Representation of hypergeometric products of higher nesting depths in difference rings
aa r X i v : . [ c s . S C ] N ov RISC-Linz Report Series No. 20-19
Representation of hypergeometric products of higher nesting depths indifference rings
Evans Doe Ocansey a,b , Carsten Schneider b a Johannes Kepler University Linz, Institute for Algebra, Linz, Austria b Johannes Kepler University Linz, Research Institute for Symbolic Computation (RISC), Linz, Austria
Abstract
A non-trivial symbolic machinery is presented that can rephrase algorithmically a finite set of nestedhypergeometric products in appropriately designed difference rings. As a consequence, one obtainsan alternative representation in terms of one single product defined over a root of unity and nestedhypergeometric products which are algebraically independent among each other. In particular, one cansolve the zero-recognition problem: the input expression of nested hypergeometric products evaluatesto zero if and only if the output expression is the zero expression. Combined with available symbolicsummation algorithms in the setting of difference rings, one obtains a general machinery that canrepresent (and simplify) nested sums defined over nested products.
Keywords: difference rings, nested hypergeometric products, constant field, ring of sequences, zerorecognition, algebraic independence, roots of unity products
1. Introduction
An important problem in symbolic summation is the simplification of sums defined over productsto expressions in terms of simpler sums and products; in the best case, one might find an expressionwithout sums. A first milestone was Gosper’s algorithm [Gos78] and Zeilberger’s groundbreakingapplication for creative telescoping [Zei91] where the summand is given by one hypergeometric product.Further extensions have been accomplished for q -hypergeometric and multibasic products [PR97],their mixed versions [BP99] and ( q –)multi-summation [WZ92, Weg97, Rie03]. In addition, structuralproperties and further insight in this setting have been elaborated, e.g., in [Pau95, CFFL11, CJKS13].More generally, the holonomic system approach [Zei90] and their refinements [Chy00, Kou13, BRS18]represent, e.g., multi-sums over ( q –)hypergeometric products by systems of linear recurrence relations.In particular, Karr’s difference field approach [Kar81, Kar85] paved the way for a general frameworkto represent rather complicated product expressions in a formal way. Here the generators of his ΠΣ-field construction enables one to model (up to a certain level) indefinite nested products of the form P ( n ) = n Y k = ℓ f ( k ) · · · k m − Y k m = ℓ m f m ( k m ) (1)where the multiplicands f i ( n ) = p i ( n ) q i ( n ) for all i with 1 ≤ i ≤ m are built by polynomial expressions p i ( n ) and q i ( n ) in terms of indefinite nested products that are again of the form (1). In Karr’sseminal works [Kar81, Kar85], which can be considered as the discrete version of Risch’s integrationalgorithm [Ris69], a sophisticated algorithm is provided that enables one to test if a given product ⋆ This work was supported by the Austrian Science Fund (FWF) grant P32301 and by the Austrian Science Fund(FWF) grant F5009-N15 in the framework of the SFB “Algorithmic and Enumerative Combinatorics”.
Email addresses: [email protected] (Evans Doe Ocansey), [email protected] (Carsten Schneider)
Preprint submitted to Elsevier November 18, 2020 epresentation (and sums defined over such products) are expressed properly in his ΠΣ-field setting. Asa bonus, his toolbox and refinements in [Sch15, Sch07a, Sch08] enable one to decide constructively if agiven summand represented in a ΠΣ-field has a solution in the same field or in an appropriate extensionof it. In addition, first contributions have been provided in [Sch05, AP10] to simplify products furthersuch that the degrees in the numerators and denominators are minimal.For such complicated classes of products the following task is non-trivial: given an arbitrary ex-pression in terms of nested products (1), design algorithmically an appropriate difference field or ringin which the expression can be represented and in which one can solve, e.g., the (creative) telescop-ing problem. Already for a hypergeometric product P ( n ) = Q nk = ℓ f ( k ) with a rational function f ( x ) ∈ K ( x ) ∗ and ℓ ∈ Z ≥ chosen properly (i.e., P ( n ) is well defined and nonzero for all n ∈ Z ≥ ), ithas been shown in [Sch05] that Karr’s ΠΣ-fields are not sufficient: namely, such a product P ( n ) canbe represented in a ΠΣ-field if and only if it cannot be rewritten in the form P ( n ) = ζ n r ( n ) where r ( x ) ∈ K ( x ) and ζ ∈ K is a primitive λ -th root of unity with λ >
1. More precisely, such objectscan be only represented in ring extensions taking care of the relation ( ζ n ) λ = 1. As a consequence,zero-divisors are introduced coming from (1 + ζ n + · · · + ( ζ n ) λ − )(1 − ζ n ) = 0 . Motivated by this observation (among others) a refined difference ring theory has been elaboratedin [Sch16, Sch17] that combines big parts of Karr’s general framework together with generators of theform ζ n . In this way, not only the hypergeometric product P ( n ), but more generally any polynomialexpression in terms of hypergeometric products can be represented in the class of so-called RΠΣ-extensions. The first algorithms derived in [Sch05, Sch14] require that the input products are definedover K ( x ) where K = Q ( κ , . . . , κ u ) with u ≥ Q . More generally, a complete algorithm has been elaborated in [OS18] (utilizing ideasfrom [Ge93b, Sch14]) that can represent a finite set of hypergeometric products over K ( x ) where K = K ( y , . . . , y r ) is a rational function field defined over an algebraic number field K . In addition, usingalgorithms from [BP99] it can deal also with q -hypergeometric, multibasic and mixed hypergeometricproducts. Finally, a general framework has been elaborated in [Sch20] that considers single nestedproducts defined over a general class of difference fields; an extra bonus is that the latter approachconstructs a difference ring in which the given products are rephrased optimally with following property:the number of generators of the ring and the order λ of the used ζ n are minimal.A remarkable feature of the above algorithms [Sch05, Sch14, OS18, Sch20] is that the given inputexpression of hypergeometric products (and their generalized versions) are rephrased in terms of a finiteset of alternative products Q ( n ) , . . . , Q s ( n ) together with a distinguished root of unity product ζ n such that the sequences produced by Q ( n ) , . . . , Q s ( n ) are algebraically independent among each other.For this result we rely on ideas of [Sch17] that are inspired by [Sch10a, HS08]; compare also [CFFL11].We remark further that these results are also connected to [KZ08] that can compute all algebraicrelations of C -finite solutions (i.e., solutions of homogeneous recurrences with constant coefficients).We emphasize that the algorithms from [Sch05, AP10, Sch14, OS18, Sch20] can be utilized to simplifyhypergeometric solutions [Pet92, Hoe99, ABPS20] of linear difference equations and can be combinedwith symbolic summation algorithms [Sch07a, Sch15, Sch08] to simplify more general solutions, suchas d’Alembertian solutions [AP94, AZ96] and Liouvillian solutions [HS99, PZ13].In this article we aim at extending this toolbox significantly for the general class of nested hyper-geometric products that can be defined as follows. Definition 1.
Let K ( x ) be a rational function field and let f ( x ) , . . . , f m ( x ) ∈ K ( x ) ∗ . Furthermore,let ℓ , . . . , ℓ m ∈ Z ≥ such that for all i with 1 ≤ i ≤ m , f i ( j ) is non-zero and has no pole for all j ∈ Z ≥ with j ≥ ℓ i . Then the indefinite product expression (1) is called a hypergeometric product in n of nestingdepth m . The vector ( f ( x ) , . . . , f m ( x )) ∈ ( K ( x ) ∗ ) m is also called the multiplicand representation of P ( n ). If f i ( x ) ∈ K ∗ for 1 ≤ i ≤ m , then we call (1) a constant or geometric product in n of nesting depth m . Further, we define the set of ground expressions with K ( n ) = { f ( n ) | f ( x ) ∈ K ( x ) } . Moreover, we Throughout this article all fields and rings have characteristic 0. Their elements are considered as expressions that can be evaluated for sufficiently large n ∈ Z ≥ . n ( G ) with G ⊆ K ( x ) as the set of all such products where the multiplicand representationsare taken from G . Furthermore, we introduce the set of product monomials ProdM n ( G ) as the set ofall elements a ( n ) P ( n ) ν · · · P e ( n ) ν e with a ( x ) ∈ G , e ∈ Z ≥ , ν , . . . , ν e ∈ Z and P ( n ) , . . . , P e ( n ) ∈ Prod n ( G ). Finally, we introduce theset of product expressions ProdE n ( G ) as the set of all elements A ( n ) = X v =( ν ,...,ν e ) ∈ S a v ( n ) P ( n ) ν · · · P e ( n ) ν e (2)with e ∈ Z ≥ , S ⊆ Z e finite, a v ( x ) ∈ G for v ∈ S and P ( n ) , . . . , P e ( n ) ∈ Prod n ( G ). Note thatProd n ( G ) ⊆ ProdM n ( G ) ⊆ ProdE n ( G ).Utilizing the available algorithms from [OS18] we will obtain enhanced algorithms that can rephraseexpressions from ProdE n ( K ( x )) in the setting of R ΠΣ-extensions. As a consequence we will solve thefollowing problem; for further details see Theorem 9 and Corollary 4 below.
Problem RPE: Representation of Product Expressions.
Let K = K ( κ , . . . , κ u ) be a rational function field with e ≥ K . Given A ( n ) ∈ ProdE n ( K ( x )). Find B ( n ) ∈ ProdE n ( K ( x )) with ˜ K = ˜ K ( κ , . . . , κ u ) where ˜ K is analgebraic field extension of K , and a non-negative integer δ ∈ Z ≥ with the following properties:(1) A ( n ) = B ( n ) for all n ∈ Z ≥ with n ≥ δ ;(2) All the products P ( n ) , . . . , P s ( n ) ∈ Prod n ( K ( x )) arising in B ( n ) (apart from the distinguishedproduct ζ n with ζ a root of unity) are algebraically independent among each other.(3) The zero-recognition property holds, i.e., A ( n ) = 0 holds for all n from a certain point on ifand only if B ( n ) is the zero-expression.The full machinery have been implemented within Ocansey’s Mathematica package NestedProducts whose functionality will be illustrated in Section 6.3 below; for additional aspects we refer alsoto [Oca19]. We expect that this implementation will open up new applications, e.g., in combinatorics,such as non-trivial evaluations of determinants [MRRJ83, Zei96, Kra01]. In particular, in interactionwith the symbolic summation algorithms available in the package
Sigma [Sch07b] one obtains a fullyautomatic toolbox to tackle nested sums defined over nested hypergeometric products.The outline of the article is as follows. In Section 2 we will introduce rewrite rules that enable oneto transform expressions from ProdE n ( K ( x )) to a more suitable form (see Proposition 2 below) to solveProblem RPE. Given this tailored form, we show in Section 3 how such expressions can be rephrasedstraightforwardly in terms of multiple-chain AP-extensions. In order to solve Problem RPE, we haveto refine this difference ring construction. Namely, in Section 4 we introduce RΠ-extensions: these areAP-extensions where during the construction the set of constants remain unchanged. In particular, wewill elaborate that such rings can be straightforwardly embedded into the ring of sequences and willprovide structural theorems that will prepare the ground to solve Problem RPE. With these resultswe will present in Section 5 the main steps how nested products can be represented in RΠ-extensions.In Section 6 we will combine all these ideas yielding a complete algorithm for Problem RPE that issummarized in Theorem 9 and Corollary 4. In addition, we will illustrate with non-trivial examples howone can solve Problem RPE with the new Mathematica package NestedProducts . The conclusionsare given in Section 7. 3 . Preprocessing hypergeometric products of finite nesting depth
In order to support our machinery to solve Problem RPE, the arising products P ( n ) in A ( n ) ∈ ProdE n ( K ( x )) (e.g., given in (2)) will be transformed to a particularly nice form. We will illustrateeach preprocessing step with an example and then summarize the derived result in Proposition 2 below.Let K ( x ) be a rational function field together with the zero-function (in short Z -function ) definedby Z ( p ) = max (cid:0) { k ∈ Z ≥ | p ( k ) = 0 } (cid:1) + 1 for any p ∈ K [ x ] (3)with max( ∅ ) = −
1. We call K computable if all basic field operations are computable. Note thatif K is a rational function field over an algebraic number field, then K and also its Z -function arecomputable.We start with the hypergeometric product in n of nesting depth m ∈ Z ≥ given by P ( n ) = n Y k = ℓ f ( k ) k Y k = ℓ f ( k ) · · · k m − Y k m = ℓ m f m ( k m ) ∈ Prod n ( K ( x )) (4)where f i ( x ) ∈ K ( x ) ∗ and ℓ i ∈ Z ≥ for all 1 ≤ i ≤ m . Note that by definition P ( n ) = 0 for all n ∈ Z ≥ .In particular, no poles arise for any evaluation at n ∈ Z ≥ . We remark that the Z -function can beused to specify the lower bounds ℓ i ∈ Z ≥ such that this property holds. Then P ( n ) in Prod n ( K ( x ))is preprocessed as follows. The first transformation is based on the following simple observation.
Proposition 1.
For P ( n ) given in (4) with multiplicands f , . . . , f m ∈ K ( x ) ∗ we have P ( n ) = n Y k = ℓ f ( k ) ! n Y k = ℓ k Y k = ℓ f ( k ) ! · · · n Y k = ℓ k Y k = ℓ · · · k m − Y k m = ℓ m f m ( k m ) ! ∈ ProdM n ( K ( x )) . (5) Definition 2.
The right hand side of (5) is also called a product factored form of P ( n ). Moreover, aproduct of the form P ′ ( n ) = n Y k = ℓ k Y k = ℓ · · · k m − Y k m = ℓ m p m ( k m )is also called a product in factored form . In particular, we also call p m ( x ) ∈ K ( x ) ∗ (instead of(1 , . . . , , p m ( x ))) the multiplicand representation of P ′ ( n ).Further, for 1 ≤ i ≤ m write f i = u i f e i, i, · · · f e i,ri i,r i ∈ K ( x ) (6)in its complete factorization. This means that f i can be decomposed by u i ∈ K ∗ and irreducible monicpolynomials f i,j ∈ K [ x ] \ K with e i,j ∈ Z for some 1 ≤ j ≤ r i with r i ∈ Z ≥ . Substituting (6) into theright-hand side of (5) and expanding the product quantifiers over each factor in (6) we get P ( n ) = A ( n ) A ( n ) · · · A m ( n ) ∈ ProdM n ( K ( x ))where A i ( n ) = n Y k = l · · · k i − Y k i = ℓ i u i ! n Y k = ℓ · · · k i − Y k i = ℓ i f i, ( k i ) ! e i, · · · n Y k = ℓ · · · k i − Y k i = ℓ i f i,r i ( k i ) ! e i,ri (7)for all 1 ≤ i ≤ m . In particular, the first product on the right hand side in (7) with innermostmultiplicand u i ∈ K ∗ is a geometric product of nesting depth i in Prod n ( K ), while the rest are nestingdepth i hypergeometric products in ProdM n ( K ( x )) which are not geometric products.4 xample 1. Let K = Q ( √
3) and K ( x ) be the rational function field over K with the Z -function (3).Suppose we are given the nesting depth 2 hypergeometric product P ( n ) = n Y k =1 k + 1 −√ k Y j =3 − j − j + 2)5 ( j − j − ∈ Prod n ( K ( x )) . (8)Then with A ( n ) = n Y k =1 − ! n Y k =1 √ ! − n Y k =1 ! n Y k =1 (cid:0) k + (cid:1)! (9) A ( n ) = n Y k =1 k Y j =3 − n Y k =1 k Y j =3 − n Y k =1 k Y j =3 n Y k =1 k Y j =3 ( j − − n Y k =1 k Y j =3 ( j − n Y k =1 k Y j =3 ( j + 1) − n Y k =1 k Y j =3 ( j + 2) (10)equation (8) can be written in the form P ( n ) = A ( n ) A ( n ) ∈ ProdM n ( K ( x ))where the multiplicand representations of the products in A ( n ) and A ( n ) are either from K or areirreducible polynomials from K [ x ]. Another transformation will guarantee that all arising products have the same lower bound, i.e.,that the expression is δ -refined for some δ ∈ Z ≥ . Definition 3.
Let K ( x ) be a rational function field over a field K and δ ∈ Z ≥ . H ( n ) ∈ ProdE n ( K ( x ))is said to be δ -refined if the lower bounds in all the arising products of H ( n ) are δ .Such a transformation of a given product expression to a δ -refined version can be accomplished bytaking δ to be the maximum of all arising lower bounds within the given expression. Example 2 (Cont. Example 1). In P ( n ) (resp. A ( n ) and A ( n ) of Example 1 we choose δ = 3.Namely, for all 1 ≤ i ≤ m , rewrite each product in (7) such that the lower bounds are synchronized to δ . More precisely we apply the formula n Y k = ℓ k Y k = ℓ · · · k i − Y k i = ℓ i h ( k i ) = δ − Y k = ℓ k Y k = ℓ · · · k i − Y k i = ℓ i h ( k i ) ! n Y k = δ δ − Y k = ℓ · · · k i − Y k i = ℓ i h ( k i ) ! n Y k = δ k Y k = δ δ − Y k = ℓ · · · k i − Y k i = ℓ i h ( k i ) ! · · · n Y k = δ k Y k = δ · · · k i − Y k i − = δ δ − Y k i = ℓ i h ( k i ) ! n Y k = δ k Y k = δ · · · k i − Y k i = δ h ( k i ) ! (11)to each of the products in (7). Note that the first product on the right-hand side in (11) evaluates to aconstant in K ∗ , the last product is from Prod n ( K ( x )), and all the remaining products (after all finitemultiplications are carried out) are from Prod n ( K ). Summarizing we obtain˜ P ( n ) = ˜ A ( n ) ˜ A ( n ) · · · ˜ A m ( n ) ∈ ProdM n ( K ( x ))5ith ˜ A i ( n ) = a i n Y k = δ ˜ u i, ! · · · n Y k = δ · · · k i − Y k i = δ ˜ u i,i ! n Y k = δ · · · k i − Y k i = δ f i, ( k i ) ! e i, · · · n Y k = δ · · · k i − Y k i = δ f i,r i ( k i ) ! e i,ri (12) where a i , ˜ u i,j ∈ K ∗ for some j ∈ Z ≥ . Since δ is chosen as the maximum among all lower bounds ofthe input expression, no poles or zero-evaluations will be introduced. As a consequence, the obtainedresult is again an element from ProdM n ( K ( x )). In particular, we have that A i ( n ) = ˜ A i ( n ) for all n ≥ max(0 , δ −
1) and consequently, P ( n ) = ˜ P ( n ) holds for all n ≥ max( δ − , Example 3 (Cont. Example 2).
Synchronizing the lower bounds of each product factor in (9)and (10) to 3 computed in Example 2 and rewriting each product factor in (9) and (10) we get˜ A ( n ) = 12253 n Y k =3 − ! n Y k =3 √ ! − n Y k =3 ! n Y k =3 (cid:0) k + (cid:1)! , (13)˜ A ( n ) = n Y k =3 k Y j =3 − n Y k =3 k Y j =3 − n Y k =3 k Y j =3 n Y k =3 k Y j =3 ( j − − n Y k =3 k Y j =3 ( j − n Y k =3 k Y j =3 ( j + 1) − n Y k =3 k Y j =3 ( j + 2) . (14)In particular, for i = 1 ,
2, and for all n ≥ δ − δ = 3, A i ( n ) = ˜ A i ( n ) holds. Consequently, with˜ P ( n ) = ˜ A ( n ) ˜ A ( n ) we have that P ( n ) = ˜ P ( n ) holds for all n ≥ P ′ ( n ) = A ′ ( n ) A ′ ( n ) · · · A ′ m ( n ) ∈ ProdM n ( K ( x ))where A ′ i ( n ) = ˜ a i n Y k =1 ˜ u i, ! · · · n Y k =1 · · · k i − Y k i =1 ˜ u i,i ! n Y k = δ · · · k i − Y k i = δ f i, ( k i ) ! e i, · · · n Y k = δ · · · k i − Y k i = δ f i,r i ( k i ) ! e i,ri (15) with ˜ a i , ˜ u i,j ∈ K ∗ for some j ∈ Z ≥ . In particular we have that, A i ( n ) = A ′ i ( n ) holds for all n ≥ max( δ − ,
0) and consequently, P ( n ) = P ′ ( n ) holds for all n ≥ max(0 , δ − P ′ ( n ) we obtain the decomposition P ′ ( n ) = c G ( n ) H ( n ) (16)with c ∈ K ∗ , G ( n ) ∈ ProdM n ( K ) is composed multiplicatively by geometric products in factored formof nesting depth at most m which are 1-refined and H ( n ) ∈ ProdM n ( K ( x )) is composed multiplicativelyby hypergeometric products (which are not geometric) in factored form of nesting depth at most m which are δ -refined. In particular, the multiplicand representations are given by monic irreduciblepolynomials. 6 xample 4 (Cont. Example 3). Synchronizing the lower bounds of each geometric product in (13)and (14) to 1 and rewriting these geometric products we get A ′ ( n ) = 1225576 n Y k =1 − ! n Y k =1 √ ! − n Y k =1 ! n Y k =3 (cid:0) k + (cid:1)! ,A ′ ( n ) = − n Y k =1 ! − n Y k =1 ! n Y k =1 k Y j =1 − n Y k =1 k Y j =1 − n Y k =1 k Y j =1 n Y k =3 k Y j =3 ( j − − n Y k =3 k Y j =3 ( j − n Y k =3 k Y j =3 ( j + 1) − n Y k =3 k Y j =3 ( j + 2) . In particular, for i = 1 ,
2, and for all n ≥ A i ( n ) = A ′ i ( n ) holds. In total we obtain P ′ ( n ) = A ′ ( n ) A ′ ( n ) = c G ( n ) H ( n )with c = − , (17) G ( n ) = n Y k =1 − ! n Y k =1 √ ! − n Y k =1 ! − n Y k =1 ! n Y k =1 ! n Y k =1 k Y j =1 − n Y k =1 k Y j =1 − n Y k =1 k Y j =1 , (18) H ( n ) = n Y k =3 (cid:0) k + (cid:1)! n Y k =3 k Y j =3 ( j − − n Y k =3 k Y j =3 ( j − n Y k =3 k Y j =3 ( j + 1) − n Y k =3 k Y j =3 ( j + 2) (19)such that P ( n ) = P ′ ( n ) holds for all n ≥ Finally, we turn our focus to the class of hypergeometric products given in factored form, and whoseinnermost multiplicands are irreducible monic polynomials. In order to reduce this class of productsfurther, we will need the following definition.
Definition 4.
Two nonzero polynomials f ( x ) and h ( x ) in the polynomial ring K [ x ] are said to be shift-coprime if for all k ∈ Z we have that gcd( f ( x ) , h ( x + k )) = 1. Furthermore, f ( x ) and h ( x ) arecalled shift-equivalent if there is a k ∈ Z such that f ( x + k ) h ( x ) ∈ K .It is immediate that the shift-equivalence in Definition 4 induces an equivalence relation on theset of all irreducible polynomials. Let D = { f , . . . , f e } ⊆ K [ x ] where all elements are irreducible andshift equivalent among each other. Then we call f i ∈ D with i ∈ { , , . . . , e } the leftmost polynomial in D if for all h ∈ D there is a k ∈ Z ≥ with f i ( x + k ) h ( x ) ∈ K . It is well known that f i ( x + k ) h ( x ) ∈ K iff k ∈ Z is a root of p ( z ) = res x ( f ( x ) , f i ( x + z )) ∈ K [ z ]; compare [PWZ96, Sec. 5.3]. In particular, if K iscomputable and one can factorize univariate polynomials over K , one can determine all integer roots of p ( z ) and thus can decide constructively if there is a k ∈ Z with f i ( x + k ) h ( x ) ∈ K . All the above properties(and slight generalizations) play a crucial role in symbolic summation; compare [Abr71, Pau95, Sch05,AP10, CFFL11]. In particular, the following simple lemma is heavily used within symbolic summation;see also [Sch05, Lemma 4.12]. 7 emma 1. Let K ( x ) be a rational function field and let f ( x ) , h ( x ) ∈ K [ x ] \ K be monic irreduciblepolynomials that are shift equivalent. Then there is a g ∈ K ( x ) ∗ with h ( x ) = g ( x +1) g ( x ) f ( x ) were all themonic irreducible factors in g are shift equivalent to f ( x ) (resp. h ( x ) ). If K is computable and one canfactorize polynomials over K , then such a g can be computed. Proof.
Since f ( x ) and h ( x ) are shift equivalent and monic, there is a k ∈ Z with f ( x + k ) = h ( x ). If k ≥
0, set g := Q k − i =0 f ( x + i ). Then g ( x + 1) g ( x ) = f ( x + k ) f ( x ) = h ( x ) f ( x ) . On the other hand, if k <
0, set g := Q − ki =1 1 f ( x + i ) . Then g ( x + 1) g ( x ) = 1 /f ( x )1 /f ( x + k ) = f ( x + k ) f ( x ) = h ( x ) f ( x ) . By construction all irreducible monic factors in g ( x ) are shift equivalent to f ( x ). Furthermore, k canbe computed if K is computable and one can factorize polynomials over K . (cid:3) Example 5.
Let K ( x ) be a rational function field as defined in Example 1. Let D be the set definedby the multiplicand representations of the products in factored form given in (19). That is, D = { f ( x ) , f ( x ) , . . . , f ( x ) } ⊆ K [ x ] \ K where f ( x ) = x − , f ( x ) = x − , f ( x ) = x + 1 , f ( x ) = x + 2 and f ( x ) = x + . (20)Since f ( x ) is shift equivalent with f ( x ) , f ( x ) , f ( x ), i.e., f ( x + 1) = f ( x ) , f ( x + 3) = f ( x ) , f ( x + 4) = f ( x ) , they fall into the same equivalence class E = { f ( x ) , f ( x ) , f ( x ) , f ( x ) } . The other equivalence classis E = { f ( x ) } . For each of these equivalence classes E and E , take their leftmost elements: f ( x )and f ( x ) respectively. Then by Lemma 1, we can express the elements of each equivalence class interms of the leftmost polynomial f ( x ) or f ( x ). More precisely, we have the following relations forthe equivalence class E : f ( x ) = g ( x + 1) g ( x ) f ( x ) , with g ( x ) = ( x − , (21) f ( x ) = g ( x + 1) g ( x ) f ( x ) , with g ( x ) = ( x −
2) ( x − x, (22) f ( x ) = g ( x + 1) g ( x ) f ( x ) , with g ( x ) = ( x −
2) ( x − x ( x + 1) . (23)Finally, we reduce each component of the hypergeometric product expression H ( n ) given by (19).We will begin with the nesting depth 2 hypergeometric products in factored form whose innermostmultiplicand corresponds to the polynomial f ( x ). Using (23) the product in factored form reduces asfollows: n Y k =3 k Y j =3 f ( j ) = n Y k =3 k Y j =3 ( j + 2) = n Y k =3 k Y j =3 g ( j + 1) g ( j ) ! n Y k =3 k Y j =3 f ( j ) = n Y k =3 g ( k + 1) g (3) ! n Y k =3 k Y j =3 f ( j ) ! = n Y k =3 ! n Y k =3 ( k − ! n Y k =3 k ! n Y k =3 ( k + 1) ! n Y k =3 ( k + 2) ! n Y k =3 k Y j =3 ( j − ! = 576 n Y k =1 ! n Y k =3 ( k − ! n Y k =3 k ! n Y k =3 ( k + 1) ! n Y k =3 ( k + 2) ! n Y k =3 k Y j =3 ( j − ! . (24) H ( n ) whose innermost multiplicands correspond to the polynomials f ( x ) and f ( x ) respectively. In particular, we have the following: n Y k =3 k Y j =3 f ( j ) = n Y k =3 k Y j =3 ( j + 1) = 36 n Y k =1 ! n Y k =3 ( k − ! n Y k =3 k ! n Y k =3 ( k + 1) ! n Y k =3 k Y j =3 ( j − , (25) n Y k =3 k Y j =3 f ( j ) = n Y k =3 k Y j =3 ( j −
1) = n Y k =3 ( k − ! n Y k =3 k Y j =3 ( j − . (26) Remark 1.
Suppose we are given an expression A ( n ) ∈ ProdE n ( K ( x )) (e.g., given in (2)) in termsof δ -refined hypergeometric products of finite nesting depth in factored form where all multiplicandrepresentations are irreducible monic polynomials. Choose P ′ ( n ) = n Y k = δ · · · k m − Y k m − = δ k m − Y k m = δ h ( k m ) (27)from A ( n ) with the multiplicand representation h ( x ) ∈ K [ x ]. Furthermore, among all shift-equivalentmultiplicand representations within the given product expression A ( n ), let f ( x ) be the leftmost poly-nomial which lies in the same equivalence class with h ( x ). By assumption f ( x ) and h ( x ) are monicirreducible with f ( n ) = 0 and h ( n ) = 0 for all n ≥ δ . Take k ∈ Z ≥ with h ( x + k ) = f ( x ). Then byLemma 1 we can take g := Q k − i =0 f ( x + i ) ∈ K [ x ] such that h ( x ) = g ( x +1) g ( x ) f ( x ) holds. Thus P ′ ( n ) = n Y k = δ · · · k m − Y k m − = δ k m − Y k m = δ g ( k m + 1) g ( k m ) f ( k m ) = n Y k = δ k Y k = δ · · · k m − Y k m − = δ g ( k m − + 1) g ( δ ) k m − Y k m = δ f ( k m )= n Y k = δ · · · k m − Y k m − = δ g ( δ ) − | {z } = G ′ ( n ) n Y k = δ · · · k m − Y k m − = δ g ( k m − + 1) | {z } = H ′ ( n ) n Y k = δ · · · k m − Y k m = δ f ( k m ) . Note that this reduction of a product of nesting depth m leads to a new hypergeometric product H ′ ( n )in factored form of nesting depth less than m where for the multiplicand representation h ′ ( x ) := g ( x +1)we have that h ′ ( n ) = 0 for all n ≥ max( δ − , h ′ ( x ) consists of monic irreduciblefactors which are again shift-equivalent to f ( x ). In addition taking all these new factors togetherwith f ( x ), it follows that f ( x ) remains the leftmost polynomial factor. Thus repeating the steps inSubsection 2.1 to H ′ ( n ) yields again products of the form (27) with nesting depth m − f ( x ) is still theleftmost polynomial.Note further that also the new geometric product G ′ ( n ) occurs with lower bound δ . In order to turnit to a 1-refined product, we may apply the transformations introduced in Section 2.2.Finally, observe that in the special case m = 1, we get R ( n ) = H ′ ( n ) G ′ ( n ) = g ( n + 1) g ( δ ) ∈ K [ n ] . Since the product P ′ ( n ) itself might arise in the expression under consideration in the form P ′ ( n ) z with z ∈ Z , we might introduce the factor R ( n ) z in the final expression. However, since R ( n ) = 0 for all n ≥ max( δ − , n ≥ max( δ − , n ≥ max( δ − , xample 6 (Cont. of Ex. 5). After reducing all nesting depth 2 hypergeometric products in fac-tored form in the expression H ( n ), the new polynomial f ( x ) = x emerges. It falls into the equivalenceclass E , and the leftmost polynomial of this equivalence class remains unchanged. We get f ( x ) = g ( x + 1) g ( x ) f ( x ) with g ( x ) = ( x −
2) ( x − . (28)by Lemma 1. Using the relations (23), (22), (28), and (21) we can reduce all nesting depth 1 hyper-geometric products whose multiplicand representations are f ( x ), f ( x ), f ( x ), and f ( x ) respectively.More precisely we have the following: n Y k =3 f ( k ) = n Y k =3 ( k + 2) = ( n − n ( n + 1) ( n + 2)24 n Y k =3 ( k −
2) (29) n Y k =3 f ( k ) = n Y k =3 ( k + 1) = ( n − n ( n + 1)6 n Y k =3 ( k −
2) (30) n Y k =3 f ( k ) = n Y k =3 k = ( n − n n Y k =3 ( k −
2) (31) n Y k =3 f ( k ) = n Y k =3 ( k −
1) = ( n − n Y k =3 ( k − . (32)Substituting (29), (30), (31), and (32) into (24), (25), and (26) and afterwards into the expression (19)gives ˆ H ( n ) = 23 ( n − n ( n + 1) ( n + 2) n Y k =1 ! − n Y k =1 ! n Y k =3 ( k − ! n Y k =3 (cid:0) k + (cid:1)! n Y k =3 k Y j =3 ( j − . (33) Note that H ( n ) = ˆ H ( n ) for all n ≥
2. Furthermore, the distinct irreducible monic polynomials: ( x − x + ), that corresponds to the distinct innermost multiplicands of the products in factored formin ˆ H ( n ) are shift-coprime among each other. Putting (17) and (18) in Example 4 and (33) together,we have that P ( n ) = ˜ P ( n ) = ˜ c ˜ r ( n ) ˜ G ( n ) ˜ H ( n ) (34)holds for all n ∈ Z ≥ with n ≥
2, where the components of ˜ P ( n ) are as follows:˜ c = − , (35)˜ r ( n ) = ( n − n ( n + 1) ( n + 2) , (36)˜ G ( n ) = n Y k =1 − ! n Y k =1 √ ! − n Y k =1 ! − n Y k =1 ! n Y k =1 ! n Y k =1 k Y j =1 − n Y k =1 k Y j =1 − n Y k =1 k Y j =1 , (37)˜ H ( n ) = n Y k =3 ( k − ! n Y k =3 (cid:0) k + (cid:1)! n Y k =3 k Y j =3 ( j − . (38)For further considerations the following definition will be convenient. Definition 5.
Let K ( x ) be a rational function field and let H ( n ) , . . . , H e ( n ) ∈ ProdM n ( K ( x )) wherethe arising hypergeometric products are in factored form. We say that H ( n ) , . . . , H e ( n ) are in shift-coprime product representation form if 101) the multiplicand representation of each product in H i ( n ) for 1 ≤ i ≤ e is an irreducible monicpolynomial in K [ x ] \ K ;(2) the distinct multiplicand representations in H ( n ) , . . . , H e ( n ) are shift-coprime among eachother.Then the above symbolic manipulations can be summarized by the following method. Remark 2.
We are given H ( n ) , . . . , H e ( n ) in ProdM n ( K ( x )) where the products are in factored formand are all δ -refined for some δ ∈ Z ≥ . In particular, suppose that all multiplicand representationsare monic and that each H i ( n ) can be evaluated for n ≥ ν for some ν ∈ Z ≥ with ν ≥ δ ; note thatsuch a ν can be derived by applying the available Z -function to each rational function factor of H i ( n )and taking its maximum value. Then one can follow the steps below to rewrite H ( n ) , . . . , H e ( n ) in ashift-coprime product representation form that yield the same evaluations for all n ≥ ν .(1) Factor the innermost multiplicand representations of all products in H ( n ) , . . . , H e ( n ) into ir-reducible monic polynomials in K [ x ] \ K as in (6), and expand the product quantifier over thefactorization as in (7). Let D be the set of all the irreducible monic polynomials.(2) Among all the irreducible monic polynomials in D ⊆ K [ x ] \ K , compute the shift equivalence classessay, E , . . . , E v with respect to the automorphism σ ( x ) = x + 1, and let R be the set of the leftmostpolynomial of each equivalent class. Thus, the elements of the set R are shift-coprime among eachother and each element represents exactly one equivalence class.(3) Among all the products over the irreducible monic polynomials obtained in step (1), take thosewith the highest nesting depth and reduce them with the elements in R following the constructionof Remark 1. During this rewriting one also obtains extra constants, geometric products andrational expressions from K ( n ) that are collected accordingly.(4) Go to step (1) and update the corresponding product expressions until the multiplicand repre-sentations of all products are in R .In particular, if K is computable and one can factorize polynomials over K , all the above steps can becarried out explicitly.Summarizing, given { P ( n ) , . . . , P e ( n ) } ⊆ Prod n ( K ( x )), we can bring each P i ( n ) to the form (16)with P ′ i ( n ) := c i G i ( n ) H i ( n ) with c i ∈ K ∗ , G i ( n ) ∈ ProdM n ( K ) and H i ( n ) ∈ ProdM n ( K ( x )) such that P i ( n ) = P ′ i ( n ) holds for all n ≥ max(0 , δ − H ( n ) , . . . , H e ( n ) we get the output ˜ H ( n ) , . . . , ˜ H e ( n ) where the c i and H i ( n ) are correspondinglyupdated to ˜ c i and ˜ H i ( n ) within Step 3 of Remark 2. In particular, we obtain for each component anextra factor ˜ r i ( x ) ∈ K ( x ) where for n ∈ Z ≥ with n ≥ max(0 , δ − r i ( n ). Afterwards, another synchronization will be necessary to bring the new geometricproducts in H i ( n ) to 1-refined form (by again updating the ˜ c i accordingly). The final result can besummarized in the following proposition. Proposition 2.
Let K ( x ) be a rational function field and suppose that we are given the hypergeometricproducts { P ( n ) , . . . , P e ( n ) } ⊆ Prod n ( K ( x )) of nesting depth at most d ∈ Z ≥ . Then there is a δ ∈ Z ≥ and there are(1) ˜ c , . . . , ˜ c e ∈ K ∗ ;(2) for all ≤ ℓ ≤ e rational functions r ℓ ( x ) ∈ K ( x ) ∗ ; As observed in Remark 1 the set R of the leftmost polynomials in step (2) does not change
3) geometric product expressions ˜ G ( n ) , . . . , ˜ G e ( n ) ∈ ProdM n ( K ) which are all -refined;(4) hypergeometric product expressions ˜ H ( n ) , . . . , ˜ H e ( n ) ∈ ProdM n ( K ( x )) which are δ -refined andare in shift-coprime product representation formsuch that for ≤ ℓ ≤ e and for all n ≥ max(0 , δ − we have P ℓ ( n ) = ˜ c ℓ ˜ r ℓ ( n ) ˜ G ℓ ( n ) ˜ H ℓ ( n ) = 0 . (39) If K is computable and one can factorize polynomials in K , then δ and the above representation can becomputed. From now on, we assume that the arising hypergeometric products P ( n ) , . . . , P e ( n ) ∈ Prod n ( K ( x ))have undergone the preprocessing steps discussed above yielding the representation given in (39). Ingeneral, there are still algebraic relations among the products that occur in the derived expressions (39)with 1 ≤ ℓ ≤ e , i.e., statements (2) and (3) of Problem RPE do not hold yet. In order to accomplishthis task, extra insight from difference ring theory will be utilized. More precisely, we will showthat the hypergeometric products coming from the ˜ H l are already algebraically independent, but therepresentation of the geometric products have to improved to establish a solution of Problem RPE.
3. A naive difference ring approach: AP-extensions
Inspired by [Kar81, Sch16, Sch17], this Section focuses on an algebraic setting of difference rings(resp. fields) in which expressions of ProdE n ( K ( x )) can be naturally rephrased. A difference ring (resp. field ) ( A , σ ) is a ring (resp. field) A together with a ring (resp. field)automorphism σ : A → A . Subsequently, all rings (resp. fields) are commutative with unity; inaddition they contain the set of rational numbers Q , as a subring (resp. subfield). The multiplicativegroup of units of a ring (resp. field) A is denoted by A ∗ . A difference ring (resp. field) ( A , σ ) is called computable if A and σ are both computable. In the following we will introduce AP-extensions thatwill be the foundation to represent hypergeometric products of finite nesting depth in difference rings.A-extensions will be used to cover objects like ζ k where ζ is a root of unity. In general, let ( A , σ )be a difference ring and let ζ ∈ A ∗ be a λ -th root of unity with λ > λ ∈ Z ≥ with ζ λ = 1).Take the uniquely determined difference ring extension ( A [ y ] , σ ) of ( A , σ ) where y is transcendentalover A and σ ( y ) = ζ y . Now consider the ideal I := (cid:10) y λ − (cid:11) and the quotient ring E := A [ y ] /I . Since I is closed under σ and σ − i.e., I is a reflexive difference ideal, we can define the map σ : E → E with σ ( h + I ) = σ ( h ) + I which forms a ring automorphism. Note that by this construction the ring A can naturally be embedded into the ring E by identifying a ∈ A with a + I ∈ E , i.e., a a + I .Now set ϑ := y + I . Then ( A [ ϑ ] , σ ) is a difference ring extension of ( A , σ ) subject to the relations ϑ λ = 1 and σ ( ϑ ) = ζ ϑ . This extension is called an algebraic extension (in short A -extension ) of order λ . The generator ϑ is called an A -monomial with its order λ = min { n > | ϑ n = 1 } . Note that thering A [ ϑ ] is not an integral domain (i.e., it has zero-divisors) since ( ϑ −
1) ( ϑ λ − + · · · + ϑ + 1) = 0 but( ϑ − = 0 = ( ϑ λ − + · · · + ϑ + 1). In this setting, the A-monomial ϑ with the relations ϑ λ = 1 and σ ( ϑ ) = ζ ϑ with ζ := e π i λ = ( − λ , models ζ k subject to the relations ( ζ k ) λ = 1 and ζ k +1 = ζ ζ k .In addition, we define P-extensions in order to treat products of finite nesting depth whose multi-plicands are not given by roots of unity. Let ( A , σ ) be a difference ring, α ∈ A ∗ be a unit, and considerthe ring of Laurent polynomials A [ t, t − ] (i.e., t is transcendental over A ). Then there is a uniquedifference ring extension ( A [ t, t − ] , σ ) of ( A , σ ) with σ ( t ) = α t and σ ( t − ) = α − t − . The extensionhere is called a product-extension (in short P -extension ) and the generator t is called a P -monomial .We introduce the following notations for convenience. Let ( E , σ ) be a difference ring extension of( A , σ ) with t ∈ E . A h t i denotes the ring of Laurent polynomials A [ t, t ] (i.e., t is transcendental over12 ) if ( A [ t, t ] , σ ) is a P-extension of ( A , σ ). Lastly, A h t i denotes the ring A [ t ] with t / ∈ A but subjectto the relation t λ = 1 if ( A [ t ] , σ ) is an A-extension of ( A , σ ) of order λ .We say that the difference ring extension ( A h t i , σ ) of ( A , σ ) is an AP-extension (and t is an AP-monomial) if it is an A- or a P-extension. Finally, we call ( A h t i . . . h t e i , σ ) a (nested) A-/P-/AP-extension of ( A , σ ) it is built by a tower of such extensions.In the following we will restrict to the subclass of ordered simple AP-extension. Here, the followingdefinitions are useful. Definition 6.
Let ( E , σ ) be a (nested) AP-extension of ( A , σ ) with E = A h t i . . . h t e i where σ ( t i ) = α i t i for 1 ≤ i ≤ e . We define the depth function of elements of E over A , d : E → Z ≥ as follows:(1) For any h ∈ A , d A ( h ) = 0.(2) If d A is defined for ( A h t i . . . h t i − i , σ ) with i >
1, then we define d A ( t i ) := d A ( α i ) + 1 and for f ∈ A h t i . . . h t i i , we define d A ( f ) := max (cid:0) { d A ( t i ) | t i occurs in f } ∪ { } (cid:1) .The extension depth of ( E , σ ) over A is given by d A ( E ) := (cid:0) d A ( t ) , . . . , d A ( t e ) (cid:1) . We call such anextension ordered , if d A ( t ) ≤ d A ( t ) ≤ · · · ≤ d A ( t e ). In particular, we say that ( E , σ ) is of monomialdepth m if m = max(0 , d A ( t ) , . . . , d A ( t e )). If A is clear from the context, we write d A as d .Now, let ( E , σ ) with E = A h t i . . . h t e i be a nested A-/P-/AP-extension of a difference ring ( A , σ )and let G be a multiplicative subgroup of A ∗ . Following [Sch16, Sch17] we call G EA := { g t v · · · t v e e | g ∈ G, and v i ∈ Z } (40)the product group over G with respect to A-/P-/AP-monomials for the nested A-/P-/AP-extension( E , σ ) of ( A , σ ). In the following we will restrict ourselves to the following subclass of AP-extensions. Definition 7.
Let ( A , σ ) be a difference ring and let G be a subgroup of A ∗ . Let ( E , σ ) be an A-/P-/AP-extension of ( A , σ ) with E = A h t i . . . h t e i . Then this extension is called G -simple if for all1 ≤ i ≤ e , σ ( t i ) t i ∈ G A h t i ... h t i − i A . In addition such a G -simple extension is called G -basic , if for any A-monomial t i we have σ ( t i ) t i ∈ const( A , σ ) ∗ and for any P-monomial t i we have that σ ( t i ) t i ∈ G A h t i ... h t i − i A is free of A-monomials. If G = A ∗ , such extensions are also called simple (resp. basic ) instead of A ∗ -simple ( A ∗ -basic).In particular, we will work with the following class of simple A-/P-/AP-extensions that are closelyrelated to the products in factored form given in (5); for concrete constructions see Example 7 below. Definition 8.
Let ( A , σ ) be a difference ring and G be a subgroup of A ∗ . We call ( A h t i . . . h t e i , σ ) a single chain A -/ P -/ AP -extension of ( A , σ ) over G if for all 1 ≤ k ≤ e , σ ( t k ) = c k t · · · t k − t k , with c k ∈ G . We call c also the base of the single chain A-/P-/AP-extension. If G = A ∗ , we also say that( A h t i . . . h t e i , σ ) is a single chain A -/ P -/ AP -extension of ( A , σ ). Further, we call ( E , σ ) a multi-ple chain A -/ P -/ AP -extension of ( A , σ ) over G with base ( c , . . . , c m ) ∈ G m if it is a tower of m singlechain A-/P-extensions over G with the bases c , . . . , c m , respectively. If G = A ∗ , we simply call it amultiple chain A-/P-/AP-extension. In other words, products whose multiplicands are roots of unity have nesting depth 1 (and the roots of unity arefrom the constant field), whiles the remaining products do not depend on these products over roots of unity. emark 3. Let ( A h t i . . . h t e i , σ ) be a single chain A-/P-/AP-extension of ( A , σ ) as given in Defini-tion 8 and let d : A h t i . . . h t e i → Z ≥ be the depth function over A . Then we have d ( t k ) = k for all1 ≤ k ≤ e . In particular, the extension is ordered, its extension-depth is (1 , , . . . , e ) and the monomialdepth is e . Furthermore observe that for 2 ≤ i ≤ e we have σ ( t i ) = σ ( t i − ) t i ⇔ t i σ − ( t i ) = t i − . For a field K we denote by K Z ≥ the set of all sequences (cid:10) a ( n ) (cid:11) n ≥ = h a (0) , a (1) , a (2) , . . . i (41)whose terms are in K . Equipping K Z ≥ with component-wise addition and multiplication, we get acommutative ring. In this ring, the field K can be naturally embedded into K Z ≥ as a subring, byidentifying any c ∈ K with the constant sequence h c, c, c, . . . i ∈ K Z ≥ . Following the construction in[PWZ96, Section 8.2], we turn the shift operator S : K Z ≥ → K Z ≥ with S : h a (0) , a (1) , a (2) , . . . i 7→ h a (1) , a (2) , a (3) , . . . i (42)into a ring automorphism by introducing an equivalence relation ∼ on sequences in K Z ≥ . Two se-quences A = (cid:10) a ( n ) (cid:11) n ≥ and B = (cid:10) b ( n ) (cid:11) n ≥ are said to be equivalent (in short A ∼ B ) if and only ifthere exists a non-negative integer δ such that ∀ n ≥ δ : a ( n ) = b ( n ) . The set of equivalence classes form a ring again with component-wise addition and multiplication whichwe will denote by S ( K ) := K Z ≥ (cid:30) ∼ . Now it is obvious that S : S ( K ) → S ( K ) with (42) is bijectiveand thus a ring automorphism. We call ( S ( K ) , S ) also the difference ring of sequences over K . Forsimplicity, we denote the elements of S ( K ) by the usual sequence notation as in (41) above.We will follow the convention introduced in [PS19] to illustrate how the indefinite products of finitenesting depth covered in this article are modelled by expressions in a difference ring. Definition 9.
Let ( A , σ ) be a difference ring with a constant field K = const( A , σ ). An evaluationfunction ev : A × Z ≥ → K for ( A , σ ) is a function which satisfies the following three properties:(i) for all c ∈ K , there is a natural number δ ≥ ∀ n ≥ δ : ev( c, n ) = c ; (43)(ii) for all f, g ∈ A there is a natural number δ ≥ ∀ n ≥ δ : ev( f g, n ) = ev( f, n ) ev( g, n ) , ∀ n ≥ δ : ev( f + g, n ) = ev( f, n ) + ev( g, n ); (44)(iii) for all f ∈ A and i ∈ Z , there is a natural number δ ≥ ∀ n ≥ δ : ev( σ i ( f ) , n ) = ev( f, n + i ) . (45)We say a sequence (cid:10) F ( n ) (cid:11) n ≥ ∈ S ( K ) is modelled by f ∈ A in the difference ring ( A , σ ), if there is anevaluation function ev such that F ( k ) = ev( f, k )holds for all k ∈ Z ≥ from a certain point on. 14n this article, our base field is a rational function field K ( x ) which is equipped with the evaluationfunction ev : K ( x ) × Z ≥ → K defined as follows. For f = gh ∈ K ( x ) with h = 0 where g and h arecoprime (if g = 0 we take h = 1) we haveev( f, k ) := ( h ( k ) = 0 , g ( k ) h ( k ) if h ( k ) = 0 . (46)Here, g ( k ) and h ( k ) are the usual polynomial evaluation at some natural number k .Then given a tower of AP-extension defined over ( K ( x ) , σ ), one can define an appropriate evaluationfunction by iterative applications of the following lemma that is implied by [Sch17, Lemma 5.4]. Lemma 2.
Let ( A , σ ) be a difference ring with constant field K and let ev : A × Z ≥ → K be anevaluation function for ( A , σ ) . Let ( A h t i , σ ) be an AP -extension of ( A , σ ) with σ ( t ) = α t ( α ∈ A ∗ ) andsuppose that there is a δ ∈ Z ≥ such that ev( α, n ) = 0 for all n ≥ δ . Further, take u ∈ K ∗ ; if t λ = 1 for some λ > , we further assume that u λ = 1 holds. Consider the map ev ′ : A h t i × Z ≥ → K definedby ev ′ ( X i h i t i , n ) = X i ev( h i , n ) ev ′ ( t, n ) i with ev ′ ( t, n ) = u Q nk = δ ev( α, k − . Then ev ′ is an evaluation function for ( A h t i , σ ) . We summarize the above constructions with the following example.
Example 7.
Let K = Q ( √
3) and take the difference field ( K ( x ) , σ ) where the automorphism is definedby σ ( x ) = x + 1 and σ | K = id. Furthermore, take the evaluation function ev : K ( x ) × Z ≥ → K givenby (46). Then we can construct the following single chain extensions of ( K ( x ) , σ ) in order to modelthe geometric product ˜ G ( n ) and the hypergeometric product ˜ H ( n ) given in (37) and (38).(1) We define the single chain A-extension ( K ( x ) h ϑ , ih ϑ , i , σ ) of ( K ( x ) , σ ) over K of order 2, basedat − σ ( ϑ , ) = − ϑ , , σ ( ϑ , ) = − ϑ , ϑ , . (47)In addition by applying Lemma 2 twice we extend the evaluation function to ev : K ( x ) h ϑ , ih ϑ , i× Z ≥ → K with ev( ϑ , , n ) = n Y k =1 − , ev( ϑ , , n ) = n Y k =1 k Y j =1 − . (48)(2) Similarly, define the single chain P-extension ( K ( x ) h y , i , σ ) of ( K ( x ) , σ ) over K based at √ K ( x ) h y , i × Z ≥ → K (using Lemma 2) by σ ( y , ) = √ y , , and ev( y , , n ) = n Y k =1 √ . (49)(3) Define the single chain P-extension ( K ( x ) h y , ih y , i , σ ) of ( K ( x ) , σ ) over K based at 2 equippedwith the evaluation function ev : K ( x ) h y , ih y , i × Z ≥ → K by σ ( y , ) = 2 y , , σ ( y , ) = 2 y , y , , andev( y , , n ) = n Y k =1 , ev( y , , n ) = n Y k =1 k Y j =1 . (50)154) Define the single chain P-extension ( K ( x ) h y , i , σ ) of ( K ( x ) , σ ) over K based at 3 together withthe evaluation function ev : K ( x ) h y , i × Z ≥ → K by σ ( y , ) = 3 y , , and ev( y , , n ) = n Y k =1 . (51)(5) Define the single chain P-extension ( K ( x ) h y , ih y , i , σ ) of ( K ( x ) , σ ) over K with 5 as its baseaccompanied with the evaluation function ev : K ( x ) h y , ih y , i × Z ≥ → K by σ ( y , ) = 5 y , , σ ( y , ) = 5 y , y , , andev( y , , n ) = n Y k =1 , ev( y , , n ) = n Y k =1 k Y j =1 . (52)(6) Define the single chain P-extension ( K ( x ) h y , i , σ ) of ( K ( x ) , σ ) over K with 25 as its base togetherwith the evaluation function ev : K ( x ) h y , i × Z ≥ → K by σ ( y , ) = 25 y , , and ev( y , , n ) = n Y k =1 . (53)(7) Define the single chain P-extension ( K ( x ) h z , ih z , i , σ ) of ( K ( x ) , σ ) over K ( x ) based at ( x − K ( x ) h z , ih z , i × Z ≥ → K by σ ( z , ) = ( x − z , , σ ( z , ) = ( x − z , z , , andev( z , , n ) = n Y k =3 ( k − , ev( z , , n ) = n Y k =3 k Y j =3 ( j − . (54)(8) Define the single chain P-extension ( K ( x ) h z , i , σ ) of ( K ( x ) , σ ) over K ( x ) with (cid:0) x + (cid:1) as itsbase together with the evaluation function ev : K ( x ) h z , i × Z ≥ → K by σ ( z , ) = (cid:0) x + (cid:1) z , , and ev( z , , n ) = n Y k =3 (cid:0) k + (cid:1) . (55)Putting everything together, we have constructed the multiple chain AP-extension ( A , σ ) of ( K ( x ) , σ )with A = K ( x ) h ϑ , ih ϑ , ih y , ih y , ih y , ih y , ih y , ih y , ih y , ih z , ih z , ih z , i (56)based at (cid:0) − , √ , , , , , , , x − , x − , x + (cid:1) where (1 , , , , , , , , , , ,
1) is theextension depth. In this ring, the geometric product ˜ G ( n ) and the hypergeometric product ˜ H ( n )defined in (37) and (38) are modelled by g = ϑ , y , y , ϑ , y , y , y , y , and h = z , z , z , (57)respectively. That is, ˜ G ( n ) = ev( g, n ) holds for all n ≥ H ( n ) = ev( h, n ) holds for all n ≥
2. Asa consequence, the indefinite hypergeometric product expression ˜ P ( n ) defined in (34) is modelled bythe expression ˜ p = 254 ( x − x ( x + 1) ( x + 2)432 g h ∈ A . (58)This means that ˜ P ( n ) = ev(˜ p, n ) holds for all n ∈ Z ≥ with n ≥ emark 4. We can carry out the following construction to model the given hypergeometric products P ( n ) , . . . , P e ( n ) ∈ Prod n ( K ( x )) of finite nesting depth and an expression in terms of these productsin a P-extension . Here we start with the difference field ( K ( x ) , σ ) given by σ | K = id and σ ( x ) = x + 1which is equipped with the evaluation function ev given in (46) and the zero-function (3).(1) For 1 ≤ i ≤ e , rewrite P i ( n ) such that it is composed multiplicatively by products in factoredform and such that all products are δ -refined (in its strongest form, one may use the represen-tation given in Proposition 2). Let P be the set of all products that occur in the rewritten P i ( n ).(2) Among all nested products of P with the same multiplicand representation c ( x ) ∈ K ( x ), takeone of the products, say F ( n ), with the highest nesting depth m . Construct the correspondingsingle chain P-extension ( K ( x ) h p i . . . h p m i , σ ) of ( K ( x ) , σ ) over K ( x ) and extend the evaluationfunction accordingly such that the outermost P-monomial p m models F ( n ) with ev( p µ , n ) = F ( n )for all n ≥ δ . In particular, any arising product with the same multiplicand representation anddepth µ < m is modelled by p µ . Thus we can remove all the products from P which have thesame multiplicand c ( x ).(3) Repeat step (2) for the remaining elements of P .(4) Combine the constructed single chain P-extensions of ( K ( x ) , σ ) over K ( x ) to obtain a multiplechain P-extension ( A , σ ) of ( K ( x ) , σ ) over K ( x ). In addition, combine the evaluation functionsto one extended version.In this way, all products arising in the rewritten P ( n ) , . . . , P e ( n ) can be modelled by a P-monomialwithin the constructed difference ring ( A , σ ). Furthermore, consider any expression A ( n ) of the form (2)in terms of the original products P ( n ) , . . . , P e ( n ) and let λ = max { Z ( a v ) | v ∈ S } , i.e., the evaluation a v ( n ) does not introduce poles for any n ∈ Z ≥ with n ≥ λ . Then replacing the rewritten productsin A ( n ) and afterwards replacing the involved products by the corresponding P-monomials yields agiven a ∈ A with ev( a, n ) = A ( n ) for all n ≥ max( δ, λ ). In particular, if K is computable and thezero-function Z is computable, this construction can be given explicitly.
4. A refined difference ring approach: RΠ-extensions
In general, the naive construction of an (ordered) multiple chain P-extension ( A , σ ) of ( K ( x ) , σ )following Remark 4 or a slightly refined construction of an AP-extension like in Example 7 introducealgebraic relations among the monomials. In order to tackle Problem RPE above, we will refine AP-extensions further to the class of RΠ-extensions. In this regard, the set of constantsconst( A , σ ) = { c ∈ A | σ ( c ) = c } of a difference ring (field) ( A , σ ) plays a decisive role. In general it forms a subring (subfield) of A which contains the rational numbers Q as subfield. In this article, const( A , σ ) will always be a fieldalso called the constant field of ( A , σ ), which we will also denote by K . We note further that one candecide if c ∈ A is a constant if ( A , σ ) is computable.We are now ready to refine AP-extensions as follows. Definition 10.
Let ( A h t i , σ ) be an A-/P-/AP-extension of ( A , σ ). Then it is called an R -/ Π -/ RΠ -extension if const( A h t i , σ ) = const( A , σ ) holds. Depending on the type of extension, we call thegenerator t an R -/ Π -/ RΠ -monomial , respectively. A (nested) A-/P-/AP-extension ( A h t i . . . h t e i , σ )of ( A , σ ) with const( A h t i . . . h t e i , σ ) = const( A , σ ) is also called a (nested) R -/ Π -/ RΠ –extension . Similarly, one can carry out such a construction for A-extensions or AP-extensions by checking in addition if thearising multiplicands are built by roots of unity.
Theorem 1.
Let ( A , σ ) be a difference ring. Then the following statements hold.1. Let ( A [ t, t ] , σ ) be a P -extension of ( A , σ ) with σ ( t ) = α t where α ∈ A ∗ . Then this is a Π -extension (i.e., const( A [ t, t ] , σ ) = const( A , σ ) ) iff there are no g ∈ A \ { } and v ∈ Z \ { } with σ ( g ) = α v g .2. Let ( A [ ϑ ] , σ ) be an A -extension of ( A , σ ) of order λ > with σ ( ϑ ) = ζ ϑ where ζ ∈ A ∗ . Thenthis is an R -extension (i.e., const( A [ ϑ ] , σ ) = const( A , σ ) ) iff there are no g ∈ A \ { } and v ∈ { , . . . , λ − } with σ ( g ) = ζ v g . If it is an R -extension, ζ is a primitive λ -th root of unity. We remark that the above definitions and also Theorem 1 are inspired by Karr’s ΠΣ-field exten-sions [Kar81, Sch01]. Since we will use this notion later (see Theorem 4 and 5 below) we will introducethem already here.
Definition 11.
Let ( F ( t ) , σ ) be a difference field extension of a difference field ( F , σ ) with t tran-scendental over F and σ ( t ) = α t + β with α ∈ F ∗ and β ∈ F . This extension is called a Σ -fieldextension if α = 1 and const( F ( t ) , σ ) = const( F , σ ), and it is called a Π -field extension if β = 0and const( F ( t ) , σ ) = const( F , σ ). A difference field ( K ( t ) . . . ( t e ) , σ ) is called a ΠΣ -field over K if( K ( t ) . . . ( t i ) , σ ) is a ΠΣ-extension of ( K ( t ) . . . ( t i − ) , σ ) for 1 ≤ i ≤ e with const( K , σ ) = K .Throughout this article, our base case difference field is ( K ( x ) , σ ) with the automorphism σ ( x ) = x + 1 and σ | K = id which in fact is a Σ-extension of ( K , σ ), i.e., const( K ( x ) , σ ) = K . In particular( K ( x ) , σ ) is a ΠΣ-field over K . We conclude this subsection by observing that the check if an A-extension is an R-extension (see part 2 of Theorem 1) is not necessary if the ground field is a ΠΣ-field;compare [OS18, Lemma 2.1]. Lemma 3.
Let ( F , σ ) be a ΠΣ -field over K . Then any A -extension ( F [ ϑ ] , σ ) of ( F , σ ) with order λ > is an R -extension.4.1. Embedding into the ring of sequences In this subsection, we will discuss the connection between RΠ-extensions and the difference ringof sequences. More precisely, we will show how RΠ-extensions can be embedded into the differencering of sequences [Sch17]; compare also [PS97]. This feature will enable us to handle condition (2)of Problem RPE in the sections below.
Definition 12.
Let ( A , σ ) and ( A ′ , σ ′ ) be two difference rings. The map τ : A → A ′ is called a dif-ference ring homomorphism if τ is a ring homomorphism, and for all f ∈ A , τ ( σ ( f )) = σ ′ ( τ ( f )). If τ is injective, then it is called a difference ring monomorphism or a difference ring embedding . If τ is a bijection, then it is a difference ring isomorphism and we say ( A , σ ) and ( A ′ , σ ′ ) are isomorphic;we write ( A , σ ) ≃ ( A ′ , σ ′ ). Let ( E , σ ) and (˜ E , ˜ σ ) be difference ring extensions of ( A , σ ). Then a differ-ence ring-homomorphism/isomorphism/monomorphism τ : E → ˜ E is called an A -homomorphism / A -isomorphism / A -monomorphism , if τ | A = id.Let ( A , σ ) be a difference ring with constant field K . A difference ring homomorphism (resp. monomor-phism) τ : A → S ( K ) is called K - homomorphism (resp. - monomorphism ) if for all c ∈ K we have that τ ( c ) = c := h c, c, c, . . . i .The following lemmas provide the key property that will enable us to embed RΠ-extensions intothe ring of sequences. First, we recall that the evaluation function of a difference ring establishesnaturally a K -homomorphism. More precisely, by [Sch01, Lemma 2.5.1] we get In this case, ( τ ( A ) , σ ) is a sub-difference ring of ( A ′ , σ ′ ) where ( A , σ ) and ( τ ( A ) , σ ) are the same up to renamingwith respect to τ . emma 4. Let ( A , σ ) be a difference ring with constant field K . Then the map τ : A → S ( K ) is a K -homomorphism if and only if there is an evaluation function ev : A × Z ≥ → K for ( A , σ ) (seeDefinition 9) with τ ( f ) = h ev( f, , ev( f, , . . . i . Starting with our ΠΣ-field ( K ( x ) , σ ) over K and the evaluation function (46) we can construct foran AP-extension an appropriate evaluation function by iterative application of Lemma 2. In particularthis yields a K -homomorphism from the given AP-extension into the ring of sequences by Lemma 4.Finally, we utilize the following result from [Sch17]; compare [OS18, Lemma 2.2]. Theorem 2.
Let ( A , σ ) be a difference field with constant field K and let ( E , σ ) be a basic RΠ -extensionof ( A , σ ) , Then any K -homomorphism τ : E → S ( K ) is injective. In other words, if we succeed in modeling our nested products within a basic RΠ-extension (inparticular, a multiple chain RΠ-extension) over ( K ( x ) , σ ) with an appropriate evaluation function,then we automatically obtain a K -embedding. Π -extensions In part 1 of Theorem 1 a criterion is given that enables one to check with, e.g., the algorithmsfrom [Kar81, Sch16] whether a P-extension is a Π-extension. In [OS18, Lemma 5.1] (based on [Sch10a,Sch17]) this criterion has been generalized for “single nested” P-extensions as follows.
Lemma 5.
Let ( F , σ ) be a difference field and let f , . . . , f s ∈ F ∗ . Then the following statements areequivalent.(1) There do not exist ( v , . . . , v s ) ∈ Z s \ { s } and g ∈ F ∗ such that σ ( g ) g = f v · · · f v s s holds.(2) The P -extension ( F [ z , z − ] . . . [ z s , z − s ] , σ ) of ( F , σ ) with σ ( z i ) = f i z i for ≤ i ≤ s is a Π -extension.(3) The difference field extension ( F ( z ) . . . ( z s ) , σ ) of ( F , σ ) with z i transcendental over F ( z ) . . . ( z i − ) and σ ( z i ) = f i z i for ≤ i ≤ s is a Π -field extension. In Theorem 3 we will extend this result further to multiple-chain P-extensions. Here we utilize thata solution for a certain class of homogeneous first-order difference equations has a particularly simpleform; this result is a specialization of [Sch16, Corollary 4.6].
Corollary 1.
Let ( E , σ ) be a Π -extension of a difference field ( A , σ ) with E = A h t i . . . h t e i . Then forany g ∈ E \ { } with σ ( g ) = u t z · · · t z e e g for some u ∈ A ∗ and z i ∈ Z we have g = h t v · · · t v e e with h ∈ A ∗ and v i ∈ Z . Now we are ready to prove a general criterion that enables one to check if a multiple chain P-extension forms a Π-extension. This result will be heavily used within the next two sections.
Theorem 3.
Let ( H , σ ) be a difference field and let ( H ℓ , σ ) with H ℓ = H h t ℓ, i . . . h t ℓ,s ℓ i for ≤ ℓ ≤ m be single chain P -extensions of ( H , σ ) over H with base c ℓ ∈ H ∗ where s ≥ s ≥ · · · ≥ s m . Inparticular, the automorphisms are given by σ ( t ℓ,k ) = α ℓ,k t ℓ,k where α ℓ,k = c ℓ t ℓ, · · · t ℓ,k − ∈ ( H ∗ ) H h t ℓ, i ... h t ℓ,k − i H . (59) Let ( A , σ ) be the ordered multiple chain P -ext. of ( H , σ ) with A = H h t , i . . . h t w , i . . . h t ,d i . . . h t w d ,d i of monomial depth d := max( s , . . . , s m ) with m = w ≥ w ≥ · · · ≥ w d composed by the single chain -extensions ( H ℓ , σ ) of ( H , σ ) with the automorphism (59) . Then ( A , σ ) is a Π -extension of ( H , σ ) ifand only if there does not exist a g ∈ H ∗ and ( v , . . . , v m ) ∈ Z m \ { m } such that σ ( g ) g = c v · · · c v m m . (60) Proof. “ = ⇒ ” Suppose that ( A , σ ) is a Π-extension of ( H , σ ). Then, it is a tower of Π-extensions( A i , σ ) of ( H , σ ) where A i = A i − h t ,i i . . . h t w i ,i i for 1 ≤ i ≤ d with A = H . Since ( A , σ ) is aΠ-extension of ( H , σ ), it follows by Lemma 5 that there does not exist a g ∈ H and ( v , . . . , v w ) ∈ Z w \ { w } with w = m such that (60) holds.“ ⇐ = ” Conversely, suppose that there does not exist a g ∈ H and ( v , . . . , v w ) ∈ Z w \ { w } with w = m such that (60) holds. Let ( A , σ ) with A = H h t , i . . . h t w , i be a P-extension of ( H , σ ) with σ ( t j, ) = α j, t j, for all 1 ≤ j ≤ w . By Lemma 5, ( A , σ ) is a Π-extension of ( H , σ ). Let ( A i , σ )with A i = A i − h t ,i i . . . h t w i ,i i be the multiple chain P-extension of ( H , σ ) with d ( t ,i ) = · · · = d ( t w i ,i )for all 1 ≤ i ≤ d with the automorphism (59). Assume that ( A k , σ ) is a Π-extension of ( H , σ ) for all1 ≤ k ≤ δ with d > δ ≥ A δ +1 , σ ) is not a Π-extension of ( A δ , σ ). Then by Lemma 5, wecan take a g ∈ A δ \ { } and ( υ , υ , . . . , υ w δ +1 ) ∈ Z w δ +1 \ { w δ +1 } such that σ ( g ) = α υ ,δ +1 α υ ,δ +1 · · · α υ wδ +1 w δ +1 ,δ +1 g (61)holds. By Corollary 1, it follows that g = h t v , , t v , , · · · t v w , w , · · · t v ,δ ,δ t v ,δ ,δ · · · t v wδ,δ w δ ,δ with h ∈ H ∗ and v i,j ∈ Z . For the left hand side of (61) we have that σ ( g ) = γ t v ,δ ,δ t v ,δ ,δ · · · t v wδ,δ w δ ,δ where γ ∈ A δ − and for the right hand side of (61) we have that α υ ,δ +1 α υ ,δ +1 · · · α υ wδ +1 w δ +1 ,δ +1 g = ω t v ,δ + υ ,δ t v ,δ + υ ,δ · · · t v wδ +1 ,δ + υ wδ +1 w δ +1 ,δ t v wδ +1+1 ,δ w δ +1 +1 ,δ . . . t v wδ,δ w δ ,δ where ω ∈ A δ − . Consequently, v k,δ = v k,δ + υ k and thus υ k = 0 for all 1 ≤ k ≤ w δ +1 which is acontradiction to the assumption that ( υ , . . . , υ w δ +1 ) = w δ +1 . Thus ( A d , σ ) is a Π-extension of ( H , σ ). (cid:3)
5. The main building blocks to represent nested products in RΠ-extensions
Suppose that we are given a finite set of hypergeometric products of finite nesting depth which havebeen brought into the form as given in Proposition 2. In the following we will show in Sections 5.1and 5.2 how these hypergeometric and geometric products can be modelled in RΠ-extensions. For thetreatment of geometric products one has to deal in addition with products defined over roots of unityof finite nesting depth. This extra complication will be treated in Section 5.3. Finally, in Section 6below we will combine all these techniques to represent the full class of hypergeometric products offinite nesting depth in RΠ-extensions.
In Proposition 2 we showed that a finite set of hypergeometric products of finite nesting depth canbe brought in a shift-coprime product representation form. In the setting of ΠΣ-field extensions theunderlying Definition 4 can be generalized as follows.
Definition 13.
Let ( F ( t ) , σ ) be a ΠΣ-field extension of ( F , σ ). We call two polynomials f, g ∈ F [ t ]shift-coprime (or σ -coprime) if for all k ∈ Z we have that gcd( f, σ k ( h )) = 1. Note that ( c , . . . , c m ) = ( α , , . . . , α w , ). Theorem 4.
Let ( F ( t ) , σ ) be a ΠΣ -extension of ( F , σ ) . Let f , . . . , f s ∈ F [ t ] \ F be irreducible monicpolynomials. Then the following statements are equivalent.(1) For all i, j with ≤ i < j ≤ s , f i and f j are shift-coprime.(2) There does not exist ( v , . . . , v s ) ∈ Z s \ { s } and g ∈ F ( t ) ∗ with σ ( g ) g = f v · · · f v s s .(3) The P -extension ( F ( t )[ z , z − ] . . . [ z s , z − s ] , σ ) of ( F ( t ) , σ ) with σ ( z i ) = f i z i for ≤ i ≤ s is a Π -extension. With this result we are now in the position to refine Theorem 3 in order to construct a Π-extensionin which we can model hypergeometric products of finite nesting depth that are in shift-coprimerepresentation form.
Theorem 5.
Let ( F ( t ) , σ ) be a ΠΣ -extension of ( F , σ ) Let f = ( f , . . . , f m ) ∈ ( F [ t ] \ F ) m be irreduciblemonic polynomials. For all ≤ ℓ ≤ m , let ( F ℓ , σ ) with F ℓ := F ( t ) h z ℓ, i . . . h z ℓ,s ℓ i be a single chain P -extension of ( F ( t ) , σ ) with base f ℓ ∈ F [ t ] \ F with the automorphism σ ( z ℓ,k ) = α ℓ,k z ℓ,k where α ℓ,k = f ℓ z ℓ, · · · z ℓ,k − ∈ ( F ∗ ) F h z ℓ, i ... h z ℓ,k − i F (62) and with s ≥ s ≥ · · · ≥ s m . Let ( H b , σ ) with H b = F ( t ) h z i . . . h z b i = F ( t ) h z , i . . . h z w , i . . . h z ,b i . . . h z w b ,b i be an ordered multiple chain P -extension of ( F ( t ) , σ ) of monomial depth b = max( s , . . . , s m ) with bases f , . . . , f m where m = w ≥ w ≥ · · · ≥ w b which is composed by the single chain P -extensions ( F ℓ , σ ) of ( F ( t ) , σ ) . Then ( H b , σ ) is a Π -extension of ( F ( t ) , σ ) if and only if for all i, j with ≤ i < j ≤ m the f i and f j are shift-coprime. Proof. “ = ⇒ ” If ( H b , σ ) is a Π-extension of ( F ( t ) , σ ), then by Theorem 3 there does not exist a g ∈ F ( t ) ∗ such that σ ( g ) g = f v · · · f v m m holds, and by Theorem 4 for all i, j with 1 ≤ i < j ≤ m , f i and f j are shift-coprime.“ ⇐ = ” Conversely, if for all i, j with 1 ≤ i < j ≤ m , f i and f j are shift-coprime, then by Theorem 4there does not exist a g ∈ F ( t ) ∗ such that σ ( g ) g = f v · · · f v m m holds, and by Theorem 3 ( H b , σ ) is aΠ-extension of ( F ( t ) , σ ). (cid:3) Summarizing, we obtain the following crucial result.
Corollary 2.
Let ( K ( x ) , σ ) be a rational difference field with the automorphism σ ( x ) = x + 1 andthe evaluation function ev : K ( x ) × Z ≥ → K given by (46) . Let ˜ H ( n ) , . . . , ˜ H e ( n ) be hypergeometricproducts in Prod n ( K ( x )) of nesting depth at most b which are all in shift-coprime representation form(see Definition 5) and which are all δ -refined for some δ ∈ Z ≥ . Then one can construct an orderedmultiple chain Π -extension ( ˜ H b , σ ) of ( K ( x ) , σ ) with ˜ H b = K ( x ) h ˜ z i . . . h ˜ z b i = K ( x ) h ˜ z , i . . . h ˜ z p , i . . . h ˜ z ,b i . . . h ˜ z p b ,b i (63) which is composed by the single chain Π -extensions (˜ F ℓ , σ ) of ( K ( x ) , σ ) with ˜ F ℓ = K ( x ) h ˜ z ℓ, i . . . h ˜ z ℓ,s ℓ i with(1) the automorphism σ : ˜ F ℓ → ˜ F ℓ defined by σ (˜ z ℓ,k ) = ˜ α ℓ,k ˜ z ℓ,k where ˜ α ℓ,k = ˜ f ℓ ˜ z ℓ, · · · ˜ z ℓ,k − ∈ ( K ( x ) ∗ ) K ( x ) h ˜ z ℓ, i ... h ˜ z ℓ,k − i K ( x ) (64) for ≤ ℓ ≤ p and ≤ k ≤ s ℓ where ˜ f ℓ ∈ K [ x ] \ K is an irreducible monic polynomial, and
2) the evaluation function ˜ev : ˜ F ℓ × Z ≥ → K given by ˜ev | K ( x ) = ev with (46) and ˜ev(˜ z ℓ,k , n ) = n Y j = δ ˜ev(˜ α ℓ,k , j −
1) (65) for ≤ ℓ ≤ p and ≤ k ≤ s ℓ with the following property: for all ≤ i ≤ e there are k, j such that ev(˜ z k,j , n ) = ˜ H i ( n ) , ∀ n ≥ max(0 , δ − . (66) Furthermore, for all g ∈ ˜ H b , the map ˜ τ : ˜ H b → S ( K ) defined by ˜ τ ( g ) = (cid:10) ˜ev( g, n ) (cid:11) n ≥ (67) is a K -embedding. If K is computable, the above construction can be given explicitly. Proof.
By the procedure outlined in Remark 4 (skipping step (1) since the input is already in theright form) we obtain the ordered multiple chain P-extension ( ˜ H b , σ ) of ( K ( x ) , σ ) with (63) and (65)such that (66) holds for all 1 ≤ i ≤ e for some j, k . Since the bases ˜ f , . . . , ˜ f p of the single chainP-extensions (˜ F , σ ) , . . . , (˜ F p , σ ) that composes ( ˜ H b , σ ) are shift-coprime, it follows by Theorem 5 that( ˜ H b , σ ) is a Π-extension of ( K ( x ) , σ ). Since ( ˜ H b , σ ) is a basic Π-extension of the rational differencefield ( K ( x ) , σ ), it follows by Theorem 2 that the K -homomorphism ˜ τ : ˜ H b → S ( K ) defined by (67) isinjective. Since K is computable, all the above ingredients can be constructed explicitly. (cid:3) Example 8 (Cont. Example 7).
Consider the ordered multiple chain P-extension ( ˜ H , σ ) of the ra-tional difference field ( K ( x ) , σ ) of monomial depth 2 with ˜ H = K ( x ) h z , ih z , ih z , i where ( ˜ H , σ ) iscomposed by the single chain Π-extensions of ( K ( x ) , σ ) constructed in parts (7) and (8) of Example 7.Since the bases of ( ˜ H , σ ) given by ( x −
2) and (cid:0) x + (cid:1) are shift-coprime with respect to the auto-morphism σ ( x ) = x + 1, it follows that the ordered multiple chain P-extension ( ˜ H , σ ) of the rationaldifference field ( K ( x ) , σ ) of monomial depth 2 is a Π-extension. Furthermore, it follows by Theorem 2that the map ˜ τ : ˜ H → S ( K ) defined by ˜ τ ( f ) = h ˜ev( f, n ) i n ≥ for all f ∈ ˜ H is a K -embedding where˜ev = ev defined in (54), and (55). In particular, for the expression h given by (57) we have that˜ H ( n ) = ˜ev( h, n ) holds for all n ≥ In Karr’s algorithm [Kar81] and all the improvements [KS06, Sch07a, Sch08, AP10, Sch15, Sch16,Sch17] one relies on certain algorithmic properties of the constant field K . Among those, one needs tosolve the following problem. Problem GO for α , . . . , α w ∈ K ∗ Given a field K and α , . . . , α w ∈ K ∗ . Compute a basis of the submodule V := (cid:8) ( u , . . . , u w ) ∈ Z w (cid:12)(cid:12)(cid:12) w Y i =1 α iu i = 1 (cid:9) of Z w over Z . In our approach Problem GO is crucial to solve Problem RPE, but one has to solve it not only in agiven field K (compare the definition of σ -computable in [Sch05, KS06]) but one must be able to solveit in any finite algebraic field extension of K . This gives rise to the following definition. Definition 14.
A field K is strongly σ -computable if the standard operations in K can be performed,multivariate polynomials can be factored over K and Problem GO can be solved for K and any finitealgebraic field extension of K . 22ote that Ge’s algorithm [Ge93a] (see also [Kau05, Algorithm 7.16, page 84]) solves Problem GOover an arbitrary number field K . Since any finite algebraic extension of an algebraic number fieldis again an algebraic number field, we obtain the following result; for a weaker result see [Sch05,Theorem 3.5]. Lemma 6.
An algebraic number field K is strongly σ -computable. By [OS18, Theorem 5.4] and the consideration of [OS18, pg. 204] (see also [Oca19, Lemma 5.2.2])we provided an algorithm that enabled us to handle geometric products of nested depth 1. Moreprecisely, given a P-extension that models such products, Lemma 7 states that one can construct anRΠ-extension in which the products can be rephrased.
Lemma 7.
Let K = K ( κ , . . . , κ u ) be a rational function field over a field K and ( K , σ ) be a differencefield with σ ( c ) = c for all c ∈ K . Let ( K h x i . . . h x e i , σ ) be a P -extension of ( K , σ ) with σ ( x i ) = γ i x i for ≤ i ≤ e where γ i ∈ K ∗ . Let ev : K h x i . . . h x e i × Z ≥ → K be the evaluation function defined by ev( x i , n ) = γ ni for ≤ i ≤ e . Then:(1) One can construct an RΠ -extension ( ˜ K h ϑ ih ˜ y i . . . h ˜ y s i , σ ) . of ( ˜ K , σ ) with σ ( ϑ ) = ζ ϑ and σ (˜ y k ) = α k ˜ y k (68) for ≤ k ≤ s where ˜ K = ˜ K ( κ , . . . , κ u ) and ˜ K is a finite algebraic field extension of K with ζ ∈ ˜ K being a primitive λ -th root of unity and α k ∈ ˜ K ∗ ;(2) one can construct the evaluation function ˜ev : ˜ K h ϑ ih ˜ y i . . . h ˜ y s i × Z ≥ → ˜ K defined as ˜ev( ϑ, n ) = ζ n and ˜ev(˜ y k , n ) = α nk ; (69) (3) one can construct a difference ring homomorphism ϕ : K h x i . . . h x e i → ˜ K h ϑ ih ˜ y i . . . h ˜ y s i with ϕ ( x i ) = ϑ µ i ˜ y v i = ϑ µ i ˜ y v i, · · · ˜ y v i,s s (70) for ≤ i ≤ e where ≤ µ i < λ and v i,k ∈ Z for ≤ k ≤ s such that for all f ∈ K h x i . . . h x e i and for all n ∈ Z ≥ , ev( f, n ) = ˜ev( ϕ ( f ) , n ) holds. If K is strongly σ -computable, then the above constructions are computable. Using this result we will derive an extended version in Lemma 9 that deals with the class of orderedmultiple chain AP-extensions that models geometric products of arbitrary but finite nesting depth.In the following let m ∈ Z ≥ , and for 1 ≤ ℓ ≤ m let ( K ℓ , σ ) with K ℓ = K h y ℓ i = K h y ℓ, i . . . h y ℓ,s ℓ i be a single chain P-extension of ( K , σ ) with base h ℓ ∈ K ∗ where σ ( y ℓ,i ) = α ℓ,i y ℓ,i with α ℓ,i = h ℓ y ℓ, · · · y ℓ,i − ∈ ( K ∗ ) K h y ℓ, i ... h y ℓ,i − i K . (71)In particular, we assume that s ≥ s ≥ · · · ≥ s m . Let ev : K ℓ × Z ≥ → K be the evaluation functiondefined by ev( y ℓ,i , n ) = n Y j =1 ev( α ℓ,i , j −
1) = n Y j =1 α ℓ,i ; (72) For concrete instances the R-monomial ϑ might be obsolete. In particular, if µ i = 0 for all 1 ≤ i ≤ e in (70) it canbe removed.
23n particular, for all c ∈ K and n ≥ c, n ) = c . Let ( A , σ ) be the multiple chain P-extensionof ( K , σ ) built by the single chain Π-extensions ( K ℓ , σ ) of ( K , σ ) over K . That is, A = K h y ih y i . . . h y m i = K h y , i . . . h y ,s ih y , i . . . h y ,s i . . . h y m, i . . . h y m,s m i . We emphasize that all the y ℓ,i model 1-refined geometric products in product factored form of anarbitrary but finite nesting depth. Depending on the context, y ℓ denotes ( y ℓ, , . . . , y ℓ,s ℓ ) or y ℓ, , . . . , y ℓ,s ℓ or y ℓ, · · · y ℓ,s ℓ . Note that the P-monomials y ℓ,i can be ordered in increasing order of their depths.Namely, take the depth function d : A → Z ≥ over K of ( A , σ ) and let d = max( s , s , . . . , s m ) be themaximal depth. Then taking A = K we can consider the tower of P-extensions ( A i , σ ) of ( A i − , σ )with A i = A i − h y i i = A i − h y ,i ih y ,i i . . . h y w i ,i i for 1 ≤ i ≤ d where m = w ≥ w ≥ · · · ≥ w d and with the automorphism (71) for 1 ≤ ℓ ≤ w i . Inthis way, the P-monomials at the i -th extension have the depth d ( y ,i ) = d ( y ,i ) = . . . d ( y w i ,i ) = i .Further, the ring A d is isomorphic to A up to reordering of the P-monomials. In particular, ( A d , σ )is an ordered multiple chain P -extension of ( K , σ ) of monomial depth at most d induced by the singlechain Π-extensions ( K ℓ , σ ) of ( K , σ ) for 1 ≤ ℓ ≤ m with (71) and (72). Observe that since A d ≃ A , theevaluation function ev : A i × Z ≥ → K for all i with 1 ≤ i ≤ d is also defined by (72).In order to derive the main result of this subsection in Lemma 9, we need following simple con-struction. Lemma 8.
Let ( A h t i , σ ) be a Π -extension of ( A , σ ) with σ ( t ) = α t and let ( H , σ ) be a difference ring.Let ˜ ρ : A → H be a difference ring homomorphism and let ρ : A h t i → H be a ring homomorphismdefined by ρ | A = ˜ ρ and ρ ( t ) = g for some g ∈ H . If σ ( g ) = ρ ( α ) g , then ρ is a difference ringhomomorphism. Proof.
Suppose that σ ( g ) = ρ ( α ) g holds. Then σ ( ρ ( t )) = σ ( g ) = ρ ( α ) g = ρ ( α t ) = ρ ( σ ( t )).Consequently, σ ( ρ ( f )) = ρ ( σ ( f )) for all f ∈ A h t i . (cid:3) Lemma 9.
For ≤ ℓ ≤ m , let ( K ℓ , σ ) with K ℓ = K h y ℓ, i . . . h y ℓ,s ℓ i be single chain P -extensions of ( K , σ ) over a rational function field K = K ( κ , . . . , κ u ) with base h ℓ ∈ K ∗ , the automorphisms (71) and the evaluation functions (72) . Let d := max( s , . . . , s m ) and A = K . Consider the tower ofdifference ring extensions ( A i , σ ) of ( A i − , σ ) where A i = A i − h y ,i ih y ,i i . . . h y w i ,i i for ≤ i ≤ d with m = w ≥ · · · ≥ w d , the automorphism (72) and the evaluation function (72) . In particular, one gets ( A d , σ ) as the ordered multiple chain P -extension of ( K , σ ) of monomial depth at most d composedby the single chain P -extensions ( K ℓ , σ ) of ( K , σ ) for ≤ ℓ ≤ m with (71) and (72) . Then one canconstruct(a) an ordered multiple chain AP -extension ( G d , σ ) of ( ˜ K , σ ) of monomial depth at most d with ˜ K = ˜ K ( κ , . . . , κ u ) where ˜ K is a finite algebraic field extension of K , with G d = ˜ K h ϑ , i . . . h ϑ υ , ih ˜ y , i . . . h ˜ y e , i . . . h ϑ ,d i . . . h ϑ υ d ,d ih ˜ y ,d i . . . h ˜ y e d ,d i (73) where υ i ≥ , e i ≥ . Here the automorphism is defined for the A -monomials by σ ( ϑ ℓ,k ) = γ ℓ,k ϑ ℓ,k where γ ℓ,k = ζ µ ℓ ϑ ℓ, · · · ϑ ℓ,k − ∈ U ˜ K [ ϑ ℓ, ] ... [ ϑ ℓ,k − ]˜ K (74) for ≤ k ≤ d and ≤ ℓ ≤ υ k where U = h ζ i is a multiplicative cyclic subgroup of ˜ K ∗ generatedby a primitive λ -th root of unity ζ ∈ ˜ K ∗ , and the automorphism is defined for the P -monomialsby σ (˜ y ℓ,k ) = ˜ α ℓ,k ˜ y ℓ,k where ˜ α ℓ,k = ˜ h ℓ ˜ y ℓ, · · · ˜ y ℓ,k − ∈ ( ˜ K ∗ ) ˜ K h ˜ y ℓ, i ... h ˜ y ℓ,k − i ˜ K (75) for ≤ k ≤ d and ≤ ℓ ≤ e k ; Note that if υ i = 0 or e i = 0, then there is no depth- i A-monomial or P-monomial of depth i , respectively. b) an evaluation function ˜ev : G d × Z ≥ → ˜ K defined by ˜ev( ϑ ℓ,k , n ) = n Y j =1 ˜ev( γ ℓ,k , j − and ˜ev(˜ y ℓ,k , n ) = n Y j =1 ˜ev(˜ α ℓ,k , j − (c) a difference ring homomorphism ρ d : A d → G d defined by ρ d | K = id K and ρ d ( y ℓ,k ) = ϑ µ ℓ , k k ˜ y v ℓ , k k = ϑ µ ℓ, ,k ,k · · · ϑ µ ℓ,υk,k υ k ,k ˜ y v ℓ, ,k ,k · · · ˜ y v ℓ,ek,k e k ,k (77) for ≤ ℓ ≤ m and ≤ k ≤ s ℓ with µ ℓ,i,k ∈ Z ≥ for ≤ i ≤ υ k and v ℓ,i,k ∈ Z for ≤ i ≤ e k such that the following properties hold:(1) There does not exist a ( v , . . . , v e ) ∈ Z e \ { e } with ˜ h v . . . ˜ h v e e = 1 .(2) The P -extension ( ˜ A d , σ ) of ( ˜ K , σ ) with ˜ A d = ˜ K h ˜ y ih ˜ y i . . . h ˜ y d i = ˜ K h ˜ y , i . . . h ˜ y e , ih ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i (78) and the automorphism given in (75) is a Π -extension. In particular, it is an ordered multiplechain Π -extension of monomial depth d .(3) For all f ∈ A d and for all n ∈ Z ≥ we have ev( f, n ) = ˜ev( ρ d ( f ) , n ) . (79) If K is strongly σ -computable, then the above constructions are computable. Proof.
Let ( A d , σ ) with A d = A d − h y ,d ih y ,d i . . . h y w d ,d i be the ordered multiple chain P-extension of( K , σ ) of monomial depth d ∈ Z ≥ as described above with the automorphism (72) and the evaluationfunction (72). We prove the Lemma by induction on the monomial depth d .If d = 1, statements (2) and (3) of the Lemma hold by Lemma 7. Hence by Lemma 5 there are no g ∈ ˜ K ∗ and ( v , . . . , v e ) ∈ Z e \ { e } with ˜ h v . . . ˜ h v e e = σ ( g ) g = 1 and thus also statement (1) of theLemma holds.Now let d ≥ d −
1. That is, we can construct ( G d − , σ ) with G d − = ˜ K h ϑ , i . . . h ϑ υ , ih ˜ y , i . . . h ˜ y e , i . . . h ϑ ,d − i . . . h ϑ υ d − ,d − ih ˜ y ,d − i . . . h ˜ y e d − ,d − i which is an ordered multiple chain AP-extension of ( ˜ K , σ ) of monomial depth at most d − ≤ k ≤ d − ≤ ℓ ≤ υ k and given by (75) for 1 ≤ k ≤ d − ≤ ℓ ≤ e k . In addition, we get the evaluation function ˜ev : G d − × Z ≥ → ˜ K defined as (76) andthe difference ring homomorphism ρ d − : A d − → G d − defined by ρ d − | K = id K and (77) such thatstatements (1), (2), and (3) of the Lemma hold. We prove the Lemma for the ordered multiple chainP-extension ( A d , σ ) of ( K , σ ) with A d = A d − h y ,d ih y ,d i . . . h y w d ,d i where d ( y ,d ) = · · · = d ( y w d ,d ) = d .Since the shift quotient of these P-monomials is contained in A d − , i.e., σ ( y ℓ,d ) y ℓ,d = α ℓ,d ∈ A ∗ d − , we can iteratively apply the difference ring homomorphism ρ d − : A d − → G d − to rephrase each α ℓ,d in G d − . In particular, by Remark 3 we have σ − ( α ℓ,d ) = y ℓ,d − and thus by (77) we get h ℓ,d := ρ d − ( σ − ( α ℓ,d )) = ρ d − ( y ℓ,d − ) = ϑ µ ℓ , d − d − ˜ y v ℓ , d − d − (80) For all c ∈ K , we set ˜ev( c, n ) = c for all n ≥ Note that any P-monomial y ℓ,k with depth k is mapped to a power product of AP-monomials having all depth k . ϑ u ℓ , d − d − = ϑ u ℓ, ,d − ,d − · · · ϑ u ℓ,υd − ,d − υ d − ,d − and ˜ y v ℓ , i d − = ˜ y v ℓ, ,d − ,d − · · · ˜ y v ℓ,ed − ,d − e d − ,d − for 1 ≤ ℓ ≤ w d with µ ℓ,k,d − ∈ Z ≥ for 1 ≤ k ≤ υ d − and v ℓ,k,d − ∈ Z for 1 ≤ k ≤ e d − .If h ℓ,d = 1, it follows with (79) ( d replaced by d −
1) that for all n ∈ Z ≥ we haveev( y ℓ,d , n ) = n Y j =1 ev( α ℓ,d , j −
1) = n Y j =1 ev( σ − ( α ℓ,d ) , j ) = n Y j =1 ˜ev( ρ ( σ − ( α ℓ,d )) , j ) = n Y j =1 ˜ev( h ℓ,d , j ) = 1 . In particular, if h ℓ,d = 1 holds for all 1 ≤ ℓ ≤ w d , we can set G d := G d − and extend ρ d − to ρ d : A d → G d − with ρ d ( y ℓ,d ) = 1 for 1 ≤ ℓ ≤ w d . Thus the lemma is proven.Otherwise, take all AP-monomials in (80) for 1 ≤ ℓ ≤ w d with non-zero integer exponents. Then theybelong to at least one of the single chain AP-extensions of ( ˜ K , σ ) in ( G d − , σ ). Suppose there are e d ≥ υ d ≥ H b , σ ); notethat we have e d + υ d ≥
1. By appropriate reordering of ( G d − , σ ) we may suppose that these e d singlechain Π-extensions ( F r , σ ) of ( ˜ K , σ ) with 1 ≤ r ≤ e d are given by F r = ˜ K h ˜ y r, ih ˜ y r, i . . . h ˜ y r,d − i andthe υ d A-extensions ( H b , σ ) of ( ˜ K , σ ) with 1 ≤ b ≤ υ d can be given by H b = ˜ K h ϑ b, ih ϑ b, i . . . h ϑ b,d − i .Now adjoin the P-monomials ˜ y r,d to F r with (74) where k = d and ℓ = r yielding the single chainP-extensions ( F ′ r , σ ) of ( ˜ K , σ ) of monomial depth d where F ′ r = F r h ˜ y r,d i = ˜ K h ˜ y r, ih ˜ y r, i . . . h ˜ y r,d − ih ˜ y r,d i and adjoin the A-monomial ϑ b,d with (75) where k = d and ℓ = b yielding the single chain A-extensions( H ′ b , σ ) of ( ˜ K , σ ) of monomial depth d where H ′ b = H b h ϑ b,d i = ˜ K h ϑ b, ih ϑ b, i . . . h ϑ b,d − ih ϑ b,d i . Furthermore extend the evaluation functions ˜ev : F ′ r × Z ≥ → ˜ K and ˜ev : H ′ b × Z ≥ → ˜ K with (76)where k = d, ℓ = r or k = d, ℓ = b , respectively. Now consider the multiple chain P-extension ( ˜ A d , σ )of ( ˜ K , σ ) with ˜ A d = ˜ K h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d − i . . . h ˜ y e d − ,d − ih ˜ y ,d ih ˜ y ,d i . . . h ˜ y e d ,d i which one gets by taking all P-monomials in G d − and the new P-monomials in F ′ r = F r h ˜ y r,d i with 1 ≤ r ≤ e d . Here the automorphism is given by (74) and (75) and the equipped evalua-tion function is given by (76). Since there does not exist a g ∈ ˜ K and ( v , . . . , v d ) ∈ Z d \ { d } with σ ( g ) g = ˜ h v · · · ˜ h v d d , it follows by Theorem 3 that ( ˜ A d , σ ) is a Π-extension of ( ˜ K , σ ). In par-ticular, it is an ordered multiple chain Π-extension of monomial depth d by construction. Thusstatements (1) and (2) of the Lemma hold. Finally, take the AP-extension ( G d , σ ) of ( G d − , σ ) with G d = G d − h ϑ ,d ih ϑ ,d i · · · h ϑ υ d ,d ih ˜ y ,d ih ˜ y ,d i · · · h ˜ y e d ,d i .Let 1 ≤ ℓ ≤ w d and consider h ℓ,d in (80). If h ℓ,d = 1, we define g ℓ = 1. In this case, 1 = ρ ( h ℓ,d ) = ρ d − ( σ − ( α ℓ,d )) = σ − ( ρ d − ( α ℓ,d )), and thus σ ( g ℓ ) g ℓ = 1 = ρ d − ( α ℓ,d ) holds. Otherwise, if h ℓ,d = 1, weset g ℓ := ϑ µ ℓ, ,d ,d · · · ϑ µ ℓ,υd,d υ d ,d ˜ y v ℓ, ,d ,d · · · ˜ y v ℓ,ed,d e d ,d ∈ ( ˜ K ∗ ) G d ˜ K . (81)Then based on the Remark 3 it follows that ˜ y j,d σ − (˜ y j,d ) = ˜ y j,d − and ϑ j,d σ − ( ϑ j,d ) = ϑ j,d − and thus g ℓ σ − ( g ℓ ) = ϑ µ ℓ, ,d − ,d − · · · ϑ µ ℓ,υd − ,d − υ d − ,d − ˜ y v ℓ, ,d − ,d − · · · ˜ y v ℓ,ed − ,d − e d − ,d − = ϑ µ ℓ , d − d − ˜ y v ℓ , d − d − = ρ d − ( y l,d − ) . Hence also in this case we get σ ( g ℓ ) g ℓ = σ ( g ℓ σ − ( g ℓ ) ) = σ ( ρ d − ( y l,d − )) = ρ d − ( σ ( y l,d − )) = ρ d − ( α ℓ,d ) .
26y iterative application of Lemma 8, the difference ring homomorphism ρ d − : A d − → G d − can beextended to ρ d : A d − h y ,d ih y ,d i · · · h y w d ,d i → G d − h ϑ ,d ih ϑ ,d i · · · h ϑ υ d ,d ih ˜ y ,d ih ˜ y ,d i · · · h ˜ y e d ,d i with ρ d | A d − = ρ d − and ρ d ( y ℓ,d ) = g ℓ for 1 ≤ ℓ ≤ w d . Finally, we show that for all f ∈ A d and n ∈ Z ≥ we have ev( f, n ) = ˜ev( ρ d ( f ) , n ). First note that for all n ≥ y ℓ,d , n + 1) = ev( σ ( y ℓ,d ) , n ) = ev( α ℓ,d , n ) ev( y ℓ,d , n ) . (82)On the other hand, since ρ d is a difference ring homomorphism, we have that σ ( ρ d ( y ℓ,d )) = ρ d ( σ ( y ℓ,d )) = ρ d ( α ℓ,d ) ρ d ( y ℓ,d ) = ρ d − ( α ℓ,d ) ρ d ( y ℓ,d ) (83)for all n ≥
0. Thus we get˜ev( ρ d ( y ℓ,d ) , n + 1) = ˜ev( σ ( ρ d ( y ℓ,d )) , n ) (83) = ˜ev( ρ d − ( α ℓ,d ) , n ) ˜ev( ρ d ( y ℓ,d ) , n ) . (84)By the induction hypothesis, ev( α ℓ,d , n ) = ˜ev( ρ d − ( α ℓ,d ) , n ) holds for all n ∈ Z ≥ . Therefore with (82)and (84) it follows that ev( y ℓ,d , n ) and ˜ev( ρ ( y ℓ,d ) , n ) satisfy the same first-order recurrence relation.With ev( y ℓ,d ,
0) = 1 and˜ev( ρ d ( y ℓ,d ) ,
0) = ˜ev( g l ,
0) = ˜ev( ϑ ,d , µ ℓ, ,d · · · ˜ev( ϑ υ d ,d , µ ℓ,υd,d ˜ev(˜ y ,d , v ℓ, ,d · · · ˜ev(˜ y e d ,d , v ℓ,ed,d = 1it follows then that ev( y ℓ,d , n ) = ˜ev( ρ d ( y ℓ,d , n ) holds for all n ≥
0. Together with the inductionhypothesis ev( f, n ) = ˜ev( ρ d − ( f ) , n ) for all f ∈ A d − and n ∈ Z ≥ we get (79) for all f ∈ A d and forall n ≥
0. Consequently also statement (1) of the Lemma holds.Finally, if K is strongly σ -computable, the base case d = 1 can be executed explicitly by activatingLemma 7. In particular the induction step can be performed iteratively and thus the difference ring( G d , σ ) with (73) together with (74), (75) and (76) can be computed. In addition, the difference ring( G d , σ ), the difference ring homomorphism ρ d : A d → G d defined by (77) and the evaluation function˜ev can be computed. This completes the proof. (cid:3) Remark 5.
Note that the generators of G d with (73) constructed in Lemma 9 can be rearrangedto get the AΠ-extension ( ˜ K [ ϑ , ] . . . [ ϑ υ , ] . . . [ ϑ ,d ] . . . [ ϑ υ d ,d ] h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i , σ ) of( ˜ K , σ ). Furthermore, a consequence of statement (3) of Lemma 9 is that the diagram A S ( K ) G d S ( ˜ K ) ψρ ρ ′ ˜ τ commutes where A = A d , ρ = ρ d , ρ ′ = id and the difference ring homomorphism ˜ τ and ψ are definedby ˜ τ ( f ) = (cid:10) ˜ev( f, n ) (cid:11) n ≥ and ψ ( g ) = (cid:10) ev( g, n ) (cid:11) n ≥ respectively. Example 9 (Cont. Example 7).
Take the ordered multiple chain AP-extension ( A ′ , σ ) of ( K , σ )with monomial depth 2 with A ′ = K h ϑ , ih y , ih y , ih y , ih y , ih y , ih ϑ , ih y , ih y , i where ( A ′ , σ )is composed by the single chain AP-extensions of ( K , σ ) constructed in parts (1), (2), (3), (4), (5),and (6) of Example 7. By Lemma 9 and Remark 5 we can construct the AP-extension ( G , σ ) of ( K , σ )where G = K [ ϑ , ][ ϑ , ] h ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , i (85)with the automorphism σ and evaluation function ˜ev : G × Z ≥ → K given by (47), (48) and σ (˜ y , ) = √ y , , σ (˜ y , ) = 2 ˜ y , , σ (˜ y , ) = 5 ˜ y , , σ (˜ y , ) = 2 ˜ y , ˜ y , , σ (˜ y , ) = 5 ˜ y , ˜ y , , ˜ev(˜ y , , n ) = n Y k =1 √ , ˜ev(˜ y , , n ) = n Y k =1 , ˜ev(˜ y , , n ) = n Y k =1 , ev(˜ y , , n ) = n Y k =1 k Y j =1 , ev(˜ y , , n ) = n Y k =1 k Y j =1 D , σ ) of the differencering ( G , σ ) with ˜ D = K h ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , i is an ordered multiple chain Π-extension of ( K , σ )with the automorphism σ and the evaluation function ˜ev : ˜ D × Z ≥ → K defined in (86). In additionby part (3) of Lemma 9 we get the difference ring homomorphism ρ : A ′ → G defined by ρ | K [ ϑ , ][ ϑ , ] =id K [ ϑ , ][ ϑ , ] and ρ ( y , ) = ˜ y , , ρ ( y , ) = ˜ y , , ρ ( y , ) = ˜ y , , ρ ( y , ) = ˜ y , ,ρ ( y , ) = ˜ y , , ρ ( y , ) = ˜ y , , ρ ( y , ) = ˜ y , (87)such that ˜ev( ρ ( f ) , n ) = ev( f, n ) holds for all n ∈ Z ≥ and f ∈ A ′ , Throughout this Subsection, K is a field containing Q , K m is a splitting field for the polynomial x m − K (i.e., all roots of the polynomial x m − K m ) for some m ∈ Z ≥ and U m is theset of all m -th roots of unity over K in K m (which forms a multiplicative subgroup of K ∗ ). ThenProd n ( U m ) is the set of all geometric products over roots of unity in U m . For G ⊆ K m ( x ) we defineProdE n ( G , U m ) as the set of all elements X v =( v ,...,v e ) ∈ S a v ( n ) P ( n ) v · · · P e ( n ) v e with e ∈ Z ≥ , S ⊆ Z e ≥ finite, a v ( x ) ∈ G for v ∈ S and P ( n ) , . . . , P e ( n ) ∈ Prod n ( U m ).The main result of this Subsection in Theorem 6 states that products over roots of unity with finitenesting depth can be represented by the single product ζ nλ where ζ λ ∈ U λ for some λ ≥ λ -th root of unity. Theorem 6.
Suppose we are given the geometric products A ( n ) , . . . , A e ( n ) ∈ Prod n ( U m ) in n ofnesting depth r i ∈ Z ≥ with A i ( n ) = n Y k = ℓ i, ζ i, k Y k = ℓ i, ζ i, · · · k ri − Y k ri = ℓ i,ri ζ i,r i (88) for ≤ i ≤ e where ζ i,j ∈ U m , ℓ i,j ∈ Z ≥ for ≤ j ≤ r i . Then there exist a λ ∈ Z ≥ with m | λ anda primitive λ -th root of unity ζ λ ∈ K ∗ λ satisfying the following property. For all ≤ i ≤ e there exist f i,j ∈ K λ for ≤ j < λ such that for B i ( n ) = λ − X j =0 f i,j ( ζ nλ ) j ∈ ProdE n ( K λ , U λ ) (89) we have A i ( n ) = B i ( n ) (90) for all n ≥ max( ℓ i, , . . . , ℓ i,r i ) − . In particular, if K is computable and one can solve Problem O (seebelow), the above construction can be given explicitly.5.3.1. The period and algorithmic aspects For the treatment of Theorem 6 we will introduce the period of a difference ring element introducedin [Kar81]. In particular, we will use the algorithms from [Sch16] that enable one to calculate the periodwithin nested R-extensions, resp. A-extensions.
Definition 15.
Let ( A , σ ) be a difference ring. The period of α ∈ A ∗ is defined byper( α ) = ( ∄ n > σ n ( α ) = α min { n > | σ n ( α ) = α } otherwise .
28s it turns out, this task is connected to compute the order of a ring.
Definition 16.
Let A be a ring and let α ∈ A \ { } . Then the order of α is defined byord( α ) = ( , if ∄ n > α n = 1 . min { n > | α n = 1 } , otherwise . Namely, if we can solve the following problem (which is Problem GO with w = 1): Problem O for α ∈ K ∗ Given a field K and α ∈ K ∗ . Find ord( α ).then we can also compute the period by the following lemma. Lemma 10.
Let ( E , σ ) with E = K m [ ϑ ] . . . [ ϑ e ] be a simple A -extension of ( K m , σ ) . Then the follow-ing statements hold.(1) per( ϑ i ) > for all ≤ i ≤ e .(2) If K m is computable and Problem O is solvable, then per( ϑ i ) is computable for all ≤ i ≤ e . Proof.
Statement (1) follows by [Sch16, Proposition 5.5] (compare [Oca19, Proposition 6.2.20]) andstatement (2) follows by [Sch16, Corollary 5.6] (compare [Oca19, Corollary 6.2.21]). (cid:3)
Example 10 (Cont. Example 9).
Consider the sub-difference ring ( K [ ϑ , ][ ϑ , ] , σ ) of ( G , σ ) with(85) with the automorphism and the evaluation function defined in (47) and (48). By construction itis a simple A-extension of the difference field ( K , σ ). Since per( c ) = 1 for all c ∈ K , it follows thatper( −
1) = 1. Applying the algorithms from [Sch16] (see the comments in the proof of Lemma 10) wecompute for the depth-1 A-monomial ϑ , the period per( ϑ , ) = 2, while for the depth-2 A-monomial ϑ , we get per( ϑ , ) = 4. RP -extensions In order to prove Theorem 6 we rely on the property that elements in basic RP-extensions can beexpressed by idempotent elements. We start with the following basic facts inspired by [PS97].
Lemma 11.
Let F be a field and let ζ be a primitive λ -th root of unity. Let F [ ϑ ] be a polynomial ringsubject to the relation ϑ λ = 1 . Then the following statements hold.(1) The elements e , . . . , e λ − ∈ F [ ϑ ] with e k = e k ( ϑ ) := λ − Y i =0 i = λ − − k ϑ − ζ i ζ λ − − k − ζ i (91) are idempotent and for all ≤ k < λ we have e k ( ζ j ) = ( if j = λ − − k if j = λ − − k and e k ( ζ ϑ ) = e k +1 mod λ . (92) (2) The idempotent elements defined in (91) are pairwise orthogonal and e + · · · + e λ − = 1 .
29n Proposition 3 we state that a simple RP-extension ( E , σ ) of a difference field ( F , σ ) with E = F [ ϑ ] h t i . . . h t e i can be decomposed in terms of these idempotent elements. For more details in thegeneral setting of RΠΣ-extensions we refer to [Sch17, Theorem 4.3]; compare also [PS97, Corollary1.16], and [HS08, Lemma 6.8]. Proposition 3.
Let ( E , σ ) with E = F [ ϑ ] h t i . . . h t e i be an RP -extension of a difference field ( F , σ ) where ϑ is an R -monomial of order λ with ζ = σ ( ϑ ) ϑ ∈ const( F , σ ) ∗ and the t i are P -monomials. Let e , . . . , e λ − be the idempotent, pairwise orthogonal elements in (91) that sum up to one. Then thefollowing statement holds:(1) The ring E can be written as the direct sum E = e E ⊕ · · · ⊕ e λ − E (93) where e k E forms for all ≤ k < λ a ring with e k being the multiplicative identity element.(2) We have that e k E = e k ˜ E for ≤ k < λ where ˜ E = F h t i . . . h t e i . We are now ready to obtain the following key result; for the corresponding result for nested A-extensions of monomial depth 1 we refer to [Sch17, Lemma 2.22].
Theorem 7.
Let m ∈ Z ≥ and take a primitive m -th root of unity ζ m ∈ K ∗ m . Let ( K m [ ϑ ] . . . [ ϑ e ] , σ ) be a simple A -extension of ( K m , σ ) with σ ( ϑ i ) = α i ϑ i for ≤ i ≤ e where α i = ζ u i m ϑ z i, · · · ϑ z i,i − i − with u i , z i,j ∈ Z ≥ . Furthermore, let ev m : K m [ ϑ ] . . . [ ϑ e ] × Z ≥ → K m be the evaluation function definedby ev m ( ϑ i , n ) = n Y j =1 ev m ( α i , j − , (94) and let τ m : K m [ ϑ ] . . . [ ϑ e ] → S ( K ) be the K m -homomorphism given by τ m ( f ) = (cid:10) ev m ( f, n ) (cid:11) n ≥ Thenthe following statements hold.(1) Define λ := lcm( m, per( ϑ ) , . . . , per( ϑ e )) > . Then there is an R -extension ( K λ [ ϑ ] , σ ) of ( K λ , σ ) of order λ with ζ = σ ( ϑ ) ϑ ∈ K ∗ λ such that ϕ : K m [ ϑ ] . . . [ ϑ e ] → K λ [ ϑ ] = e K λ ⊕ · · · ⊕ e λ − K λ (95) defined with ϕ ( f ) = f e + · · · + f λ − e λ − (96) where f i = ev m ( f, λ − − i ) ∈ K m ⊆ K λ for ≤ i < λ is a difference ring homomorphism; herethe e k are the idempotent orthogonal elements defined in (91) . In particular, ϕ | K m = id K m .(2) Take the evaluation function ev λ : K λ [ ϑ ] × Z ≥ → K λ defined by ev λ | K λ = id and ev λ ( ϑ, n ) = ζ n and consider the K λ -homomorphism τ λ : K λ [ ϑ ] → S ( K λ ) defined by τ λ ( f ) = (cid:10) ev λ ( f, n ) (cid:11) n ≥ . Thenfor the pairwise orthogonal idempotent elements e k defined in (91) with ≤ k < λ , we have that ev λ ( e k , n ) = ( if λ | n + k + 1 , if λ ∤ n + k + 1 . (97) (3) The K λ -homomorphism τ λ : K λ [ ϑ ] → S ( K λ ) with the evaluation function defined in part (2) isinjective. K λ is a finite algebraic extension of K m and ζ ∈ K m is a primitive λ -th root of unity.
4) The diagram K m [ ϑ ] . . . [ ϑ e ] S ( K m ) K λ [ ϑ ] ≃ e K λ ⊕ · · · ⊕ e λ − K λ S ( K λ ) τ m ϕ ϕ ′ τ λ (98) commutes where ϕ ′ : S ( K m ) → S ( K λ ) is the injective difference ring homomorphism defined by ϕ ′ ( a ) = a .If K m is computable and Problem O is solvable in K m , then the above constructions are computable. Proof. (1) Since ζ u i m ∈ K ∗ m , per( ζ u i m ) = 1 > ≤ u i ≤ e . In addition, it follows by statement (1) ofLemma 10 that per( ϑ i ) > ≤ i ≤ e . Define λ := lcm( m, per( ϑ ) , . . . , per( ϑ e )) >
1. Notethat m | λ , i.e., K λ is an algebraic field extension of K m . Finally, take ζ := e π i λ = ( − λ ∈ K ∗ λ and construct the A-extension ( K λ [ ϑ ] , σ ) of ( K λ , σ ) with σ ( ϑ ) = ζ ϑ . By Lemma 3 it follows that( K λ [ ϑ ] , σ ) is an R-extension of ( K λ , σ ). By Proposition 3 we have that K λ [ ϑ ] = e K λ ⊕· · ·⊕ e λ − K λ where the e k for 0 ≤ k < λ are the orthogonal idempotent elements defined in (91). Now considerthe map (95) defined by (96). We will now show that ϕ is a ring homomorphism. Observe thatfor any c ∈ K m , ev m ( c, i ) = c for all i ∈ Z ≥ and with statement (2) of Lemma 11 we have that ϕ ( c ) = c e + · · · + c e λ − = c ( e + · · · + e λ − ) = c. Further, let f, g ∈ K m [ ϑ ] . . . [ ϑ e ] with f := a ϑ v · · · ϑ v e e and g := b ϑ z · · · ϑ z e e where a, b ∈ K m and v i , z i ∈ Z ≥ for 1 ≤ i ≤ e . Define f k := ev m ( f, λ − − k ) and g k := ev m ( g, λ − − k ) for 0 ≤ k < λ .Then, ϕ ( f + g ) = ev m ( f + g, λ − e + · · · + ev m ( f + g, e λ − = (cid:0) ev m ( f, λ −
1) + ev m ( g, λ − (cid:1) e + · · · + (cid:0) ev m ( f,
0) + ev m ( g, (cid:1) e λ − = (cid:0) ev m ( f, λ − e + · · · + ev m ( f, e λ − (cid:1) + (cid:0) ev m ( g, λ − e + · · · + ev m ( g, e λ − (cid:1) = (cid:0) f e + · · · + f λ − e λ − (cid:1) + (cid:0) g e + · · · + g λ − e λ − (cid:1) = ϕ ( f ) + ϕ ( g ) . Similarly, ϕ ( f g ) = ev m ( f g, λ − e + · · · + ev m ( f g, e λ − = (cid:0) ev m ( f, λ −
1) ev m ( g, λ − (cid:1) e + · · · + (cid:0) ev m ( f,
0) ev m ( g, (cid:1) e λ − = f g e + f g e + · · · + f λ − g λ − e λ − = (cid:0) f e + · · · + f λ − e λ − (cid:1) (cid:0) g e + · · · + g λ − e λ − (cid:1) = ϕ ( f ) ϕ ( g ) . The first equality follows since the e i are idempotent. Thus, ϕ is a ring homomorphism. Nextwe show by induction on the number of A-monomials, e ∈ Z ≥ , that ϕ is a difference ring ho-momorphism. For the base case, i.e., e = 0, there are no A-monomials. Since σ ( ϕ ( c )) = σ ( c ) = c = ϕ ( c ) = ϕ ( σ ( c )) for all c ∈ K m , ϕ is a difference ring-homomorphism. Now assume that thestatement holds for all A-monomials ϑ i with 0 ≤ i < e , and consider an A-monomial ϑ e with σ ( ϑ e ) = ˜ α ϑ e where ˜ α ∈ ( K ∗ m ) K m [ ϑ ] ··· [ ϑ e − ] K m . Then we will show that σ ( ϕ ( ϑ e )) = ϕ ( σ ( ϑ e )) (99)31olds. For the left hand side of (99), we have that ϕ ( ϑ e ) = γ e + · · · + γ λ − e λ − where γ i =ev m ( ϑ e , λ − − i ) ∈ K m for 0 ≤ i < λ are λ -th roots of unity. Thus, σ ( ϕ ( ϑ e )) = σ ( γ ) σ ( e ) + · · · + σ ( γ λ − ) σ ( e λ − ) . By (92) we have that σ ( e λ − ) = e and σ ( e i ) = e i +1 for 0 ≤ i < λ −
1. In addition, for 1 ≤ i < λ we get σ ( γ i ) = ev m ( ϑ e , λ − i ) = γ i − . For i = λ observe that per( ϑ e ) | λ by definition and thus σ λ ( ϑ e ) = ϑ e . Consequently σ ( γ ) = ev m ( ϑ e , λ ) = ev m ( σ λ ( ϑ e ) ,
0) = ev m ( ϑ e ,
0) = γ λ − . Therefore, σ ( ϕ ( ϑ e )) = ˜ γ e + · · · + ˜ γ λ − e λ − (100)where ˜ γ = γ λ − and ˜ γ i = γ i − for 1 ≤ i ≤ λ − ϕ ( σ ( ϑ e )) = ϕ (˜ α ϑ e ) = ϕ (˜ α ) ϕ ( ϑ e )= ( α e + · · · + α λ − e λ − )( γ e + · · · + γ λ − e λ − )= α γ e + · · · + α λ − γ λ − e λ − (101)where α i = ev m (˜ α, λ − − i ) and γ i = ev m ( ϑ e , λ − − i ) for 0 ≤ i < λ are λ -th roots of unity.Again (101) holds since the e i are idempotent. Finally observe that for 0 ≤ i < λ we have α i γ i = ev m ( ϑ e , λ − − i ) ev m (˜ α, λ − − i ) = ev m (˜ α ϑ e , λ − − i )= ev m ( σ ( ϑ e ) , λ − − i ) = ev m ( ϑ e , λ − i ) = ˜ γ i . With (100) we conclude that (99) holds. Thus, ϕ is a difference ring homomorphism.(2) By Lemma 2 we can define the evaluation function ev λ : K λ [ ϑ ] × Z ≥ → K λ with ev λ | K λ = id andev λ ( ϑ, n ) = ζ n and by Lemma 4 we get the K λ -homomorphism τ λ : K λ [ ϑ ] → S ( K λ ) defined by τ λ ( f ) = (cid:10) ev λ ( f, n ) (cid:11) n ≥ . Statement (97) follows by (92).(3) Since ( K λ [ ϑ ] , σ ) is an R-extension of a difference field ( K λ , σ ) it follows by Theorem 2 that τ λ isinjective.(4) Let α ∈ K m [ ϑ ] . . . [ ϑ e ] and let ev m , ev λ be evaluation functions for K m [ ϑ ] . . . [ ϑ e ] and K λ [ ϑ ]defined by (94) and (97), respectively. We will show ϕ ′ ( τ m ( α )) = τ λ ( ϕ ( α )) . (102)For the left hand side of (102), we have ϕ ′ ( τ m ( α )) = τ m ( α ) = (cid:10) ev m ( α, n ) (cid:11) n ≥ ∈ S ( K m ) ⊆ S ( K λ ) . For the right hand side of (102) we note by (96) that ϕ ( α ) = α e + · · · + α λ − e λ − holds where α i = ev m ( α, λ − − i ) ∈ K m ⊆ K λ for 0 ≤ i < λ . Thus, τ λ ( ϕ ( α )) = (cid:10) ev λ ( α e + · · · + α λ − e λ − , n ) (cid:11) n ≥ = (cid:10) ev λ ( α e , n ) (cid:11) n ≥ + · · · + (cid:10) ev λ ( α λ − e λ − , n ) (cid:11) n ≥ = α (cid:10) ev λ ( e , n ) (cid:11) n ≥ + · · · + α λ − (cid:10) ev λ ( e λ − , n ) (cid:11) n ≥ = (cid:10) ev m ( α, n ) (cid:11) n ≥ . The last equality follows by (97). This implies that the diagram (98) commutes.32inally, if K m is computable and Problem O is solvable in K m , then by statement (2) of Lemma 10per( ϑ i ) is computable for all 1 ≤ i ≤ e . Consequently, the R-extension ( K λ [ ϑ ] , σ ) of ( K λ , σ ), theevaluation function ev λ : K λ [ ϑ ] × Z ≥ → K λ given in statement (2) and the injective K λ -homomorphism τ λ : K λ [ ϑ ] → S ( K λ ) given in statement (4) can be constructed explicitly. (cid:3) Remark 6.
By statement (4) of Theorem 7 and (97) we observe that for a fixed k ∈ Z ≥ and α ∈ K m [ ϑ ] . . . [ ϑ e ] we getev m ( α, k ) = ev λ ( ϕ ( α ) , k ) = α ev λ ( e , k ) + · · · + α λ − ev λ ( e λ − , k )= α j ev λ ( e j , k ) = α j = ev m ( α, j ) (103)for some j ∈ { , , . . . , λ − } with λ | k − j . In other words, the sequence repeats periodically. Example 11 (Cont. Example 10).
Consider the U -simple A-extension ( K [ ϑ , ][ ϑ , ] , σ ) of ( K , σ )with the automorphism and the evaluation function given in (47) and (48), which was constructed inExamples 9 and 10 with K = Q ( √ K ). From Example 10 we already know the period of theA-monomials ϑ , and ϑ , in K [ ϑ , ][ ϑ , ]. Set λ = lcm( m, per( ϑ , ) , per( ϑ , )) = 4 with m = 2, takea primitive 4th root of unity, say ζ := e π i = ( − = i and define ˜ K = Q ( i , √ K ). Then bystatement (1) of Theorem 7 there is an R-extension ( ˜ K [ ϑ ] , σ ) of ( ˜ K , σ ) of order 4 with the automorphism σ ( ϑ ) = i ϑ (104)and the evaluation function ˜ev : ˜ K [ ϑ ] × Z ≥ → ˜ K given by˜ev( ϑ, n ) = n Y k =1 i = i n . (105)We have ˜ K [ ϑ ] = e ˜ K ⊕ e ˜ K ⊕ e ˜ K ⊕ e ˜ K where the idempotent elements e k for 0 ≤ k ≤ e = i (cid:0) ϑ + i ϑ − ϑ − i (cid:1) , e = (cid:0) − ϑ + ϑ − ϑ (cid:1) , e = i (cid:0) − ϑ + i ϑ + ϑ − i (cid:1) , e = (cid:0) ϑ + ϑ + ϑ (cid:1) (106)with e + e + e + e = 1. Furthermore, the ring homomorphism ϕ : K [ ϑ , ][ ϑ , ] → ˜ K [ ϑ ] defined by ϕ | K = id K and ϕ ( ϑ ,i ) = β i, e + β i, e + β i, e + β i, e where β i,j = ev( ϑ i,j , − j ) for i ∈ { , } and0 ≤ j ≤ ϕ ( ϑ , ) = − e + e − e + e = ϑ ,ϕ ( ϑ , ) = − e − e + e + e = (1 − i )2 ϑ ( ϑ + i ) . (107)Given ev and ˜ev we obtain the difference ring homomorphisms τ : K [ ϑ , ][ ϑ , ] → S ( K ) defined by τ ( f ) = (ev( f, n )) n ≥ and τ : ˜ K [ ϑ ] → S ( ˜ K ) defined by τ ( f ) = ( ˜ev( f, n )) n ≥ . In particular, bystatement (3) of Theorem 7 τ is injective. Finally, by defining the embedding ϕ ′ : S ( K ) → S ( ˜ K ) with ϕ ′ ( a ) = a for all a ∈ S ( K ) we conclude by statement (4) of the Theorem 7 that the following diagramcommutes K [ ϑ , ][ ϑ , ] S ( ˜ K )˜ K [ ϑ ] = e ˜ K ⊕ e ˜ K ⊕ e ˜ K ⊕ e ˜ K S ( ˜ K ) . τ ϕ ϕ ′ τ We are finally ready to obtain the 33 roof of Theorem 6.
Suppose we are given the geometric products A ( n ) , . . . , A e ( n ) ∈ Prod n ( U m )in n with (88) where ζ i,r i = 1 for 1 ≤ i ≤ e . As elaborated in Section 2.2 we can rewrite each A i ( n ) as A i ( n ) = u i ˜ A i ( n ) where ˜ A i ( n ) = n Y k =1 ζ i, k Y k =1 ζ i, · · · k ri − Y k ri =1 ζ i,r i (108)and u i ∈ U m which holds for all n ≥ max( ℓ i, , . . . , ℓ i,r i ) − δ i . Similar to Remark 4 we can rephrasethe products in a simple A-extension ( A , σ ) of ( K m , σ ) with A = K m [ ϑ , ] . . . [ ϑ ,r ] , . . . , [ ϑ e, ] . . . [ ϑ e,r e ]where α i,j := σ ( ϑ i,j ) ϑ i,j = ζ u i,j m ϑ i, · · · ϑ i,j − for 1 ≤ i ≤ e and 1 ≤ j ≤ r i with u i,i ∈ Z ≥ equipped withthe evaluation function ev m : A × Z ≥ → K m defined by ev m ( ϑ i,j , n ) = n Y k =1 ev m ( α i,j , k −
1) with thefollowing property. For all i with 1 ≤ i ≤ e , there are ν i , µ i such that the geometric product ˜ A i ( n ) ismodelled by ϑ ν i ,µ i , i.e., ev m ( ϑ ν i ,µ i , n ) = ˜ A i ( n ) (109)holds for all n ≥
0. In particular, we get the K -homomorphism τ m : A → S ( K λ ). By Theorem 7,there is a single R-extension ( K λ [ ϑ ] , σ ) of ( K λ , σ ) subject to the relations ϑ λ = 1 and σ ( ϑ ) = ζ λ ϑ where λ := lcm( m, per( γ ) , . . . , per( γ s )) > ζ λ := e π i λ = ( − λ ∈ K λ and K λ is some algebraicextension of K m with m | λ . Furthermore, there is a difference ring homomorphism ϕ : A → K λ [ ϑ ] anda K λ -embedding τ λ : K λ [ ϑ ] → S ( K λ ) with τ λ ( ϑ ) = (cid:10) ζ nλ (cid:11) n ≥ such that τ m ( f ) = τ λ ( ϕ ( f )) holds for all f ∈ A . In particular, we get ev m ( ϑ ν i ,µ i , n ) = ev λ ( ϕ ( ϑ ν i ,µ i ) , n ) (110)for 1 ≤ i ≤ e . Now define g i,k ∈ K λ by ϕ ( ϑ ν i ,µ i ) = P λ − k =0 g i,k ϑ k ∈ K λ [ ϑ ]. Then for G i ( n ) := P λ − k =0 g i,k ( ζ nλ ) k with 1 ≤ i ≤ e we getev m ( ϑ ν i ,µ i , n ) (110) = ev λ ( ϕ ( ϑ ν i ,µ i ) , n ) = G i ( n ) ∀ n ≥ . With (108) and (109) we conclude that A i ( n ) = u i ˜ A i ( n ) = u i G i ( n )holds for all n ≥ δ i . In particular, for B i ( n ) given in (89) with f i,k := u i g i,k ∈ K λ we get (90).If K m is computable and Problem O can be solved, Theorem 7 is constructive and all the aboveingredients can be given explicitly. (cid:3) Example 12 (Cont. Example 11).
Consider the product expression A ( n ) = √ n Y i =1 ( −
1) + 2 n Y k =1 k Y i =1 ( −
1) + 3 n Y i =1 ( − ! n Y k =1 k Y i =1 ( − ∈ ProdE n (cid:16) Q ( √ , U (cid:17) . For this instance we follow the construction in Example 11 and get f = √ ϑ , + 2 ϑ , + 3 ϑ , ϑ , ∈ K [ ϑ , ][ ϑ , ] with ev( f, n ) = A ( n ) for all n ∈ Z ≥ . As a consequence we obtain ˜ f := ϕ ( f ) = (cid:0) − i (cid:1) ϑ (cid:0) (2 + 3 i ) ϑ + (1 + i ) √ ϑ + (3 + 2 i ) (cid:1) ∈ ˜ K [ ϑ ] yielding for n ∈ Z ≥ the identity A ( n ) = ev( f, n ) = ˜ev( ˜ f , n ) = (cid:0) − i (cid:1) i n (cid:16) (2 + 3 i )( i n ) + (1 + i ) √ i n + (3 + 2 i ) (cid:17) . In particular, as claimed in Theorem 6, each of the products in A ( n ) can be expressed in terms of i n .Namely, for all n ∈ Z ≥ we obtain n Y k =1 ( −
1) = ev( ϑ , , n ) = ˜ev( ϕ ( ϑ ) , n ) = ˜ev( ϑ , n ) = ( i n ) , n Y k =1 k Y i =1 ( −
1) = ev( ϑ , , n ) = ˜ev( ϕ ( ϑ ) , n ) = ˜ev( (1 − i )2 ϑ ( ϑ + i ) , n ) = (1 − i )2 i n (( i n ) + i ) . . A complete solution of Problem RPE We are now ready to combine the building blocks from the previous section to solve Problem RPEin Sections 6.1 and 6.2 below. Afterwards we apply in Section 6.3 the machinery implemented withinthe package
NestedProducts to concrete examples.
First we combine Lemma 9 discussed in Subsection 5.2 and Theorem 7 discussed in Subsection 5.3.As a consequence, we will obtain the necessary difference ring tools for the full treatment of geometricproducts of arbitrary but finite nesting depth.
Theorem 8.
For ≤ ℓ ≤ m , let ( K ℓ , σ ) with K ℓ = K h y ℓ, i . . . h y ℓ,s ℓ i be the single chain P -extensionsof ( K , σ ) over K = K ( κ , . . . , κ u ) with base h ℓ ∈ K ∗ for ≤ ℓ ≤ m , the automorphisms (71) and theevaluation functions (72) . Let d := max( s , . . . , s m ) and A = K . Consider the tower of differencering extensions ( A i , σ ) of ( A i − , σ ) where A i = A i − h y ,i ih y ,i i . . . h y w i ,i i for ≤ i ≤ d with m = w ≥ w ≥ · · · ≥ w d and the automorphism (71) and the evaluation function (72) . This yields the orderedmultiple chain P -extension ( A d , σ ) of ( K , σ ) of monomial depth at most d composed by the single chain Π -extensions ( K ℓ , σ ) of ( K , σ ) for ≤ ℓ ≤ m with (71) and (72) . Then one can construct(1) an RΠ -extension ( D , σ ) of ( K ′ , σ ) with D = K ′ [ ϑ ] h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i , where K ′ = K ′ ( κ , . . . , κ u ) and K ′ is a finite algebraic field extension of K and with the automor-phism σ ( ϑ ) = ζ ′ ϑ and σ (˜ y ℓ,d ) = ˜ α ℓ,k ˜ y ℓ,k where ζ ′ ∈ K ′ is a λ ′ -th root of unity and ˜ α ℓ,k = ˜ h ℓ ˜ y ℓ, · · · ˜ y ℓ,k − ∈ ( K ′∗ ) K ′ h ˜ y ℓ, i ... h ˜ y ℓ,k − i K ′ for ≤ k ≤ d and ≤ ℓ ≤ e k ;(2) an evaluation function ˜ev : D × Z ≥ → K ′ defined as ˜ev( ϑ, n ) = n Y j =1 ζ ′ and ˜ev(˜ y ℓ,k , n ) = n Y j =1 ˜ev(˜ α ℓ,k , j − (3) a difference ring homomorphism ϕ : A d → D defined by ϕ | K = id K and ϕ ( y ℓ,k ) = γ ℓ,k ˜ y v ℓ, ,k ,k · · · ˜ y v ℓ,ek,k e k ,k (112) for ≤ ℓ ≤ m and ≤ k ≤ s ℓ with γ ℓ,k ∈ K ′ [ ϑ ] and v ℓ,i,k ∈ Z for ≤ i ≤ e k such that for all f ∈ A d and n ∈ Z ≥ we have ev( f, n ) = ˜ev( ϕ ( f ) , n ) . (113) If K is strongly σ -computable, then the constructions above are computable. For concrete instances the R-monomial ϑ might not be needed. In particular, if ν ℓ,k = 0 in (112), it can be removed. Note that for all c ∈ K ′ , ˜ev( c, n ) = c for all n ≥ roof. Let ( A d , σ ) be an ordered multiple chain P-extension of ( K , σ ) of monomial depth at most d with the automorphism σ : A d → A d defined by (71) and the evaluation function ev : A d × Z ≥ → K defined by (72). Then by Lemma 9 we can construct an ordered multiple chain AP-extension ( G d , σ ) of( ˜ K , σ ) of monomial depth at most d with ˜ K = ˜ K ( κ , . . . , κ u ), ˜ K being a finite algebraic field extensionof K where where G d is given by (73) with the automorphism (74) and (75), the evaluation functionev ′ : G d × Z ≥ → ˜ K with (76) (where ˜ev is replaced by ev ′ ), and the difference ring homomorphism ρ d : A d → G d defined by ρ d | K = id K and (77) with the following properties: the sub-difference ring( ˜ A d , σ ) of ( G d , σ ) where ˜ A d is given by (78) is a Π-extension of ( ˜ K , σ ). Furthermore, for all f ∈ A d and for all n ∈ Z ≥ we have ev( f, n ) = ev ′ ( ρ d ( f ) , n ) . (114)If υ = 0 (and thus υ = · · · = υ d = 0), i.e., no A-monomials are involved, we are essentially done. Wesimply adjoin a redundant R-monomial (compare the footnote in Theorem 8). Otherwise υ ≥ G d can be rearranged to get the AΠ-extension ( ˜ G , σ ) of ( ˜ K , σ ) where˜ G = ˜ K [ ϑ , ] . . . [ ϑ υ , ] . . . [ ϑ ,d ] . . . [ ϑ υ d ,d ] h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i (115)with the automorphism given by (74) and (75) and the evaluation function given by (76) (where ˜evis replaced by ev ′ ) satisfying properties (1) and (2) of Lemma 9. Now consider the sub-difference ring( L , σ ) of ( ˜ H , σ ) with L = ˜ K [ ϑ , ] . . . [ ϑ υ , ] . . . [ ϑ ,d ] . . . [ ϑ υ d ,d ], which is a difference ring extension of( ˜ K , σ ), with the automorphism defined by σ ( ϑ ℓ,k ) = γ ℓ,k ϑ ℓ,k where γ ℓ,k = ζ µ ℓ ϑ ℓ, · · · ϑ ℓ,k − ∈ U ˜ K [ ϑ ℓ, ] ... [ ϑ ℓ,k − ]˜ K (116)for 1 ≤ k ≤ d and for 1 ≤ ℓ ≤ υ k where U = h ζ i is the multiplicative cyclic subgroup of ˜ K generatedby a primitive λ -th root of unity, ζ ∈ ˜ K ∗ . Observe that the difference ring extension ( L , σ ) of ( ˜ K , σ )with (116) is a simple A-extension to which statement (1) of Theorem 7 can be applied. Thus there isan R-extension ( K ′ [ ϑ ] , σ ) of ( K ′ , σ ) with σ ( ϑ ) = ζ ′ ϑ (117)of order λ ′ where K ′ = K ′ ( κ , . . . , κ u ), ζ ′ is a primitive λ ′ -th root of unity in K ′ and K ′ is a finitealgebraic field extension of ˜ K . Note that the difference ring ( ˜ D , σ ) where ˜ D is given by˜ D = K ′ h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i (118)with the automorphism defined by (75) is a Π-extension of ( K ′ , σ ). Thus by Lemma 3 it follows thatthe A-extension ( ˜ D [ ϑ ] , σ ) of ( ˜ D , σ ) with (117) of order λ ′ is an R-extension. Note that the generatorsin the ring ˜ D [ ϑ ] can be rearranged to get ( D , σ ) where D = K ′ [ ϑ ] h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i and σ is defined by (117) and (75). Since this rearrangement does not change the set of constants, ( D , σ )is an RΠ-extension of ( K ′ , σ ). By statement (1) of Proposition 3 D = e D ⊕ · · · ⊕ e λ ′ − D and bystatement (2) of the same proposition, e k D = e k ˜ D for 0 ≤ k < λ ′ . Thus D = e ˜ D ⊕ e ˜ D ⊕ · · · ⊕ e λ ′ − ˜ D holds. Now we show that φ : ˜ G → D defined by φ | ˜ K = id ˜ K with φ (˜ y ℓ,k ) = ˜ y ℓ,k , (119) φ ( ϑ ℓ,k ) = β ℓ,k, e + · · · + β ℓ,k,λ ′ − e λ ′ − (120)where β ℓ,k,i = ev( ϑ ℓ,k , λ ′ − − i ) for 0 ≤ i < λ ′ is a difference ring homomorphism. By statement (1)of Theorem 7, φ | L which is defined by (120) is a difference ring homomorphism. Since φ maps ˜ y ℓ,k toitself, also φ is a difference ring homomorphism. Furthermore, for all f ∈ ˜ G and for all n ∈ Z ≥ , wehave ev ′ ( f, n ) = ˜ev( φ ( f ) , n ) . (121)Putting everything together, the map ϕ : A d → D with ϕ = φ ◦ ρ d is a difference ring homomorphism.It is uniquely determined by ϕ | K = id K and ϕ ( y ℓ,d ) = φ ( ρ d ( y ℓ,d )) = γ ℓ,d ˜ y v ℓ, ,d ,d · · · ˜ y v ℓ,ed,d e d ,d γ ℓ,d = β ℓ,d, e + · · · + β ℓ,d,λ ′ − e λ ′ − ∈ K ′ [ ϑ ]. Furthermore, by (114) and (121) it follows that forall f ∈ A d and for all n ∈ Z ≥ we getev( f, n ) = ev ′ ( ρ d ( f ) , n ) = ˜ev( φ ( ρ d ( f )) , n ) = ˜ev( ϕ ( f ) , n ) . Finally if K is strongly σ -computable, then by Lemma 9 the difference ring ( ˜ G , σ ) with (115) togetherwith automorphism (74) and (75), evaluation function (76) (where ˜ev is replaced by ev ′ ) and thedifference ring homomorphism ρ d : A d → ˜ G with 77 can be computed. Further, by Theorem 7 thedifference ring ( ˜ D [ ϑ ] , σ ) with the automorphism σ ( ϑ ) = ζ ′ ϑ and (75), the evaluation function (111)and the difference ring homomorphism ϕ : ˜ G → D given by (119) and (120) can be computed. Inparticular, ϕ and all the components stated in the theorem can be given explicitly. (cid:3) Example 13 (Cont. Example 9, 11).
Take the AΠ-extension ( G , σ ) of ( K , σ ) with (85) constructedin Example 9 with the automorphism defined in (47) and (86), and consider the sub-difference ring( K [ ϑ , ][ ϑ , ] , σ ) of ( G , σ ) with the automorphism σ given in (47), which is a simple A-extension of( K , σ ) where K = Q ( √ K [ ϑ ] , σ ) of ( ˜ K , σ ) of order 4 with the automorphism (104) and theevaluation function ˜ev : ˜ K [ ϑ ] × Z ≥ → ˜ K given by (105) where ˜ K = Q ( i , √ D , σ )where D = ˜ K [ ϑ ] h ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , i with the automorphism and evaluation function givenby (104) and (105) for the R-monomial ϑ and (86) for the Π-monomials ˜ y ℓ,k . By Theorem 8 ( D , σ ) isan RΠ-extension of ( ˜ K , σ ) where the ring D can be written as the direct sum D = e ˜ D ⊕ e ˜ D ⊕ e ˜ D ⊕ e ˜ D with ˜ D = ˜ K h ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , ih ˜ y , i ; here the idempotent elements e k for 0 ≤ k ≤ φ : G → D defined by φ | ˜ D = id ˜ D and (107) is adifference ring homomorphism.Finally, consider the AP-extension ( A ′ , σ ) of ( K , σ ) as given in Example 9 and consider the differ-ence ring homomorphism ρ : A ′ → G given in (87). Then with the difference ring homomorphism ϕ : A ′ → D defined by φ ( ρ ( f )) for f ∈ A ′ we get (113) for all n ∈ Z ≥ and f ∈ A ′ . Given this explicitconstruction we can choose for instance g ∈ A ′ defined in (57) that models ˜ G ( n ) given in (37). Thismeans that ev( g, n ) = ˜ G ( n ) for all n ≥
0. Thus˜ g := ϕ ( g ) = ϕ ( ϑ , ) ϕ ( y , ) ϕ ( y , ) ϕ ( ϑ , ) ϕ ( y , ) ϕ ( y , ) ϕ ( y , ) ϕ ( y , ) = (1 − i ) ϑ ( i ϑ + 1) ˜ y , ˜ y , ˜ y , y , ˜ y , ∈ D (122)yields for n ≥ G ( n ) = ev( g, n ) = ˜ev(˜ g, n ) = 12 (1 − i ) ( i ) n ( i ( i n ) + 1) (cid:0) √ (cid:1) n (cid:0) n (cid:1) n +12 )2 n n +12 ) . So far we have treated hypergeometric products over monic irreducible polynomials of finite nestingdepth, say b , that are δ -refined for some δ ∈ Z ≥ ; see Definition 5. Given such hypergeometric products,it follows by Corollary 2 that we can construct an ordered multiple chain Π-extension ( ˜ H b , σ ) of( K ( x ) , σ ) with K = K ( κ , . . . , κ u ) and˜ H b = K ( x ) h ˜ z i . . . h ˜ z b i = K ( x ) h ˜ z , i . . . h ˜ z p , i . . . h ˜ z ,b i . . . h ˜ z p b ,b i . (123)In particular, ( ˜ H b , σ ) is composed by the single chain Π-extensions (˜ F ℓ , σ ) of ( K ( x ) , σ ) for 1 ≤ ℓ ≤ p with ˜ F ℓ = K ( x ) h ˜ z ℓ, ih ˜ z ℓ, i . . . h ˜ z ℓ,s ℓ i , ≤ k ≤ s ℓ given by the automorphism σ : ˜ F ℓ → ˜ F ℓ defined by σ (˜ z ℓ,k ) = ˜ α ℓ,k ˜ z ℓ,k where ˜ α ℓ,k = ˜ f ℓ ˜ z ℓ, · · · ˜ z ℓ,k − ∈ ( K ( x ) ∗ ) K ( x ) h ˜ z ℓ, i ... h ˜ z ℓ,k − i K ( x ) (124)37nd the evaluation function ˜ev : ˜ F ℓ × Z ≥ → K defined by˜ev(˜ z ℓ,k , n ) = n Y j = δ ˜ev(˜ α ℓ,k , j − . (125)On the other hand, geometric products over the contents were treated in Subsection 5.2. In Theorem 8we constructed a simple RΠ-extension ( D , σ ) of ( ˜ K , σ ) with ˜ K = ˜ K ( κ , . . . , κ u ) where ˜ K is a finitealgebraic field extension of K in which the geometric products can be modelled. To accomplish thistask, we set up a ring of the form D = ˜ K [ ϑ ] h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i (126)with(a) the automorphism σ : D → D defined by σ ( ϑ ) = ζ ϑ, (127) σ (˜ y ℓ,k ) = ˜ γ ℓ,k ˜ y ℓ,k (128)where ζ ∈ ˜ K ∗ is a λ -th root of unity and ˜ γ ℓ,k = ˜ h ℓ ˜ y ℓ, · · · ˜ y ℓ,k − ∈ ( ˜ K ∗ ) ˜ K h ˜ y ℓ, i ... h ˜ y ℓ,k − i ˜ K for1 ≤ k ≤ d and 1 ≤ ℓ ≤ e k and(b) the evaluation function ˜ev : D × Z ≥ → ˜ K defined by˜ev( ϑ, n ) = n Y j =1 ζ, (129)˜ev(˜ y ℓ,k , n ) = n Y j =1 ˜ev(˜ γ ℓ,k , j − . (130)In particular, by reordering we obtain the difference ring extension ( ˜ A d , σ ) of ( ˜ K , σ ) with˜ A d = ˜ K h ˜ y i . . . h ˜ y d i = ˜ K h ˜ y , i . . . h ˜ y e , i . . . h ˜ y ,d i . . . h ˜ y e d ,d i , (131)the automorphism (128) and the evaluation function (130) which is a sub-difference ring of ( D , σ ).Furthermore, it is an ordered multiple chain Π-extension of ( ˜ K , σ ) and is composed by the single chainΠ-extensions ( ˜ K ℓ , σ ) of ( ˜ K , σ ) where ˜ K ℓ = ˜ K h ˜ y ℓ, ih ˜ y ℓ, i . . . h ˜ y ℓ, ˜ e ℓ i for 1 ≤ ℓ ≤ e .Putting the two difference rings ( ˜ H b , σ ) with (123) and ( D , σ ) with (126) together, we will obtaina difference ring in which we can model any finite set of hypergeometric product expressions of finitenesting depth coming from ProdE n ( K ( x )). Before we can complete this final argument, we have totake care that the two combined extensions yield again an RΠ-extension. Here we utilize the followingresult from [OS18, Lemma 5.6] that holds for single nested RΠ-extensions. Lemma 12.
Let ( K ( x ) , σ ) be the rational difference field with σ ( x ) = x +1 and let ( K ( x ) h z i . . . h z s i , σ ) be a Π -extension of ( K ( x ) , σ ) with σ ( z k ) z k ∈ K [ x ] \ K . Further, let K ′ be an algebraic field extension of K and let ( K ′ h y i . . . h y w i , σ ) be a Π -extension of ( K ′ , σ ) with σ ( y i ) y i ∈ K ′ \ { } . Then the differencering ( E , σ ) with E = K ′ ( x ) h y i . . . h y w ih z i . . . h z s i is a Π -extension of ( K ′ ( x ) , σ ) . Furthermore, the A -extension ( E [ ϑ ] , σ ) of ( E , σ ) with σ ( ϑ ) = ζ ϑ of order λ is an R -extension. Namely, we can enhance the above lemma to nested RΠ-extensions.38 orollary 3.
Let ( K ( x ) , σ ) be a rational difference field over K with σ ( x ) = x + 1 and let the dif-ference ring ( ˜ H b , σ ) with (123) be an ordered multiple chain Π -extension of ( K ( x ) , σ ) with the auto-morphism (124) . Further, let ˜ K be an algebraic field extension of K and let the difference ring ( ˜ A d , σ ) with (131) be the ordered multiple chain Π -extension of ( ˜ K , σ ) with the automorphism (128) . Then thedifference ring (˜ E , σ ) with ˜ E = ˜ K ( x ) h ˜ y ih ˜ z i . . . h ˜ y d ih ˜ z b i (132) where h ˜ y i i = h ˜ y ,i i . . . h ˜ y e i ,i i for ≤ i ≤ d and h ˜ z k i = h ˜ z ,k i . . . h ˜ z p k ,k i for ≤ k ≤ b is an orderedmultiple chain Π -extension of ( ˜ K ( x ) , σ ) . Furthermore, the A -extension ( E , σ ) of (˜ E , σ ) where E = ˜ E [ ϑ ] with (127) of order λ is an R -extension. Proof.
Take the Π-extensions ( ˜ H , σ ) of ( K ( x ) , σ ) with ˜ H = K ( x ) h z , i . . . h z p , i and ( ˜ A , σ ) of ( ˜ K , σ )with ˜ A = ˜ K h ˜ y , i . . . h ˜ y e , i which are both of monomial depth 1. By Lemma 12 the difference ring( E , σ ) with ˜ E = ˜ K ( x ) h ˜ y , i . . . h ˜ y e , ih ˜ z , i . . . h ˜ z p , i is a Π-extension of ( ˜ K ( x ) , σ ) of monomial depth1. Consider the ordered multiple chain P-extension (˜ E , σ ) of ( ˜ K ( x ) , σ ) with (132) which is composedby the single chain Π-extensions in the ordered multiple chains ( ˜ H b , σ ) and ( ˜ A d , σ ). By Theorem 3together with Lemma 5 it follows that (˜ E , σ ) is a Π-extension of ( ˜ K ( x ) , σ ). The quotient field of ˜ E gives the rational function field H = K ( x )( ˜ y )( ˜ z ) . . . ( ˜ y d )( ˜ z b ) and one can extend the automorphism σ from ˜ E to H accordingly. Then by Lemma 5 ((2) ⇔ (3)) it follows that ( H , σ ) is a Π-field extensionof ( K ( x ) , σ ). In particular, ( H , σ ) is a ΠΣ-field over K . Thus by Lemma 3 the A-extension ( H [ ϑ ] , σ )of ( H , σ ) of order λ with the automorphism (127) is an R-extension. Therefore const( H [ ϑ ]) = K andwith K ⊆ ˜ E [ ϑ ] ⊆ H it follows that const(˜ E [ ϑ ]) = K . But this implies that the A-extension (˜ E [ ϑ ] , σ ) of(˜ E , σ ) of order λ with the automorphism (127) is an R-extension. (cid:3) Finally, we arrive at the following main result.
Theorem 9.
Let K = K ( κ , . . . , κ u ) be a rational function field, and let ( K ( x ) , σ ) with σ ( x ) = x + 1 be a rational difference field with the evaluation function ev : K ( x ) × Z ≥ → K defined by (46) , andthe Z -function given by (3) . Suppose we are given a finite set of hypergeometric product expressions { A ( n ) , . . . , A m ( n ) } ⊆ ProdE n ( K ( x )) (133) of nesting depth at most d for some d ∈ Z ≥ . Then there is a δ ∈ Z ≥ and an RΠ -extension ( E , σ ) of ( ˜ K ( x ) , σ ) of monomial depth at most d where ˜ K is a finite algebraic field extension of K equipped withan evaluation function ˜ev : E × Z ≥ → ˜ K with respect to δ with the following properties:(1) The map τ : E → S ( ˜ K ) with τ ( f ) = (cid:10) ˜ev( f, n ) (cid:11) n ≥ is a ˜ K -embedding.(2) There are elements a , . . . , a e ∈ E ∗ such that for j with ≤ j ≤ e and for all n ≥ δ we have A j ( n ) = ˜ev( a j , n ) . If K is a strongly σ -computable, such a δ , ( E , σ ) with the evaluation function ˜ev , and the a , . . . , a m ∈ E can be computed. Proof. (a) We are given the hypergeometric product expressions in (133) where A j ( n ) = X v =( ν ,...,ν e ) ∈ S j a j, v ( n ) P ( n ) ν · · · P e ( n ) ν e with S j ⊆ Z e finite, a j, v ( x ) ∈ K ( x ) and P ( n ) , . . . , P e ( n ) ∈ Prod n ( K ( x )). Now we follow theconstruction in Proposition 2. There we can take a δ ∈ Z ≥ and construct for all 1 ≤ j ≤ e , c j ∈ K ∗ , rational functions r j ∈ K ( x ) ∗ , 1-refined geometric product expressions ˜ G j ( n ) ∈ ProdM n ( K )39nd δ -refined hypergeometric product expressions in shift-coprime product representation form˜ H j ( n ) ∈ ProdM n ( K ( x )) such that P j ( n ) = ˜ c j ˜ r j ( n ) ˜ G j ( n ) ˜ H j ( n ) = 0 (134)holds for all n ≥ max(0 , δ − H ( n ) , . . . , ˜ H e ( n ) in (134) we have˜ H i ( n ) = ˜ H i, ( n ) n i, · · · ˜ H i,l i ( n ) n i,li for some l i ∈ Z ≥ with n i,j ∈ Z for 1 ≤ j ≤ l i where all the arising hypergeometric products˜ H i,j ( n ) are δ -refined and in shift-coprime product representation form. By Corollary 2 we canconstruct an ordered multiple chain Π-extension ( ˜ H b , σ ) of ( K ( x ) , σ ) with (123) which is composedby the single chain Π-extensions (˜ F ℓ , σ ) of ( K ( x ) , σ ) with 1 ≤ ℓ ≤ p for some p ∈ Z ≥ with˜ F ℓ = K ( x ) h ˜ z ℓ, ih ˜ z ℓ, i . . . h ˜ z ℓ,s ℓ i , the automorphism σ : ˜ F ℓ → ˜ F ℓ given in (124) and the evaluationfunction ˜ev : ˜ F ℓ × Z ≥ → K defined by ˜ev | K ( x ) × Z ≥ = ev and (125). In particular, there are ν i,j , µ i,j such that ˜ H i,j ( n ) = ev(˜ z ν i,j ,µ i,j , n ) holds for all n ≥ max(0 , δ − h i = ˜ z n i, ν i, ,µ i, . . . ˜ z n i,li ν i,li ,µ i,li ∈ ˜ H b with˜ev(˜ h j , n ) = ˜ H j ( n ) ∀ n ≥ δ. (135)(c) Next we treat the geometric product expressions ˜ G ( n ) , . . . , ˜ G e ( n ) in (134). Following Remark 4we can construct a multiple chain P-extension ( A , σ ) of ( K , σ ) where the bases are from K ∗ suchthat there are g , . . . , g e ∈ A with ev( g i , n ) = ˜ G i ( n ) for all n ∈ Z ≥ . Then by Theorem 8 we canconstruct an RΠ-extension ( D , σ ) of ( ˜ K , σ ) with (126) together with the automorphism σ : D → D given in (127) and (128), with the evaluation function ˜ev : D × Z ≥ → ˜ K defined by ˜ev( c, n ) = c for all c ∈ ˜ K , n ∈ Z ≥ , (129) and (130), and with a difference ring homomorphism ϕ : A d → D such that for all f ∈ A and n ∈ Z ≥ we have (113). Thus for ˜ g j := ϕ ( g j ) with 1 ≤ j ≤ e we get˜ev(˜ g j , n ) = ˜ ev ( ϕ ( g j ) , n ) (113) = ev( g j , n ) = ˜ G j ( n ) ∀ n ≥ . (136)(d) By Corollary 3 we can merge these two difference rings to obtain an RΠ-extension ( E , σ ) of ( ˜ K ( x ) , σ )with (132) and the automorphism σ : E → E and the evaluation function ˜ev : E × Z ≥ → ˜ K definedaccordingly. As ( E , σ ) is an RΠ-extension of the rational difference field ( ˜ K ( x ) , σ ), it follows byTheorem 2 that ˜ τ : E → S ( ˜ K ) defined by (67) is a ˜ K -embedding. For 1 ≤ j ≤ e , define p j := ˜ c j ˜ r j ˜ g j ˜ h j ∈ E . With ˜ev(˜ r j , n ) = ˜ r j ( n ) and the evaluations of ˜ h j and ˜ g j given in (135) and (136) together with (134)it follows that for all 1 ≤ j ≤ e and for all n ≥ max(0 , δ −
1) we have P j ( n ) = ˜ c j ˜ r j ( n ) ˜ G j ( n ) ˜ H j ( n ) = ˜ev(˜ c j , n ) ˜ev(˜ r j , n ) ˜ev(˜ g j , n ) ˜ev(˜ h j , n ) = ˜ev(˜ c j ˜ r j ˜ g j ˜ h j , n ) = ˜ev( p j , n ) . (e) Finally, we can define a j = X v =( ν ,...,ν e ) ∈ S j a j, v p ν · · · p e ( n ) ν e ∈ E for 1 ≤ j ≤ m and get ev( a j , n ) = A j ( n ) forall n ≥ max( δ − , K is strongly σ -computable, all the ingredients delivered by Corollary 2 and Theorem 8can be computed. In particular, ( E , σ ), and τ with ˜ev and a , . . . , a m can be computed explicitly. (cid:3) As a consequence we are now in the position to solve Problem RPE as follows.40 orollary 4.
Let A ( n ) ∈ ProdE n ( K ( x )) with (2) . For A ( n ) = A ( n ) with m = 1 let δ ∈ Z ≥ , ( E , σ ) with the evaluation function ˜ev and the a := a ∈ E be the ingredients as provided in Theorem 9. Inparticular, let E = ˜ K ( x )[ ϑ ] h p i . . . h p s i where ϑ is the R -monomial with σ ( ϑ ) = ζ ϑ and let p , . . . , p s be the Π -monomials. Furthermore, let a = X v =( ν ,...,ν s ) ∈ ˜ S b v ( n ) ϑ µ p µ · · · p µ s s with ˜ S ⊆ { , . . . , λ − } × Z s finiteand a v ( x ) ∈ ˜ K ( x ) for v ∈ ˜ S . Then the following holds:(1) ev( ϑ, n ) = ζ n for all n ∈ Z ≥ ; furthermore, for ≤ i ≤ s we have ev( p i , n ) = Q i ( n ) for all n ≥ where Q i ( n ) ∈ Prod n ( ˜ K ( x )) .(2) For B ( n ) = X v =( ν ,...,ν s ) ∈ ˜ S b v ( n ) ( ζ n ) µ Q ( n ) µ · · · Q s ( n ) µ s we have A ( n ) = B ( n ) for all n ≥ δ .(3) The subring τ ( ˜ K ( x ))[ h ζ n i n ≥ ][ h Q ( n ) i n ≥ , h Q ( n ) − i n ≥ ] . . . [ h Q s ( n ) i n ≥ , h Q s ( n ) − i n ≥ ] of S ( ˜ K ) forms a Laurent polynomial ring extension of τ ( ˜ K ( x ))[ h ζ n i n ≥ ] . Thus the sequences pro-duced by Q ( n ) , . . . , Q s ( n ) are algebraically independent among each other over τ ( ˜ K ( x ))[ h ζ n i n ≥ ] .(4) We have that A ( n ) = 0 for all n ≥ d for some d ∈ Z ≥ if and only if a = 0 if and only B ( n ) isthe zero-expression. If this holds, A ( n ) = 0 for all n ≥ δ . Proof. (1) Note that ( E , σ ) is a multiple chain Π-extension of ( ˜ K ( x )[ ϑ ] , σ ) over ˜ K ( x ) equipped by an eval-uation function given by the iterative application of Lemma 2. Thus statement (1) follows.(2) By statement (2) of Theorem 9 it follows that ˜ev( a, n ) = A ( n ) holds for all n ≥ δ . By definitionof the ˜ev function we have ˜ ev ( a, n ) = B ( n ). Thus statement (2) holds.(3) Since τ : E → S ( ˜ K ) with τ ( f ) = h ˜ev( f, n ) i n ≥ is an injective difference ring homomorphism bystatement (2) of Theorem 9, statement (3) follows.(4) Since τ is injective, it follows that0 = A ( n ) for all n ≥ d for some d ∈ Z ≥ ⇐⇒ B ( n ) = ˜ev( a, n ) for all n ≥ max( d, δ ) ⇐⇒ τ ( a ) = τ injective ⇐⇒ a = 0 ⇐⇒ B ( n ) is the zero-expression.If this is the case, A ( n ) = B ( n ) = 0 for all n ≥ δ . This proves the last statement. (cid:3) Example 14 (Cont. Examples 1, 5, 7, 8, 9).
Let ( K ( x ) , σ ) be the rational difference field with K = Q ( √
3) equipped with the field automorphism σ : K ( x ) → K ( x ) and the evaluation functionev : K ( x ) × Z ≥ → K defined by σ ( x ) = x + 1 and (46) respectively. Given the nesting depth 2hypergeometric product P ( n ) with (8) in Example 1, we computed the following 3-refined shift-coprimeproduct representation form: P ( n ) = ˜ c r ( n ) ˜ G ( n ) ˜ H ( n ) (137)where ˜ c , r ( n ), ˜ G ( n ) and ˜ H ( n ) are given in (35), (36), (37), and (38) respectively. In particular, (137)holds for all n ∈ Z ≥ with n ≥ δ − H , σ ) of ( K ( x ) , σ ) with ˜ H = K ( x ) h z , ih z , ih z , i which is composed by the single chain Π-extensions of ( K ( x ) , σ ) defined initems (7) and (8). The automorphism and the evaluation function were defined as given in (54)and (55). In particular, the hypergeometric product expression ˜ H ( n ) is modelled by the expression˜ h = z , z , z , where ˜ h = h is taken from (57).Furthermore, we constructed the RΠ-extension ( D , σ ) of ( ˜ K , σ ) with ˜ K = Q ( √ , i ) equipped with theevaluation function ˜ev from Example 13. There ˜ G ( n ) is modelled by (122).Merging the two difference rings ( ˜ H , σ ) and ( D , σ ), we get the RΠ-extension ( E , σ ) of ( ˜ K ( x ) , σ ) with E = ˜ K ( x )[ ϑ ] h ˜ y , ih ˜ y , ih ˜ y , ih z , ih z , ih ˜ y , ih ˜ y , ih z , i where the automorphism σ : E → E and theevaluation function ˜ev : E × Z ≥ → ˜ K are defined by (104), (105), (86), (54), and (55). Following theproof of Theorem 9 we set p := ˜ c r ˜ g ˜ h and get for all n ≥ P ( n ) = ˜ev( p, n ) = − r, n ) ˜ev(˜ g, n ) ˜ev(˜ h, n )= − n − n ( n + 1) ( n + 2) 12 (1 − i ) ( i ) n ( i ( i n ) + 1) (cid:0) √ (cid:1) n (cid:0) n (cid:1) n +12 )2 n n +12 ) n Y k =3 ( k − ! n Y k =3 (cid:0) k + (cid:1)! n Y k =3 k Y j =3 ( j − . Based on this representation in an RΠ-extension, we can extract the following extra property. Since τ : E → S ( K ) is a ˜ K -embedding, it follows that the sub-difference ring ( R, S ) of ( S ( ˜ K ) , S ) with R = τ ( ˜ K ( x ))[ h i n i n ≥ ][ h Q ( n ) i n ≥ , h Q ( n ) − i n ≥ ] . . . [ h Q ( n ) i n ≥ , h Q ( n ) − i n ≥ ]and Q ( n ) = (cid:0) √ (cid:1) n , Q ( n ) = 2 n , Q ( n ) = 5 n , Q ( n ) = n Y k =3 ( k − ,Q ( n ) = n Y k =3 (cid:0) k + (cid:1) , Q ( n ) = 2( n +12 ) , Q ( n ) = 5( n +12 ) , Q ( n ) = n Y k =3 k Y j =3 ( j − E , σ ). In particular, we can conclude that R is a Laurent polynomial ring extension ofthe ring G = τ ( ˜ K ( x ))[ h i n i n ≥ ]. In a nutshell, the sequences generated by the products Q ( n ) , . . . , Q ( n )are algebraically independent among each other over the ring G . NestedProducts
In the following we will demonstrate how our tools can be activated with the help of the Mathe-matica package
NestedProduct . We start with the nested hypergeometric product expression A ( n ) = 12 n − Y k =1 k − Y i =1 ( i + 1) ( i + 2)4 (2 i + 3) ! ∈ ProdE n ( Q ( x )) (138)from [Kau18, Example 3] which was guessed using the Mathematica package RATE written by Chris-tian Krattenthaler; see [Kra97]. 42fter loading the package In[1]:= < NestedProducts — A package by Evans Doe Ocansey — © RISC into Mathematica, we define the product with the command In[2]:= A = 12 FormalProduct (cid:20) (cid:16) FormalProduct h (i + 1) (i + 2)4 (2 i + 3) , { i , , k − } ] (cid:17) , { k , , n − } (cid:21) ; Here FormalProduct [f, { k,a,n+b } ] (as shortcut one can use FProduct ) defines a nested product Q n + bk = a f where a, b are integers and the multiplicand f , free of n , must be an expression in termsof nested products whose outermost upper bounds are given by k or an integer shift of k . Thenapplying the command ProductReduce to A we solve Problem RPE and get the result In[3]:= B = ProductReduce[A] Out[3]= (cid:0) n (cid:1) n Y k =1 (cid:0) k + (cid:1)! n Y k =1 k Y i =1 ( i + ) ! ( + ) (cid:0) n (cid:1) (cid:18) (cid:0) n + (cid:1)(cid:19) n Y k =1 ( k + ) ! n Y k =1 k Y i =1 (cid:0) i + (cid:1)! Mathematica Session 1 Internally, the package synchronizes in (138) the upper bounds to n and k , respectively. This yields9 ( n + 1) ( n + 2)2 (2 n + 3) n Y j =1 j + 3) ( j + 1) ( j + 2) n Y k =1 (2 k + 3) k + 1) ( k + 2) k Y j =1 ( j + 1) ( j + 2)4 (2 j + 3) . (139)Then (139) is reduced further to Out[3] in terms of the algebraically independent products2 n , n , n Y k =1 (cid:0) k + (cid:1) , n Y k =1 ( k + 1) , n +12 ) = n Y k =1 k Y i =1 , n Y k =1 k Y i =1 (cid:0) i + (cid:1) , n Y k =1 k Y i =1 ( i + 1) . (140)Note that one could have represented the product in (138) directly within a Π-extension by simplytaking the inner product as the first Π-monomial and the outermost product as the second Π-monomial.However, for more complicated expression such a representation can be rather challenging.The full capability of our machinery can be illustrated by combining, e.g., the expression (138)with the following related product (where one of the inner products is slightly modified): In[4]:= A = A + FProduct h (k + 1) (2k + 1) (k + 2) FProduct[ − (i + 1)(i + 2)4(2i − , { i , , k } ] , { k , , n } i ; Then it is not immediate how this expression can be represented in an RΠ-extension. But applyingour toolbox this task can be automatically accomplished: In[5]:= B = ProductReduce[A ] Out[5]= (cid:0) (cid:0) n + + (cid:1) + ( + i ) ( + ) ( i ) n + ( − i ) ( + ) (( i ) n ) (cid:1) n n Y k =1 k Y i =1 ( i + ) ! ( n + ) (cid:18) (cid:0) n + (cid:1)(cid:19) n Y k =1 ( k + ) ! n Y k =1 k Y i =1 (cid:0) i − (cid:1)! Mathematica Session 2 43y solving Problem RPE the expression can be rephrased in an RΠ-extension with the R-monomial i n and the Π-monomials given in (140). In short, together with i n the expression can be reduced againin terms of the algebraic independent products (140).Similar expressions as given in (138) arise during challenging evaluations of determinants; see, e.g.,[MRRJ83, Zei96, Kra01]. We expect that the new tools elaborated in this article will prove beneficialin related but more complicated product expressions.We conclude this section by combining our tools from above with the summation package Sigma .After loading in In[6]:= < Sigma - A summation package by Carsten Schneider © RISC-Linz we insert the sum In[7]:= mySum = SigmaSum[ (cid:16) − Y j=1 (1 + j) (cid:17) k Y j=1 j j Y i=1 (1 + i) − (cid:16) (3 + k) k Y j=1 − j(2 + j) (cid:17) k Y j=1 j Y i=1 − i(2 + i) , { k , , n } ]; Afterwards we can activate the available summation algorithms of Sigma with the function call SigmaReduce and succeed in eliminating the summation sign: In[8]:= SigmaReduce[mySum] Out[8]= − ( + n ) ( + n ) (cid:16) − + ( + i ) (cid:0) − i + ( i n ) (cid:1) ( + n ) i n (cid:17) ( n !) k Y i = Y j = j Mathematica Session 3 In other words, we have derived the simplification n X k =1 (cid:16) − k ) (2 + k ) k Y j =1 (1 + j ) (cid:17) k Y j =1 j j Y i =1 (1 + i ) − (cid:16) k ) (3 + k ) k Y j =1 − j (2 + j ) (cid:17) k Y j =1 j j Y i =1 − i (2 + i ) ! = 4 − 13 (1 + n ) (2 + n ) (cid:16) − i ) (cid:0) − i + ( i n ) (cid:1) (3 + n ) i n (cid:17) ( n !) k Y i =1 i Y j =1 j . We emphasize that the summand given in In[8] has been transformed internally with the package NestedProducts to the form13 (1 + k ) k Y i =1 i ! k Y i =1 i Y j =1 j − − − i ) (2 + k ) i k + ( − i ) (2 + k ) ( i k ) + (1 + k ) (2 + k ) (cid:16) − i ) (3 + k ) i k + ( − − i ) (3 + k ) ( i k ) (cid:17) k Y i =1 i ! ! . Then Sigma reads off the derived products and rephrases them directly to a tower of RΠ-extensions(without the exploitation of the available tools in Sigma that can check whether the constant fieldremains unchanged). Afterwards the underlying summation algorithms of Sigma are applied to derivethe final result. 44 . Conclusion We enhanced non-trivially the ideas from [Sch05, Sch14, OS18] (related also to [AP10, CFFL11])in order to solve Problem RPE in Theorem 9 and Corollary 4 above. There we cannot only re-duce or simplify expressions in terms of hypergeometric products of nesting depth 1 but in termsof hypergeometric products of nesting depth ≥ 1. More precisely, the expression can be reduced toan expression in terms of one root of unity product of the form ζ n and hypergeometric products Q ( n ) , . . . , Q s ( n ) ∈ Prod n ( K ( x )) of arbitrary but finite nesting depth which are algebraically indepen-dent among each other. This latter property has been extracted from results elaborated in [Sch17](which are inspired by [PS97]). Combined with the existing difference ring algorithms for symbolicsummation [Kar81, Sch01, Sch16] this yields a complete summation machinery to reduce and simplifynested sums over hypergeometric products of arbitrary but finite nesting depth.A natural future task is to enhance this combined toolbox of the packages NestedProducts and Sigma further and to tackle, e.g., definite summation problems. In particular, the interaction withthe available creative telescoping algorithms [Sch07a, Sch08, Sch10b, Sch15] and recurrence solvingalgorithms [ABPS20] should be explored further.Following the ideas from [OS18] one might extend the above machinery to the class of nested q -hypergeometric products covering also the multibasic and mixed case [BP99]. Another open task is tocombine the above ideas with contributions from [Sch20] (based on Smith normal form calculations)to find optimal representations of such nested products. This means that in the output expression theorder λ of the primitive root of unity ζ in ζ n and the number s of algebraically independent products Q ( n ) , . . . , Q s ( n ) should be minimized.Finally, it would be interesting to see if the class of hypergeometric products of finite nesting depthcan be generalized further to products of the form (1) where in the multiplicands the products do notappear only in form of Laurent polynomial expressions. References [ABPS20] Sergei A. Abramov, Manuel Bronstein, Marko Petkovˇsek, and Carsten Schneider, On Ra-tional and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ ∗ -fieldextensions , Tech. report, Arxiv, 2020, arXiv:2005.04944 [cs.SC].[Abr71] Sergej A. Abramov, On the summation of rational functions , USSR Computational Math-ematics and Mathematical Physics (1971), no. 4, 324–330.[AP94] Sergei A. Abramov and Marko Petkovˇsek, D’Alembertian solutions of linear differentialand difference equations , Citeseer, 1994.[AP10] , Polynomial ring automorphisms, rational ( w, σ ) -canonical forms, and the assign-ment problem , Journal of Symbolic Computation (2010), no. 6, 684–708.[AZ96] Sergei A. Abramov and Eugene V. Zima, D’Alembertian solutions of inhomogeneous linearequations(differential, difference, and some other) , International Conference on Symbolicand Algebraic Computation: Proceedings of the 1996 international symposium on Symbolicand algebraic computation, vol. 24, 1996, pp. 232–240.[BP99] Andrej Bauer and Marko Petkovˇsek, Multibasic and mixed hypergeometric gosper-type al-gorithms , Journal of Symbolic Computation (1999), no. 4-5, 711–736.[BRS18] Johannes Bl¨umlein, Mark Round, and Carsten Schneider, Refined holonomic summationalgorithms in particle physics , Advances in Computer Algebra. WWCA 2016. (E. Zima andC. Schneider, eds.), Springer Proceedings in Mathematics & Statistics, vol. 226, Springer,2018, arXiv:1706.03677 [cs.SC], pp. 51–91 (english).45CFFL11] Shaoshi Chen, Ruyong Feng, Guofeng Fu, and Ziming Li, On the structure of compati-ble rational functions , Proceedings of the 36th international symposium on Symbolic andalgebraic computation, ACM, 2011, pp. 91–98.[Chy00] Fr´ed´eric Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions ,Discrete Mathematics (2000), no. 1, 115 – 134.[CJKS13] Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, and Michael F. Singer, Desingular-ization explains order-degree curves for Ore operators , Proceedings of ISSAC’13 (ManuelKauers, ed.), 2013, pp. 157–164.[Ge93a] Guoqiang Ge, Algorithms Related to the Multiplicative Representation of Algebraic Num-bers , Ph.D. thesis, Univeristy of California at Berkeley, 1993.[Ge93b] , Testing equalities of multiplicative representations in polynomial time , Foundationsof Computer Science, 1993. Proceedings., 34th Annual Symposium on, IEEE, 1993, pp. 422–426.[Gos78] Ralph William Gosper, Decision procedures for indefinite hypergeometric summation , Proc.Nat. Acad. Sci. U.S.A. (1978), 40–42.[Hoe99] Mark Van Hoeij, Finite singularities and hypergeometric solutions of linear recurrence equa-tions , Journal of Pure and Applied Algebra (1999), no. 1-3, 109–131.[HS99] Peter A. Hendriks and Michael F. Singer, Solving difference equations in finite terms ,J. Symbolic Comput. (1999), no. 3, 239–259.[HS08] Charlotte Hardouin and Michael F. Singer, Differential galois theory of linear differenceequations , Mathematische Annalen (2008), no. 2, 333–377.[Kar81] Michael Karr, Summation in Finite Terms , Journal of the ACM (JACM) (1981), no. 2,305–350.[Kar85] Michael Karr, Theory of summation in finite terms , Journal of Symbolic Computation (1985), no. 3, 303 – 315.[Kau05] Manuel Kauers, Algorithms for Nonlinear Higher Order Difference Equations , Ph.D. thesis,RISC-Linz, October, 2005.[Kau18] , The guess-and-prove paradigm in action , Internationale MathematischeNachrichten (2018), 1–15.[Kou13] Christoph Koutschan, Creative telescoping for holonomic functions , Computer algebra inquantum field theory, Texts Monogr. Symbol. Comput., Springer, Vienna, 2013, pp. 171–194.[Kra97] Christian Krattenthaler, RATE – A Mathematica guessing machine , 1997, Available at https://mat.univie.ac.at/~kratt/rate/rate.html .[Kra01] , Advanced determinant calculus , The Andrews Festschrift, Springer, 2001, pp. 349–426.[KS06] Manuel Kauers and Carsten Schneider, Indefinite summation with unspecified summands ,Discrete mathematics (2006), no. 17, 2073–2083.[KZ08] Manuel Kauers and Burkhard Zimmermann, Computing the algebraic relations of c-finitesequences and multisequences , Journal of Symbolic Computation (2008), no. 11, 787–803. 46MRRJ83] William H. Mills, David P. Robbins, and Howard Rumsey Jr., Alternating sign matrices anddescending plane partitions , Journal of Combinatorial Theory, Series A (1983), no. 3,340–359.[Oca19] Evans Doe Ocansey, Difference Ring Algorithms for Nested Products , Ph.D. thesis, RISC,Johannes Kepler University Linz, Nov., 2019.[OS18] Evans Doe Ocansey and Carsten Schneider, Representing ( q -)hypergeometric products andmixed versions in difference rings , Waterloo Workshop on Computer Algebra, Springer,2018, pp. 175–213.[Pau95] Peter Paule, Greatest factorial factorization and symbolic summation , Journal of symboliccomputation (1995), no. 3, 235–268.[Pet92] Marko Petkovˇsek, Hypergeometric solutions of linear recurrences with polynomial coeffi-cients , Journal of symbolic computation (1992), no. 2-3, 243–264.[PR97] Peter Paule and Axel Riese, A Mathematica q-analogue of Zeilberger’s algorithm basedon an algebraically motivated approach to q -hypergeometric telescoping , Special Functions,q-Series and Related Topics (M. Ismail and M. Rahman, eds.), vol. 14, AMS, 1997, pp. 179–210.[PS97] Marius Van Der Put and Michael F. Singer, Galois theory of difference equations , 1997.[PS19] Peter Paule and Carsten Schneider, Towards a symbolic summation theory for unspeci-fied sequences , Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum FieldTheory, Springer, 2019, pp. 351–390.[PWZ96] Marko Petkovˇsek, Herbert S. Wilf, and Doron Zeilberger, A=B, AK Peters Ltd , Wellesley,MA (1996).[PZ13] Marko Petkovˇsek and Helena Zakrajˇsek, Solving linear recurrence equations with polynomialcoefficients , Computer Algebra in Quantum Field Theory, Springer, 2013, pp. 259–284.[Rie03] Axel Riese, qMultisum - A package for proving q -hypergeometric multiple summation iden-tities , J. Symbolic Comput. (2003), 349–377.[Ris69] Robert H. Risch, The problem of integration in finite terms , Trans. Amer. Math. Soc. (1969), 167–189. MR 237477[Sch01] Carsten Schneider, Symbolic Summation in Difference Fields , Ph.D. thesis, RISC, J. KeplerUniversity Linz, May, 2001, (Published as Technical report no. 01-17 in RISC ReportSeries.).[Sch05] , Product representations in ΠΣ -fields , Annals of Combinatorics (2005), no. 1,75–99.[Sch07a] , Simplifying sums in ΠΣ ∗ -extensions , Journal of Algebra and Its Applications (2007), no. 03, 415–441.[Sch07b] , Symbolic summation assists combinatorics , S´em. Lothar. Combin (2007), no. 1-36, B56b.[Sch08] , A refined difference field theory for symbolic summation , Journal of Symbolic Com-putation (2008), no. 9, 611–644.[Sch10a] , Parameterized telescoping proves algebraic independence of sums , Annals of Com-binatorics (2010), no. 4, 533–552, Appeared earlier in Proc. of FPSAC 2007.47Sch10b] , Structural theorems for symbolic summation , Appl. Algebra Engrg. Comm. Com-put. (2010), no. 1, 1–32 (english).[Sch14] , A streamlined difference ring theory: Indefinite nested sums, the alternating signand the parameterized telescoping problem , Symbolic and Numeric Algorithms for Scien-tific Computing (SYNASC), 2014 15th International Symposium, IEEE Computer Society,2014, arXiv:1412.2782, pp. 26–33 (english).[Sch15] , Fast algorithms for refined parameterized telescoping in difference fields , ComputerAlgebra and Polynomials, Springer, 2015, pp. 157–191.[Sch16] , A difference ring theory for symbolic summation , Journal of Symbolic Computation (2016), 82–127.[Sch17] , Summation theory II: Characterizations of RΠΣ -extensions and algorithmic as-pects , Journal of Symbolic Computation (2017), 616–664.[Sch20] , Minimal representations and algebraic relations for single nested products , Pro-gramming and Computer Software (2020), 133–161.[Weg97] Kurt Wegschaider, Computer generated proofs of binomial multi-sum identities , Master’sthesis, RISC, J. Kepler University, May 1997.[WZ92] Herbert Wilf and Doron Zeilberger, An algorithmic proof theory for hypergeometric (ordi-nary and “q”) multisum/integral identities , Invent. Math. (1992), 575–633.[Zei90] Doron Zeilberger, A holonomic systems approach to special functions identities , J. Comput.Appl. Math. (1990), 321–368.[Zei91] , The method of creative telescoping , J. Symbolic Comput. (1991), 195–204.[Zei96] , Proof of the alternating sign matrix conjecture , Electron. J. Combin3