger’s algorithm is the command
Zeilberger ,which is also in the package
SumTools[Hypergeometric] . > SumTools[Hypergeometric][Zeilberger](T,n,k,Sn); [︃ ( 𝑛 + 4 𝑛 + 4) 𝑆𝑛 + ( − 𝑛 − 𝑛 − 𝑆𝑛 − 𝑛 − 𝑛 − , − 𝑛 − 𝑘 ) ( − 𝑛 − 𝑘 ) (︂(︂ 𝑘 + (︂ − 𝑛 − )︂ 𝑘 + (︂ 𝑛 + 934 𝑛 + 392 )︂ 𝑘 − 𝑛 − 𝑛 − 𝑛 − )︂ 𝑘 binomial( 𝑛, 𝑘 ) (4 𝑛 + 8 𝑛 + 4) )︁ ]︃ ppendix A. The ShiftReductionCT
Package
ShiftMAPReduction , whichperforms the same function as applying
ModifiedAbramovPetkovsekReduction withrespect to 𝑘 to the 𝑚 -th shift 𝜎 𝑚𝑛 ( 𝑇 ) for a bivariate hypergeometric term 𝑇 ( 𝑛, 𝑘 )but in a faster way as pointed out by the remark. Moreover, this command alwaysuses the same kernel independent of the value of 𝑚 . Note that when 𝑚 = 0 thecommand is the same as the command ModifiedAbramovPetkovsekReduction .To illustrate this command, we consider the same hypergeometric term 𝑇 asbefore. > T:=binomial(n,k)^3:
Then it has a minimal additive decomposition > ModifiedAbramovPetkovsekReduction(T,k); [︃[︃ − ,
12 3 𝑘 𝑛 − 𝑘𝑛 + 𝑛 + 3 𝑘 + 3 𝑘 + 1( 𝑘 + 1) ]︃ , binomial( 𝑛, 𝑘 ) ]︃ For the first shift of 𝑇 w.r.t. 𝑛 , we have > ModifiedAbramovPetkovsekReduction(subs(n=n+1,T),k); [︃[︃ − ,
12 3 𝑘 𝑛 − 𝑘𝑛 + 𝑛 + 6 𝑘 − 𝑘𝑛 + 3 𝑛 + 3 𝑛 + 2( 𝑘 + 1) ]︃ , binomial( 𝑛 + 1 , 𝑘 ) ]︃ On the other hand, applying the command
ShiftMAPReduction gives > ShiftMAPReduction(T,n,k,1); [︃[︃ 𝑛 + 3 𝑛 + 3 𝑛 + 1( − 𝑛 − 𝑘 ) , 𝑛 + 3 𝑛 + 3 𝑛 + 1( 𝑘 + 1) ]︃ , binomial( 𝑛, 𝑘 ) ]︃ Commands related to Chapter 6
Combining the bounds given in Chapter 6, we implemented Algorithm 6.11 asthe command
BoundReductionCT . The function of this command is illustrated asfollows.Consider Example 6.12 with 𝛼 = 5. > alpha:=5: T:=1/((n-alpha*k-alpha)*(n-alpha*k-2)!); 𝑇 := 1( − 𝑘 + 𝑛 − − 𝑘 + 𝑛 − LowerBound [6] is also named
LowerBound in the package
SumTools[Hypergeometric] . With only three argu-ments, it returns a lower order bound of the telescopers for a given hypergeometricterm, > SumTools[Hypergeometric][LowerBound](T,n,k); Appendix A. The
ShiftReductionCT
Package
Moreover, by specifying a fourth and a fifth argument, the command also givesinformation about telescopers as well as certificates. > SumTools[Hypergeometric][LowerBound](T,n,k,Sn,’Zpair’); > Zpair; ⎡⎣ 𝑆𝑛 − , ⎡⎣ 𝑛 + 4) (5 𝑘 − 𝑛 ) ⎛⎝ 𝑘 − ∏︁ _ 𝑘 =0 ( − (5 _ 𝑘 − 𝑛 − 𝑘 − 𝑛 − 𝑘 − 𝑛 − 𝑘 − 𝑛 + 1)(5 _ 𝑘 − 𝑛 ) ⎞⎠⎤⎦ In the same fashion, our implementation for Algorithm 6.11, namely the com-mand
BoundReductionCT , with three arguments specified returns an upper bound aswell as a lower bound for the order of minimal telescopers for a given hypergeometricterm. > BoundReductionCT(T,n,k); [5 , ReductionCT introduced above. To be precise, we have the following commands. > BoundReductionCT(T,n,k,Sn); 𝑆𝑛 − > res:=BoundReductionCT(T,n,k,Sn,output=normalized): 𝑟𝑒𝑠 := [︃ 𝑆𝑛 − , [︃ 𝑘 − 𝑛 ) (5 𝑘 − − 𝑛 )(5 𝑘 − 𝑛 − 𝑘 − − 𝑛 )(5 𝑘 − 𝑛 + 1) , − − 𝑘 + 𝑛 − ]︂]︂ > res:=BoundReductionCT(T,n,k,Sn,output=unnormalized); 𝑟𝑒𝑠 := [︂ 𝑆𝑛 − , [︂ 𝑘 − 𝑛 + 5 − 𝑘 − 𝑛 + 1 + 20(5 𝑘 − 𝑛 + 5)(5 𝑘 − 𝑛 + 1)+ 5(5 𝑘 − 𝑛 ) (5 𝑘 − − 𝑛 )(5 𝑘 − 𝑛 − 𝑘 − − 𝑛 )(5 𝑘 − 𝑛 + 1) , − − 𝑘 + 𝑛 − ]︂]︂ ppendix B Comparison of
Memory Requirements
In this section, we collect all comparisons of memory requirements between our newprocedures from the
ShiftReductionCT package (see Appendix A) and Maple’simplementations of known algorithms. All memory requirements are obtained by theMaple command > kernelopts("bytesused"); and measured in bytes on a Linux computer with 388Gb RAM and twelve 2.80GHzDual core processors. Recall that• G : the procedure Gosper in SumTools[Hypergeometric] , which is based onGosper’s algorithm;• AP : the procedure SumDecomposition in SumTools[Hypergeometric] , whichis based on the Abramov-Petkovšek reduction;• Z : the procedure SumTools[Hypergeometric] [ Zeilberger ], which is based onZeilberger’s algorithm;• S : the procedure IsSummable in ShiftReductionCT , which determines hy-pergeometric summability in a similar way as Gosper’s algorithm;•
MAP : the procedure
ModifiedAbramovPetkovsekReduction in ShiftReductionCT ,which is based on the modified reduction.•
RCT 𝑡𝑐 : the procedure ReductionCT in ShiftReductionCT , which computes aminimal telescoper and a corresponding normalized certificate;•
RCT 𝑡 : the procedure ReductionCT in ShiftReductionCT , which computes aminimal telescoper without constructing a certificate.•
BRCT 𝑡𝑐 : the procedure BoundReductionCT in ShiftReductionCT , which com-putes a minimal telescoper and a corresponding normalized certificate;•
BRCT 𝑡 : the procedure BoundReductionCT in ShiftReductionCT , which com-putes a minimal telescoper without constructing a certificate.• LB : the lower bound for telescopers given in Theorem 6.10.• order : the order of the resulting minimal telescoper. Appendix B. Comparison of Memory Requirements
Tables for Example 3.23 and Example 3.24. ( 𝜆, 𝜇 ) G AP S MAP (0 ,
0) 1.80015e7 2.79579e7 2.19643e7 2.20057e7(5 ,
5) 6.92148e7 5.45788e8 8.31337e7 1.00876e8(10 ,
10) 1.06237e8 1.74321e9 1.63963e8 2.23078e8(10 ,
20) 3.67295e8 4.22563e9 3.41155e8 7.14421e8(10 ,
30) 9.08446e8 2.06166e10 5.73637e8 2.07008e9(10 ,
40) 1.79107e9 3.74146e10 8.60492e8 5.01724e9(10 ,
50) 3.19600e9 4.98811e10 1.16624e9 9.80644e9
Table B.1:
Memory comparison of Gosper’s algorithm, the Abramov-Petkovšek re-duction and the modified version for random hypergeometric terms (in bytes) ( 𝜆, 𝜇 ) G AP S MAP (0 ,
0) 1.49566e8 3.83358e8 1.96563e8 1.97086e8(5 ,
5) 2.76453e8 9.42523e8 2.40684e8 2.40927e8(10 ,
10) 3.15859e8 1.86511e9 2.50334e8 2.50661e8(10 ,
20) 6.81883e8 4.15802e9 3.19633e8 3.20250e8(10 ,
30) 1.48580e9 7.60674e9 3.61856e8 3.60798e8(10 ,
40) 2.66329e9 1.24394e10 3.81800e8 3.82879e8(10 ,
50) 4.96349e9 2.22568e10 4.15063e8 4.14124e8
Table B.2:
Memory comparison of Gosper’s algorithm, the Abramov-Petkovšek re-duction and the modified version for summable hypergeometric terms (in bytes) ppendix B. Comparison of Memory Requirements Tables for Example 5.11. ( 𝑑 , 𝑑 , 𝛼, 𝜆, 𝜇 ) Z RCT 𝑡𝑐 RCT 𝑡 order (1 , , , ,
5) 2.05992e9 5.36111e8 1.58646e8 4(1 , , , ,
5) 6.13485e9 3.33929e9 9.01651e8 6(1 , , , ,
5) 2.05569e10 1.12736e10 2.59005e9 7(1 , , , ,
5) 2.84955e10 1.46063e10 3.24756e9 7(2 , , , ,
10) 3.58374e10 6.87524e9 6.90891e8 4(2 , , , ,
10) 3.03599e10 4.30070e10 7.44379e9 6(2 , , , ,
10) 6.95166e10 1.29853e11 2.56292e10 7(2 , , , ,
10) 7.63196e10 1.34622e11 2.78371e10 7(2 , , , ,
15) 1.72175e11 2.44536e10 1.52217e9 4(2 , , , ,
15) 8.27362e10 1.38827e11 2.09677e10 6(2 , , , ,
15) 1.79564e11 4.57813e11 1.04973e11 7(2 , , , ,
15) 2.01763e11 4.49569e11 1.06872e11 7(3 , , , ,
10) 7.48174e11 4.17901e10 5.18114e9 6(3 , , , ,
10) 3.63162e11 2.25463e11 5.19205e10 8(3 , , , ,
10) 7.60572e11 6.16676e11 1.78310e11 9
Table B.3:
Memory comparison of Zeilberger’s algorithm to reduction-based creativetelescoping with and without construction of a certificate (in bytes) Appendix B. Comparison of Memory Requirements
Tables for Example 6.15 and Example 6.16. 𝛼 RCT 𝑡 RCT 𝑡𝑐 BRCT 𝑡 BRCT 𝑡𝑐 LB order
20 2.53275e8 2.57797e8 1.42371e8 1.46826e8 20 2030 1.04691e9 1.05593e9 4.73815e8 4.83413e8 30 3040 3.16905e9 3.18565e9 1.31468e9 1.33395e9 40 4050 7.69274e9 7.71999e9 3.12029e9 3.15161e9 50 5060 1.62442e10 1.62819e10 6.24941e9 6.28674e9 60 6070 3.15561e10 3.16084e10 1.19886e10 1.20418e10 70 70
Table B.4:
Memory comparison of two reduction-based creative telescoping with andwithout construction of a certificate for Example 6.15 (in bytes) ( 𝑚, 𝛼 ) RCT 𝑡 RCT 𝑡𝑐 BRCT 𝑡 BRCT 𝑡𝑐 LB order (1,1) 2.64768e7 3.12387e7 2.64914e7 3.12548e7 1 2(1,10) 9.91388e8 1.62603e9 8.94051e8 1.50416e9 10 11(1,15) 2.01112e10 2.32990e10 1.33834e10 1.75427e10 15 16(1,20) 2.23859e11 2.43209e11 1.13767e11 1.29430e11 20 21(2,10) 1.03547e9 1.65297e9 9.12084e8 1.52683e9 10 11(2,15) 2.70850e10 3.02579e10 1.38753e10 1.64594e10 15 16(2,20) 2.37348e11 2.48004e11 1.29174e11 1.41685e11 20 21
Table B.5:
Memory comparison of two reduction-based creative telescoping with andwithout construction of a certificate for Example 6.16 (in bytes) ibliography [1] Sergei A. Abramov. On the summation of rational functions.
Ž. Vyčisl. Mat. iMat. Fiz. , 11:1071–1075, 1971.[2] Sergei A. Abramov. The rational component of the solution of a first orderlinear recurrence relation with rational right hand side.
Ž. Vyčisl. Mat. i Mat.Fiz. , 15(4):1035–1039, 1090, 1975.[3] Sergei A. Abramov. Indefinite sums of rational functions. In
ISSAC 1995—Proceedings of the 20th International Symposium on Symbolic and AlgebraicComputation , pages 303–308. ACM, New York, 1995.[4] Sergei A. Abramov. When does Zeilberger’s algorithm succeed?
Adv. in Appl.Math. , 30(3):424–441, 2003.[5] Sergei A. Abramov, Jacques J. Carette, Keith O. Geddes, and Ha Q. Le. Tele-scoping in the context of symbolic summation in Maple.
J. Symbolic Comput. ,38(4):1303–1326, 2004.[6] Sergei A. Abramov and Ha Q. Le. On the order of the recurrence produced bythe method of creative telescoping.
Discrete Math. , 298(1-3):2–17, 2005.[7] Sergei A. Abramov and Marko Petkovšek. Minimal decomposition of indefinitehypergeometric sums. In
ISSAC 2001—Proceedings of the 26th InternationalSymposium on Symbolic and Algebraic Computation
Adv. in Appl. Math. , 29(3):386–411, 2002.[10] Sergei A. Abramov and Marko Petkovšek. Rational normal forms and minimaldecompositions of hypergeometric terms.
J. Symbolic Comput. , 33(5):521–543,2002. Computer algebra (London, ON, 2001).[11] Gert Almkvist and Doron Zeilberger. The method of differentiating under theintegral sign.
J. Symbolic Comput. , 10(6):571–591, 1990. Bibliography [12] Moa Apagodu and Doron Zeilberger. Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory.
Adv. inAppl. Math. , 37(2):139–152, 2006.[13] David Bailey, Peter Borwein, and Simon Plouffe. On the rapid computation ofvarious polylogarithmic constants.
Math. Comp. , 66(218):903–913, 1997.[14] Alin Bostan, Shaoshi Chen, Frédéric Chyzak, and Ziming Li. Complexity of cre-ative telescoping for bivariate rational functions. In
ISSAC 2010—Proceedingsof the 35th International Symposium on Symbolic and Algebraic Computation ,pages 203–210. ACM, New York, 2010.[15] Alin Bostan, Shaoshi Chen, Frédéric Chyzak, Ziming Li, and Guoce Xin. Her-mite reduction and creative telescoping for hyperexponential functions. In
IS-SAC 2013—Proceedings of the 38th International Symposium on Symbolic andAlgebraic Computation , pages 77–84. ACM, New York, 2013.[16] Alin Bostan, Pierre Lairez, and Bruno Salvy. Creative telescoping for rationalfunctions using the Griffiths-Dwork method. In
ISSAC 2013—Proceedings of the38th International Symposium on Symbolic and Algebraic Computation , pages93–100. ACM, New York, 2013.[17] Manuel Bronstein and Marko Petkovšek. An introduction to pseudo-linear al-gebra.
Theoret. Comput. Sci. , 157(1):3–33, 1996. Algorithmic complexity ofalgebraic and geometric models (Creteil, 1994).[18] Shaoshi Chen.
Some applications of differential-difference algebra to creativetelescoping . PhD thesis, February 2011.[19] Shaoshi Chen, Hui Huang, Manuel Kauers, and Ziming Li. A modifiedAbramov-Petkovšek reduction and creative telescoping for hypergeometricterms. In
ISSAC 2015—Proceedings of the 40th International Symposium onSymbolic and Algebraic Computation , pages 117–124. ACM, New York, 2015.[20] Shaoshi Chen and Manuel Kauers. Order-degree curves for hypergeometriccreative telescoping. In
ISSAC 2012—Proceedings of the 37th InternationalSymposium on Symbolic and Algebraic Computation , pages 122–129. ACM, NewYork, 2012.[21] Shaoshi Chen and Manuel Kauers. Trading order for degree in creative tele-scoping.
J. Symbolic Comput. , 47(8):968–995, 2012.[22] Shaoshi Chen, Manuel Kauers, and Christoph Koutschan. A generalizedApagodu-Zeilberger algorithm. In
ISSAC 2014—Proceedings of the 39th Inter-national Symposium on Symbolic and Algebraic Computation , pages 107–114.ACM, New York, 2014.[23] Shaoshi Chen, Manuel Kauers, and Christoph Koutschan. Reduction-basedcreative telescoping for algebraic functions. In
ISSAC 2016—Proceedings of the41st International Symposium on Symbolic and Algebraic Computation , page175–182. ACM, New York, 2016. ibliography
ISSAC 2012—Proceedings of the 37thInternational Symposium on Symbolic and Algebraic Computation , pages 130–137. ACM, New York, 2012.[25] Shaoshi Chen, Manuel Kauers, and Michael F. Singer. Desingularization of Oreoperators.
J. Symbolic Comput. , 74:617–626, 2016.[26] David V. Chudnovsky and Gregory V. Chudnovsky. Computer algebra in theservice of mathematical physics and number theory. In
Computers in mathe-matics (Stanford, CA, 1986) , volume 125 of
Lecture Notes in Pure and Appl.Math. , pages 109–232. Dekker, New York, 1990.[27] Frédéric Chyzak and Bruno Salvy. Non-commutative elimination in Ore alge-bras proves multivariate identities.
J. Symbolic Comput. , 26(2):187–227, 1998.[28] Richard M. Cohn.
Difference algebra . Interscience Publishers John Wiley &Sons, New York-London-Sydeny, 1965.[29] Mary Celine Fasenmyer. Some generalized hypergeometric polynomials.
Bull.Amer. Math. Soc. , 53:806–812, 1947.[30] Stéphane Fischler and Tanguy Rivoal. On the values of 𝐺 -functions. Comment.Math. Helv. , 89(2):313–341, 2014.[31] Stéphane Fischler and Tanguy Rivoal. Arithmetic theory of 𝐸 -operators. J.Éc. polytech. Math. , 3:31–65, 2016.[32] Philippe Flajolet, Stefan Gerhold, and Bruno Salvy. On the non-holonomiccharacter of logarithms, powers, and the 𝑛 th prime function. Electron. J. Com-bin. , 11(2):Article 2, 16 pp. (electronic), 2004/06.[33] Philippe Flajolet and Andrew Odlyzko. Singularity analysis of generating func-tions.
SIAM J. Discrete Math. , 3(2):216–240, 1990.[34] Philippe Flajolet and Robert Sedgewick.
Analytic combinatorics . CambridgeUniversity Press, Cambridge, 2009.[35] Philippe Flajolet and Brigitte Vallée. Continued fractions, comparison algo-rithms, and fine structure constants. In
Constructive, experimental, and non-linear analysis (Limoges, 1999) , volume 27 of
CMS Conf. Proc. , pages 53–82.Amer. Math. Soc., Providence, RI, 2000.[36] R. William Gosper, Jr. Decision procedure for indefinite hypergeometric sum-mation.
Proc. Nat. Acad. Sci. U.S.A. , 75(1):40–42, 1978.[37] William A. Harris, Jr. and Yasutaka Sibuya. The reciprocals of solutions oflinear ordinary differential equations.
Adv. in Math. , 58(2):119–132, 1985.[38] Hui Huang. New bounds for hypergeometric creative telescoping. In
ISSAC2016—Proceedings of the 41st International Symposium on Symbolic and Alge-braic Computation , pages 279–286. ACM, New York, 2016.00
Bibliography [39] Hui Huang and Manuel Kauers. D-finite numbers, 2016. in preparation.[40] Edward L. Ince.
Ordinary Differential Equations . Dover Publications, NewYork, 1944.[41] Manuel Kauers and Peter Paule.
The Concrete Tetrahedron . Texts and Mono-graphs in Symbolic Computation. SpringerWienNewYork, Vienna, 2011. Sym-bolic sums, recurrence equations, generating functions, asymptotic estimates.[42] Manuel Kauers and Gleb Pogudin. in preparation, 2016.[43] Maxim Kontsevich and Don Zagier. Periods. In
Mathematics unlimited—2001and beyond , pages 771–808. Springer, Berlin, 2001.[44] Ha Q. Le.
Algorithms for the construction of the minimal telescopers . PhDthesis, University of Waterloo, Waterloo, Ont., Canada, 2003. AAINQ83003.[45] Leonard Lipshitz. 𝐷 -finite power series. J. Algebra , 122(2):353–373, 1989.[46] Christian Mallinger. Algorithmic manipulations and transformations of uni-variate holonomic functions and sequences. Master’s thesis, RISC, J. KeplerUniversity, August 1996.[47] Marc Mezzarobba. NumGfun: a package for numerical and analytic compu-tation and D-finite functions. In
ISSAC 2010—Proceedings of the 2010 Inter-national Symposium on Symbolic and Algebraic Computation , pages 139–146.ACM, New York, 2010.[48] Marc Mezzarobba and Bruno Salvy. Effective bounds for P-recursive sequences.
J. Symbolic Comput. , 45(10):1075–1096, 2010.[49] Mohamud Mohammed and Doron Zeilberger. Sharp upper bounds for the ordersof the recurrences output by the Zeilberger and 𝑞 -Zeilberger algorithms. J.Symbolic Comput. , 39(2):201–207, 2005.[50] Oystein Ore. Sur la forme des fonctions hypergéométriques de plusieurs vari-ables.
J. Math. Pures Appl. (9) , 9(4):311–326, 1930.[51] Peter Paule. Greatest factorial factorization and symbolic summation.
J. Sym-bolic Comput. , 20(3):235–268, 1995.[52] Peter Paule and Markus Schorn. A Mathematica version of Zeilberger’s algo-rithm for proving binomial coefficient identities.
J. Symbolic Comput. , 20(5-6):673–698, 1995. Symbolic computation in combinatorics Δ (Ithaca, NY,1993).[53] Marko Petkovšek. Hypergeometric solutions of linear recurrences with polyno-mial coefficients. J. Symbolic Comput. , 14(2-3):243–264, 1992.[54] Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger. 𝐴 = 𝐵 . A K Peters,Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth, With aseparately available computer disk. ibliography J. Symbolic Comput. , 20(5-6):617–635, 1995. Symbolic compu-tation in combinatorics Δ (Ithaca, NY, 1993).[56] John Riordan. Combinatorial identities . Robert E. Krieger Publishing Co.,Huntington, N.Y., 1979. Reprint of the 1968 original.[57] Bruno Salvy and Paul Zimmermann. Gfun: a maple package for the manipula-tion of generating and holonomic functions in one variable.
ACM Transactionson Mathematical Software , 20(2):163–177, 1994.[58] Carl L. Siegel. Über einige Anwendungen diophantischer Approximatio-nen [reprint of Abhandlungen der Preußischen Akademie der Wissenschaften.Physikalisch-mathematische Klasse 1929, Nr. 1]. In
On some applications ofDiophantine approximations , volume 2 of
Quad./Monogr. , pages 81–138. Ed.Norm., Pisa, 2014.[59] Richard P. Stanley. Differentiably finite power series.
European J. Combin. ,1(2):175–188, 1980.[60] Richard P. Stanley.
Enumerative combinatorics. Vol. 2 , volume 62 of
CambridgeStudies in Advanced Mathematics . Cambridge University Press, Cambridge,1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.[61] Nobuki Takayama. An approach to the zero recognition problem by Buchbergeralgorithm.
J. Symbolic Comput. , 14(2-3):265–282, 1992.[62] Joris van der Hoeven. Fast evaluation of holonomic functions.
Theoret. Comput.Sci. , 210(1):199–215, 1999.[63] Joris van der Hoeven. Fast evaluation of holonomic functions near and in regularsingularities.
J. Symbolic Comput. , 31(6):717–743, 2001.[64] Joris van der Hoeven. Efficient accelero-summation of holonomic functions.
J.Symbolic Comput. , 42(4):389–428, 2007.[65] Herbert S. Wilf and Doron Zeilberger. Rational functions certify combinatorialidentities.
J. Amer. Math. Soc. , 3(1):147–158, 1990.[66] Herbert S. Wilf and Doron Zeilberger. Towards computerized proofs of identi-ties.
Bull. Amer. Math. Soc. (N.S.) , 23(1):77–83, 1990.[67] Herbert S. Wilf and Doron Zeilberger. An algorithmic proof theory for hy-pergeometric (ordinary and “ 𝑞 ”) multisum/integral identities. Invent. Math. ,108(3):575–633, 1992.[68] Herbert S. Wilf and Doron Zeilberger. Rational function certification ofmultisum/integral/“ 𝑞 ” identities. Bull. Amer. Math. Soc. (N.S.) , 27(1):148–153, 1992.[69] Doron Zeilberger. A fast algorithm for proving terminating hypergeometricidentities.
Discrete Math. , 80(2):207–211, 1990.02
Bibliography [70] Doron Zeilberger. A holonomic systems approach to special functions identities.
J. Comput. Appl. Math. , 32(3):321–368, 1990.[71] Doron Zeilberger. The method of creative telescoping.
J. Symbolic Comput. ,11(3):195–204, 1991. otation
The following list describes the most important mathematical notations that havebeen used in this thesis. For each group, the order follows roughly the order of firstappearance in the text.
Abbreviations gcd The greatest common divisormin The minimummax The maximum 𝑝 | 𝑞 A polynomial 𝑝 divides a polynomial 𝑞 over the domain wherethe polynomials live 𝑝 (cid:45) 𝑞 A polynomial 𝑝 does not divide a polynomial 𝑞 over the domainwhere the polynomials livelog The natural logarithmexp The exponential function Number Sets N , Z , Q , R , C Sets of natural, integer, rational, real, complex numbers Q ( 𝑖 ) The Gaussian rational field 𝒟 𝑅, F The set of D-finite numbers with respect to 𝑅 and F 𝒟 F The set 𝒟 F , F 𝒜 F The set of limits of convergent algebraic sequences over F ∅ The empty set 𝒞 F The set of limits of convergent C-finite sequences over F Notation
Operators 𝜎 𝑘 The shift operator w.r.t. 𝑘 which maps 𝑟 ( 𝑘 ) to 𝑟 ( 𝑘 + 1) forevery rational function 𝑟 ∈ F ( 𝑘 )Δ 𝑘 The difference of 𝜎 𝑘 and the identity map 𝑆 𝑛 The operator in the ring of linear recurrence operators over F which satisfies 𝑆 𝑛 𝑟 = 𝜎 𝑛 ( 𝑟 ) 𝑆 𝑛 for all 𝑟 ∈ F . ∑︀ 𝜌𝑗 =0 𝑝 𝑗 𝑆 𝑗𝑛 A recurrence operator with polynomial coefficients 𝑝 𝑗 𝐷 𝑧 The derivation operator w.r.t. 𝑧 which maps a power series orfunction 𝑓 ( 𝑧 ) to its derivative 𝑓 ′ ( 𝑧 ) = 𝑑𝑑𝑧 𝑓 ( 𝑧 ) ∑︀ 𝜌𝑗 =0 𝑝 𝑗 𝐷 𝑗𝑧 A differential operator with polynomial coefficients 𝑝 𝑗 Rings/Fields/Algebra
Quot( 𝑅 ) The quotient field of the ring 𝑅 K A field of characteristic zero F A field of characteristic zero, or the field K ( 𝑛 ) (Chapter 5), ora subfield of C (Chapter 8) F ( 𝑘 ) The field of univariate rational functions in 𝑘 over FF [ 𝑘 ] The ring of univariate polynomials in 𝑘 over FD A difference ring extension of F ( 𝑘 ) K ( 𝑛, 𝑘 ) The field of bivariate rational functions in 𝑛, 𝑘 over KF [ 𝑛 ] ⟨ 𝑆 𝑛 ⟩ The Ore algebra of linear recurrence operators with polyno-mial coefficients w.r.t. 𝑛 K [ 𝑛, 𝑘 ] The ring of bivariate polynomials in 𝑛, 𝑘 over KK ( 𝑛 )[ 𝑘 ] The ring of polynomials in 𝑘 over the field K ( 𝑛 ) 𝑅 A subring of C 𝑅 [[ 𝑧 ]] The ring of formal power series over 𝑅𝑅 N The ring of all sequences from N to 𝑅 F [ 𝑧 ] ⟨ 𝐷 𝑧 ⟩ The Ore algebra of linear differential operators with polyno-mial coefficients wr.t. 𝑧 ¯ F The algebraic closure of the field F otation Other Symbols ∑︀ 𝑏𝑗 = 𝑎 𝑓 ( 𝑘 ) The sum 𝑓 ( 𝑎 ) + 𝑓 ( 𝑎 + 1) + · · · + 𝑓 ( 𝑏 )deg 𝑘 ( 𝑝 ) Degree of a polynomial 𝑝 w.r.t. 𝑘 lc 𝑘 ( 𝑝 ) Leading coefficient of a polynomial 𝑝 w.r.t. 𝑘𝐴 ∖ 𝐵 The relative complement of a set 𝐵 with respect to a set 𝐴𝑘 !, (︂ 𝑛𝑘 )︂ Factorial 𝑘 ! = 1 · · . . . ( 𝑘 − · 𝑘 and binomial coefficient (︀ 𝑛𝑘 )︀ = 𝑛 ( 𝑛 − . . . ( 𝑛 − 𝑘 + 1) /𝑘 ! U 𝑇 The union of { } and the set of summable hypergeometricterms that are similar to a hypergeometric term 𝑇 V 𝐾 The set { 𝐾𝜎 𝑘 ( 𝑟 ) − 𝑟 | 𝑟 ∈ F ( 𝑘 ) } where 𝐾 is a shift-reducedrational function in F ( 𝑘 ) 𝐴 ≡ 𝑘 𝐵 mod 𝐶 𝑘 The expression 𝐴 − 𝐵 belongs to a set 𝐶 𝑘 𝜑 𝐾 The map for polynomial reduction with respect to a shift-reduced rational function 𝐾 im( 𝜑 𝐾 ) The image space of the map 𝜑 𝐾 W 𝐾 The standard complement of im( 𝜑 𝐾 ) 𝐴 ⊕ 𝐵 The direct sum of two vector spaces 𝐴 and 𝐵𝐴 ∩ 𝐵 The intersection of two sets 𝐴 and 𝐵𝐴 ∪ 𝐵 The union of two sets 𝐴 and 𝐵 |𝒫| The number of elements of the set 𝒫 (cid:74) 𝜙 (cid:75) The Iversion bracket, namely (cid:74) 𝜙 (cid:75) equals 1 if the expression 𝜙 is true, otherwise it is 0. ∏︀ 𝑏𝑗 = 𝑎 𝑓 ( 𝑘 ) The product 𝑓 ( 𝑎 ) 𝑓 ( 𝑎 + 1) . . . 𝑓 ( 𝑏 ) 𝑝 ∼ 𝑘 𝑞 A polynomial 𝑝 is shift-equivalent to a polynomial 𝑞 w.r.t. 𝑘𝑝 ≈ 𝑘 𝑞 A shift-free polynomial 𝑝 is shift-related to a shift-free poly-nomial 𝑞 w.r.t. 𝑘𝐿 ( 𝑇 ) The application of a recurrence operator 𝐿 to a hypergeomet-ric term 𝑇𝑝 ∼ 𝑛,𝑘 𝑞 A polynomial 𝑝 is shift-equivalent to a polynomial 𝑞 w.r.t. 𝑛 and 𝑘 | 𝜉 | The modulus of a complex number 𝜉 Notation dim K ( 𝑛 ) ( W 𝐾 ) The dimension of the vector space W 𝐾 over the field K ( 𝑛 ) 𝛿 ( 𝜆,𝜇 ) The operator 𝜎 𝛼𝑛 𝜎 𝛽𝑘 where 𝜆, 𝜇 are coprime integers and 𝛼𝜆 + 𝛽𝜇 = 1 with | 𝛼 | < | 𝜇 | and | 𝛽 | < | 𝜆 | ∑︀ ∞ 𝑛 =0 𝑎 𝑛 𝑧 𝑛 A power series with the coefficient sequence ( 𝑎 𝑛 ) ∞ 𝑛 =0 ( 𝑎 𝑛 ) ∞ 𝑛 =0 An infinite sequence 𝑎 , 𝑎 , 𝑎 , . . .𝑓 ′ ( 𝑧 ) The first derivative of a power series or function 𝑓 ( 𝑧 ) w.r.t. 𝑧 lc( 𝐿 ) The leading coefficient of an operator 𝐿𝐿 · 𝑎 𝑛 The application of a recurrence operator 𝐿 to an infinite se-quence ( 𝑎 𝑛 ) ∞ 𝑛 =0 𝐿 · 𝑓 ( 𝑧 ) The application of a differential operator 𝐿 to a power series 𝑓𝑓 ∘ 𝑔 The composition 𝑓 ( 𝑔 ) of functions 𝑓 and 𝑔𝐴 ⊆ 𝐵 A set 𝐴 is contained by a set 𝐵 ¯ 𝜉 The complex conjugation of a complex number 𝜉 Re( 𝜉 ) The real part of a complex number 𝜉 Im( 𝜉 ) The imaginary part of a complex number 𝜉𝑎 𝑛 ∼ 𝑏 𝑛 ( 𝑛 → ∞ ) The quotient 𝑎 𝑛 /𝑏 𝑛 converges to 1 as 𝑛 → ∞ 𝑓 ( 𝑧 ) ∼ 𝑔 ( 𝑧 ) ( 𝑧 → 𝜁 ) The quotient 𝑓 ( 𝑧 ) /𝑔 ( 𝑧 ) converges to 1 as 𝑧 approaches 𝜁 [ 𝑧 𝑛 ] 𝑓 ( 𝑧 ) The coefficient of 𝑧 𝑛 in a power series 𝑓 ( 𝑧 ) ∈ F [[ 𝑧 ]] 𝑓 ( 𝑘 ) ( 𝑧 ) The 𝑘 th derivative of a power series or function 𝑓 ( 𝑧 ) w.r.t. 𝑧 ndex Abramov-Le lower bound, 61Abramov-Petkovšek reduction, 13–15, 23,25Apagodu-Zeilberger upper bound, 59Abel’s theorem, 4additive decomposition, 13algebraiccomposition, 69extension, 80function, 4, 68, 78, 79, 87number, 4, 75–80, 86sequence, 75, 78, 87analyticcontinuation, 71, 85function, 68anti-difference, see indefinite summationapparent singularity, 69Apéry’s constant, 75Archimedes’ constant, 74asymptotically equivalent, 76Basic Abelian theorem, 76bound, see order bound, 58
BoundReductionCT (ShiftReductionCT) ,62, 98C-finite sequence, 79, 87Cauchy product, 67certificate, 3, 43, 46, 48closure properties, 69–71comparison, 25, 48, 59, 61, 101complex conjugation, 70, 80computable, 74, 82congruence, 15creative telescoping, 3–4, 41–42, 44–47,49, 58D-finitefunction, 1, 68, 81, 82, 87 number, 5, 73, 74, 80–87power series, see
D-finite functionsequence, see
P-recursive sequencedecompositionadditive, 13multiplicative, 12shift-coprime, 35shift-homogeneous, 50degree, 9derivation operator, 68desingularization, 69differencefield, 9, 42ring, 9, 42differential operator, 68discrete residual form, see residual formdispersion, 35echelon basis, 19–20Euler’sconstant, 75number, 74existence criterion for telescopers, 43exponential function, 10factorial term, 10, 43formal power series, 67functionalgebraic, 4, 68, 78, 79, 87D-finite, 1, 68, 81, 82, 87exponential, 10Gamma, 75Riemann zeta, 4G-function, 4Gamma function, 75generating function, 67generic situation, 59
Gosper (SumTools[Hypergeometric]) , 25,94 Index
Gosper’s algorithm, 3, 10, 14, 25, 94Hadamard product, 68Hermite reduction, 27, 41holonomic constant, 4, 73, 81hyperexponential function, 17, 27hypergeometricidentity, 2summability, see summablesummable, see summablesummation, 2term, 9, 42imaginary unit, 5, 79, 80, 86indefinite summation, 10integer-linear, 43, 50, 54
IsSummable (ShiftReductionCT) , 25, 94kernel, 12kernel reduction, 35leading coefficient, 9, 68Linux computer, 25, 47, 61, 101lower bound, 62
LowerBound , 61map for polynomial reduction, 17, 19
Maple , 3, 24, 47, 61, 91
Mathematica , 3memory requirement, 24, 47, 61, 101minimal telescoper, see also telescopermodified Abramov-Petkovšek reduction,17–25
ModifiedAbramovPetkovsekReduction (ShiftReductionCT) , 25, 93multiple zeta values, 4multiplicative decomposition, 12numberalgebraic, 4, 75–80, 86computable, 74, 82D-finite, 5, 73, 74, 86, 87numerical evaluation, 71, 74
NumGfun , 71operatorderivation, 68differential, 68recurrence, 43, 68 shift, 42, 68order, 48, 62, 68, 102order bound, 57–61Abramov-Le , 61Apagodu-Zeilberger , 59lower, 58upper, 57Ore algebra, 68P-recursive sequence, 1, 68period, 4polynomial reduction, 19–21proper term, 43, 59Pythagoras’ constant, 75quotient field, 70rationalnormal form, 28–29summable, 10, 11recurrence operator, 43, 68reductionAbramov-Petkovšek , 13–15, 23, 25kernel, 35modified Abramov-Petkovšek , 17–25polynomial, 19–21shell, 16reduction-based, 3, 45, 48, 58, 62, 63
ReductionCT (ShiftReductionCT) , 48, 61,95remainder, 49residual form, 18, 29, 30, 33, 37, 52Riemann zeta function, 4runtime, see timingsequence, 67algebraic, 75, 78, 87C-finite, 79, 87P-recursive, 1, 68shell, 12shell reduction, 16shift operator, 42, 68shift-coprime, 35coprime decomposition, 35equivalent, 28, 50free, 11, 33 ndex
ShiftReductionCT , 24, 47, 61, 91–99,101
BoundReductionCT , 62, 98, 101
IsSummable , 25, 94, 101
KernelReduction , 94
ModifiedAbramovPetkovsekReduction ,25, 93, 101
PolynomialReduction , 92
ReductionCT , 48, 61, 95, 101
ShellReduction , 92
ShiftMAPReduction , 97
SignificantDenom , 94
TranslateDRF , 94
VerifyMAPReduction , 93
VerifyRCT , 97significant denominator, 18, 30, 33, 37,52similar, 10singularity, 69, 86square-free, 76, 79standard complement, 18, 19strongly coprime, 15
SumDecomposition (SumTools[Hyper-geometric]) , 25, 92summability, see summablesummable, 10, 13, 15, 16, 18
SumTools[Hypergeometric]
Gosper , 25, 94, 101
LowerBound , 98
SumDecomposition , 25, 92, 101
Zeilberger , 48, 97, 101telescoper, 3, 43, 44, 46timing, 24, 47, 61Transfer theorem, 76univariate representation, 50, 51Wilf-Zeilberger’s theory, 2, 41
Zeilberger (SumTools[Hypergeometric]) , 48,97 Zeilberger’s algorithm, 3, 44, 48, 97zeta constant, 4 urriculum Vitae
Personal data
Name Hui HUANGGender FemaleDate of birth August 28, 1989Place of birth Fujian province, ChinaNationality People’s Republic of China
Contact
Address Institute for AlgebraJohannes Kepler University LinzAltenberger Straße 694040 Linz, AustriaEmail [email protected]
Education
Awards Curriculum Vitae
Scientific Work
Refereed Publications
1. Hui Huang. New bounds for hypergeometric creative telescoping. In
ISSAC2016—Proceedings of the 41st International Symposium on Symbolic and Alge-braic Computation , pages 279–286. ACM, New York, 2016.2. Shaoshi Chen, Hui Huang, Manuel Kauers, and Ziming Li. A modified Abramov-Petkovšek reduction and creative telescoping for hypergeometric terms. In
IS-SAC 2015—Proceedings of the 40th International Symposium on Symbolic andAlgebraic Computation , pages 117–124. ACM, New York, 2015.
Posters • Shaoshi Chen, Hui Huang, and Ziming Li. Improved Abramov-Petkovšek’s Re-duction and Creative Telescoping for Hypergeometric Terms (Poster at ISSAC2014). In
ACM Commun. Comput. Algebra , 48(3/4):106-108. ACM, New York,2014.
Talks
July 2016
New Bounds for Hypergeometric Creative Telescoping . ISSAC2016, Waterloo, Canada.June 2016
Reduction and Creative Telescoping for Hypergeometric Terms .Center for Combinatorics Seminar, Tianjin, China.November 2015
Two Applications of the Modified Abramov-Petkovšek Reduc-tion . CM 2015, Hefei, China.July 2015
A Modified Abramov-Petkovšek Reduction and Creative Tele-scoping for Hypergeometric Terms . ISSAC 2015, Bath, UnitedKingdom.June 2015
Abramov-Petkovšek Reduction and Creative Telescoping for Ra-tional Functions . CanaDAM 2015, Saskatoon, Canada.August 2013