Algebraic independence of sequences generated by (cyclotomic) harmonic sums
aa r X i v : . [ c s . S C ] A p r ALGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY(CYCLOTOMIC) HARMONIC SUMS
JAKOB ABLINGER AND CARSTEN SCHNEIDER
Abstract.
An expression in terms of (cyclotomic) harmonic sums can be simplified bythe quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sumsare algebraically independent within the quasi-shuffle algebra. In this article we show thatthe basis sums can be represented within a tower of difference ring extensions where theconstants remain unchanged. This property enables one to embed this difference ring forthe (cyclotomic) harmonic sums into the ring of sequences. This construction implies thatthe sequences produced by the basis sums are algebraically independent over the rationalsequences adjoined with the alternating sequence. Introduction
Special functions like the harmonic numbers and more generally indefinite nested sumsdefined over products play a dominant role in many research branches, like in combinatorics,number theory, and in particle physics. For concrete examples within these research areas inconnection with symbolic summation see, e.g., [49, 35], [32, 48] and [4, 5], respectively. Inparticular, these nested sums cover the class of d’Alembertian solutions [12], a sub-class ofLiouvillian solutions [24], of linear recurrence relations; for further details see [34].Numerous properties of such sum classes, like the harmonic sums [17, 51], cyclotomic har-monic sums [8], generalized harmonic sums [30, 9] or binomial sums [23, 22, 52, 7] have beenexplored. In particular, the connection of the nested sums to nested integrals (i.e., to multi-ple polylogarithms and generalizations of them) via the (inverse) Mellin transform [36], theanalytic continuation [13, 18] of nested sums or the calculation of asymptotic expansions ofsuch sums [21, 15, 16] has been worked out. For further details and generalizations of theseresults we refer to [8, 9, 7]. The underlying algorithms are implemented in the Mathematicapackage
HarmonicSums [2, 3].Among all these algorithmic constructions, a key technology is the elimination of alge-braic dependencies of the arising nested sums within a given expression to gain compactrepresentations. Here the Mathematica package
Sigma [38, 43] provides strong tools that cansimplify, among many other features, an expression in terms of indefinite nested product-sumsto an expression in terms of such sums that are all algebraically independent in the analysissense [40, 42, 46, 44, 47]. This means that the sequences with entries from a field K , thatare produced by the reduced sums, are algebraically independent. In order to accomplishthis task, the arising sums and products are represented in a difference ring, i.e., the sumobjects are represented in a ring A and the shift behaviour of the sums is modelled by aring automorphism σ : A → A . More precisely, the sums and products are represented in an Key words and phrases. harmonic sums, cyclotomic harmonic sums, quasi-shuffle algebra, algebraic inde-pendence, difference rings, Σ ∗ -extensions, ring of sequences, difference ring embedding.Supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15). R ΠΣ ∗ -extension [46, 44] with the distinguished property that the set of constants is preciselythe field K , i.e., { c ∈ A | σ ( c ) = c } = K . Exactly this property enables one to embed the ring A into the ring of sequences. This tech-nology has been used to show in [46] that the sequences of the generalized harmonic numbersare algebraically independent over the rational sequences. In particular, fast summation al-gorithms [41, 45, 39] in the setting of difference rings and fields support this constructionalgorithmically and expressions with up to several hundred algebraically independent sumscan be generated automatically. However, recently we were faced with QCD calculations [6]with expressions of about 1GB and more than 20000 sums. At this level, the difference ringalgorithms failed to eliminate all algebraic relations in a reasonable amount of time.In order to perform such large scale calculations, another key property of certain classesof indefinite nested sums can be utilized: they obey quasi-shuffle algebras [25, 26, 27]. Thisenables one to rewrite any polynomial expression in terms of indefinite nested sums as alinear combination of indefinite nested sums. As worked out in [14] and continued in [8, 2, 9],this feature can be used to hunt for algebraic relations among the occurring indefinite nestedsums and to express the compact result in terms of the so-called basis sums which cannot beeliminated further by the quasi-shuffle algebra. Using the HarmonicSums package expressionsas mentioned above could be reduced to several MB in terms of about several thousand basissums; for details see [6]. Summarizing, using the property of the underlying quasi-shufflealgebra one obtains dramatic compactifications within the demanding calculations in particlephysics.A natural question is if the obtained sums induced by the quasi-shuffle algebra are alsoalgebraically independent in the sense of analysis, i.e., if the sequences produced by the nestedsums are algebraically independent. A special variant for non-alternating harmonic sums hasbeen accomplished in [21] using the knowledge of certain integral representations. In thefollowing we will focus on the general case for the harmonic sums [17, 51] X n ≥ i ≥ i ≥···≥ i k ≥ sign( c ) i i | c | · · · sign( c k ) i k i | c k | k (1)with non-negative integers n and non-zero integers c i (1 ≤ i ≤ k ) and for their cyclotomicversions [8]: for K being a field containing the rational numbers the summand is of the kind z i j j ( a j i j + b j ) c j , a j , b j , c j ∈ N , z j ∈ K \ { } (2)and i j denotes the summation variable.In this article we will consider the so-called basis sums induced by the quasi-shuffle alge-bra. This means we consider a particular chosen set of nested sums that generate all othernested sums and that do not possess any further relations using the quasi-shuffle algebraoperation. Our main result is that these basis sums are also algebraically independent assequences. More precisely, consider the ring of sequences which is defined by the set of se-quences K N = {h a n i n ≥ | a n ∈ K } equipped with component-wise addition and multiplicationwhere two sequences are identified as equal if they differ only by finitely many entries. Thenwe will show that the basis sums evaluated to such elements of the ring of sequences are For the corresponding difference field theory see [28, 29]. N denotes the positive integers and N = N ∪ { } . LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 3 algebraically independent: they are algebraically independent over the sub-ring of sequencesthat is generated by all rational functions from K ( n ) and ( − n . We will derive this result byshowing that the basis sums generate an R ΠΣ ∗ -extension in the difference ring sense. Thismeans that the basis sums generate a polynomial ring equipped with a shift operator suchthat the set of constants is precisely K . Based on this particularly nice structure it will followby difference ring theory [50, 47] that this difference ring can be embedded by an injectivedifference ring homomorphism into the ring of sequences. In other words, the algebraic prop-erties of the polynomial ring (in particular, the algebraic independence of variables of thepolynomial ring, which are precisely the basis sums) carry over into the setting of sequences.The outline of the article is as follows. In Section 2 we will set up the general frameworkfor (cyclotomic) harmonic sums. In Section 3 we will present basic constructions to represent(cyclotomic) harmonic sums in a difference ring. In Section 4 we will introduce the quasi-shuffle algebra for (cyclotomic) harmonic sums and will work out various properties that linkthe quasi-shuffle algebra with our difference ring construction. In Section 5 we define thereduced difference ring for (cyclotomic) harmonic sums in which all algebraic relations areeliminated that are induced by the quasi-shuffle algebra. We will provide new structuralresults obtained by the difference ring theory of R ΠΣ ∗ -extensions in Section 6. In Section 7we will combine all these results and will show that our reduced difference ring is built by atower of R ΠΣ ∗ -extensions. As a consequence we can conclude that this ring can be embeddedinto the ring of sequences. A conclusion is given in Section 8.2. A general framework for cyclotomic harmonic sums
Throughout this article we assume that K is a field containing Q as a subfield. In particular,we assume that there is a linear ordering < on K . For a set B , B ∗ denotes the set of all finitewords over B (including the empty word), i.e., B ∗ = { b , . . . , b k | k ≥ b i ∈ B } . Furthermore, we define the alphabet A := { ( a, b, c, z ) | a, c ∈ N , b ∈ N , z ∈ K \ { } with b < a and gcd( a, b ) = 1 } as a totally ordered, graded set. More precisely, the degree of ( a, b, c, d ) ∈ A is denoted by | ( a, b, c, d ) | := c . This establishes the grading A i = { a ∈ A | | a | = i } . Moreover, we define thelinear order < on A in the following way:( a , b , c , z ) < ( a , b , c , z ) if c < c ( a , b , c, z ) < ( a , b , c, z ) if a < a ( a, b , c, z ) < ( a, b , c, z ) if b < b ( a, b, c, z ) < ( a, b, c, z ) if z < z . Furthermore, we define the function λ : A × N → K λ (( a, b, c, z ) , i ) z i ( ai + b ) c . (3) JAKOB ABLINGER AND CARSTEN SCHNEIDER
Note that λ ((1 , , c , z ) , i ) λ ((1 , , c , z ) , i ) = λ ((1 , , c + c , z z ) , i ). For arbitrary letters in A the connection is more complicated but there is always a relation of the form λ (( a , b , c , z ) , i ) λ (( a , b , c , z ) , i ) = k X j =1 r j λ (( e j , f j , g j , h j ) , i ) (4)with r j ∈ Q , ( e j , f j , g j , h j ) ∈ A and g j ≤ c + c see, e.g., [8, 2]. For n ∈ N , k ∈ N , a i ∈ A with 1 ≤ i ≤ k we define nested sums (compare [8, 2]) S a ( n ) = n X i =1 λ ( a , i ) S a ,...,a k ( n ) = n X i =1 λ ( a , i ) S a ,...,a k ( i ) . (5)Moreover, we define the weight function w on these nested sums: w ( S a ,a ,a ,...,a k ( n )) = | a | + · · · + | a k | and extend it to monomials such that the weight of a product of nested sumsis the sum of the weights of the individual sums, i.e., w ( S a ( n ) S a ( n ) · · · S a k ( n )) = w ( S a ( n )) + w ( S a ( n )) + · · · + w ( S a k ( n )) . Instead of S a ,a ,...,a k ( n ) we will also write S a a ··· a k ( n ) or S a ( n ) with a = a a . . . a k ∈ A ∗ .A product of two nested sums with the same upper summation limit can be written interms of single nested sums: for n ∈ N ,S a ,...,a k ( n ) S b ,...,b l ( n ) = n X i =1 λ ( a , i ) S a ,...,a k ( i ) S b ,...,b l ( i )+ n X i =1 λ ( b , i ) S a ,...,a k ( i ) S b ,...,b l ( i ) − n X i =1 λ ( a , i ) λ ( b , i ) S a ,...,a k ( i ) S b ,...,b l ( i ) . (6)Note that the product of the two sums within the summands of the right side can be expandedfurther by using again this product formula. Applying this reduction recursively will lead toa linear combination of sums S b ,...,b r ( n ) with b i ∈ A . In particular, the maximum of all theweights of the derived sums is precisely the weight of the left hand side expression.We can consider different subsets of A :(1) If we consider only letters of the form (1 , , c,
1) with c ∈ N , i.e., we restrict to A h := A ∩ ( { } × { } × N × { } ) , then we are dealing with harmonic sums see, e.g.,[14, 51].(2) If we consider only letters of the form (1 , , c, ±
1) with c ∈ N , i.e., we restrict to A a := A ∩ ( { } × { } × N × { , − } ) , then we are dealing with alternating harmonic sums see, e.g.,[14, 51].(3) Let M ⊂ N be a finite subset of N . If we consider only letters of the form ( a, b, c, ± a ∈ M ; c ∈ N ; b ∈ N , b ≤ a , gcd( a, b ) = 1, i.e., we restrict to A c ( M ) := A ∩ ( M × N × N × { , − } ) , LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 5 then we are dealing with cyclotomic harmonic sums see, e.g.,[8, 2].(4) If we consider only letters of the form ( a, b, c, ±
1) with a, c ∈ N ; b ∈ N , b ≤ a ,gcd( a, b ) = 1, i.e., we restrict to A c := A ∩ ( N × N × N × { , − } ) , then we are dealing with the full set of cyclotomic harmonic sums see, e.g.,[8, 2].Note that for every finite subset M of N we have A h ⊂ A a ⊂ A c ( M ) ⊂ A c . Throughout this article we will assume that
H ∈ { A h , A a , A c ( M ) } (7)holds. In particular, we call a sum S a a ...a k ( n ) with a i ∈ H also H -sum.3. A basic difference ring construction for the expression of H -sums In the following we will define a difference ring in which we will represent the expressionsof H -sums. Definition 1.
An expression of H -sums in n over a field K is built by (1) rational expressions in n with coefficients from K , i.e., elements from the rationalfunction field K ( n ) , (2) ( − n that occurs in the numerator, (3) the H -sums that occur as polynomial expressions in the numerator. If f is such an expression we use for λ ∈ K ( n ) the shortcut f ( λ ) := f | n → λ . Sometimes we also use the notation f ( n ) to indicate that the expression depends on a symbolicvariable n . We say that an expression e ( n ) of H -sums has no pole for all n ∈ N with n ≥ λ for some λ ∈ N , if the rational functions occurring in e ( n ) do not introduce poles at anyevaluation n → ν for ν ∈ N with ν ≥ λ . If this is the case, one can perform the evaluation e ( ν ) for all ν ∈ N with ν ≥ λ . For a more rigorous definition of indefinite nested product-sumexpressions (containing as special case the H -sums) in terms of term algebras, we refer to [42]which is inspired by [31].These expressions will be represented in a commutative ring A and the shift operator actingon the expressions in terms of H -sums will be rephrased by a ring automorphism σ : A → A .Such a tuple ( A , σ ) of a ring A equipped with a ring automorphism is also called differencering; if A is a field, ( A , σ ) is also called a difference field. In such a difference ring we call c ∈ A a constant if σ ( c ) = c and denote the set of constants byconst( A , σ ) = { σ ( c ) = c | c ∈ A } . In general, const( A , σ ) is a subring of ( A , σ ). But in most applications we take care thatconst( A , σ ) itself forms a field.Our construction will be accomplished step by step. Namely, suppose that we are givenalready a difference ring ( A H d , σ ) in which we succeeded in representing parts of our H -sums.In order to enrich this construction, we will extend the ring from A H d to A H d +1 and will extendthe ring automorphism σ to a ring automorphism σ ′ : A H d +1 → A H d +1 , i.e., for any f ∈ A H d we have that σ ′ ( f ) = σ ( f ). We say that such a difference ring ( A H d +1 , σ ′ ) is a difference ring JAKOB ABLINGER AND CARSTEN SCHNEIDER extension of ( A H d , σ ); in short, we also write ( A H d , σ ) ≤ ( A H d +1 , σ ′ ). Since σ and σ ′ agree on A H d , we usually do not distinguish anymore between σ and σ ′ .We start with the rational function field K ( n ) and define the field/ring automorphism σ : K ( n ) → K ( n ) with σ ( f ) = f | n → n +1 . It is easy to verify that const( K ( n ) , σ ) = const( K , σ ) = K . So far, we can model rationalexpressions in n in the field K ( n ) and can shift these elements with σ .Next, we want to model the object ( − n with the relations (( − n ) = 1 and ( − n +1 = − ( − n . Therefore we take the ring K ( n )[ x ] subject to the relation x = 1. Then one canverify that there is a unique difference ring extension ( K ( n )[ x ] , σ ) of ( K ( n ) , σ ) with σ ( x ) = − x .In particular, we have thatconst( K ( n )[ x ] , σ ) = const( K ( n ) , σ ) = K . Precisely, this difference ring ( K ( n )[ x ] , σ ) enables one to represent all rational expressions in n together with objects ( − n that are rephrased by x .Before we can continue with our construction for H -sums, we observe the following easy,but important fact. Lemma 1.
Let ( A , σ ) be a difference ring and let A [ t ] be a polynomial ring, i.e., t is tran-scendental over A , and let β ∈ A . Then there is a unique difference ring extension ( A [ t ] , σ ) of ( A , σ ) with σ ( t ) = t + β . We will use this lemma iteratively in order to adjoin all H -sums to the difference ring( K ( n )[ x ] , σ ). This construction is done inductively on the weight of the sums. It is useful todefine the following function (compare (3)):¯ λ : H → K ( n )[ x ]¯ λ (( a, b, c, z )) x z − ( an + b ) c . The base case is the already constructed difference ring ( A H , σ ) with A H = K [ x ]( n ). Nowsuppose that we constructed the difference ring ( A H d − , σ ) for all H -sums of weight < d . Thenwe will construct a difference ring extension ( A H d , σ ) which covers precisely the H -sums ofweight ≤ d . Consider all H -sums with weight d ∈ N , say S ( d )1 ( n ) , . . . , S ( d ) n d ( n ) . To these sums we attach the variables s ( d )1 , . . . , s ( d ) n d (8)of weight d , respectively. Now we define the polynomial ring A H d = A H d − [ s ( d )1 , . . . , s ( d ) n d ]. To thisend, we extend σ from A H d − to A H d . Suppose that s ( d ) i models the H -sums S ( d ) i = S a i ,...,a iri with a i + · · · + a ir = d . Note that S ( d ) i ( n + 1) = S a i ,...,a iri ( n + 1) = S a i ,...,a iri ( n ) + λ ( a i , n + 1) S a i ,...,a iri ( n + 1)and S ( d ) i ( n −
1) = S a i ,...,a iri ( n −
1) = S a i ,...,a iri ( n ) − λ ( a i , n ) S a i ,...,a iri ( n ) LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 7 where S a i ,...,a iri is a H -sum of weight d − | a i | . Let s ( d −| a i | ) i ∈ A H d −| a i | which models thissum. Therefore we extend σ from A H d − to A H d subject to the relations σ ( s ( d ) i ) = s ( d ) i + σ (¯ λ ( a i )) σ ( s ( d −| a i | ) i ) (9)by applying Lemma 1 iteratively. This means, we first adjoin s ( d )1 to A H d − and extend theautomorphism with (9) for i = 1, then we adjoin to this ring the variable s ( d )2 and extend theautomorphism with (9) for i = 2, etc. We remark that this construction implies that σ − ( s ( d ) i ) = s ( d ) i − ¯ λ ( a i ) s ( d −| a i | ) i . (10)By construction ( A H d , σ ) is a difference ring extension of ( A H d − , σ ).Finally, we define the polynomial ring A H := K ( n )[ x ][ s (1)1 , . . . , s (1) n ][ s (2)1 , . . . , s (2) n ] . . . with infinitely many variables, which represents all H -sums. In particular, we define the ringautomorphism σ ′ : A H → A H as follows. For any f ∈ A H , we can choose a d ∈ N such that f ∈ A H d . This defines σ ′ ( f ) := σ ( f ) with σ : A H d → A H d . By construction, σ ′ | A H d = σ where σ is the automorphism of ( A H d , σ ). It is easy to see that ( A H , σ ′ ) is a difference ring and thatit is a difference ring extension of ( A H d , σ ). Again we do not distinguish anymore between σ and σ ′ . To sum up, we get the chain of difference ring extensions( K ( n )[ x ] , σ ) = ( A H , σ ) ≤ ( A H , σ ) ≤ ( A H , σ ) ≤ · · · ≤ ( A H , σ ) . (11)For convenience, we will also write ¯ S a ,...,a k for the variable s ( d ) j . In this way, we may writee.g., σ ( ¯ S a ,...,a iri ) = ¯ S a i ,...,a iri + σ (¯ λ ( a i )) σ ( ¯ S a i ,...,a iri ) σ − ( ¯ S a i ,...,a iri ) = ¯ S a i ,...,a iri − ¯ λ ( a i ) ¯ S a i ,...,a iri instead of (9) and (10), respectively.To give a r´esum´e, we can express every expression of H -sums over K in ( A H , σ ). Conversely,if we are given a ring element f ∈ A H , we denote by expr( f ) the expression that is obtainedwhen all occurrences of x are replaced by ( − n and all variables s ( d ) i are replaced by theattached H -sums with upper summation range n . This will lead to an expression of H -sumsin n over K . In this way, we can jump between the function and difference ring worlds.Now let f, g ∈ A H . Thenexpr( f + g )( n ) = expr( f )( n ) + expr( g )( n ) and expr( f g )( n ) = expr( f )( n ) expr( g )( n );recall that for an expression e of H -sums, e ( n ) is used to emphasize the dependence on thesymbolic variable n . If expr( f )( n ) and expr( g )( n ) have no poles for all n ≥ δ for some δ ∈ N ,then it follows thatexpr( f + g )( λ ) = expr( f )( λ ) + expr( g )( λ ) and expr( f g )( λ ) = expr( f )( λ ) expr( g )( λ )for all λ ∈ N with λ ≥ δ . Moreover observe that we model the shift-behaviour accordingly:For any k ∈ N and any λ ≥ δ we have thatexpr( σ k ( f ))( λ ) = expr( f )( λ + k ) (12) Note that any other choice d ′ with f ∈ A H d ′ will deliver the same evaluation. JAKOB ABLINGER AND CARSTEN SCHNEIDER and for any k ∈ N and any λ ≥ δ + k we have thatexpr( σ − k ( f ))( λ ) = expr( f )( λ − k ) . (13)The main goal of this article is to construct a difference ring, which represents all H -expressions and that can be embedded into the ring of sequences. As indicated already in theintroduction, we will rely on the fact that the constants are precisely the elements K . Thefollowing example shows immediately, that ( A H , σ ) is a too naive construction. Example 1.
Take f := 4 ¯ S , + 6 ¯ S , + 4 ¯ S , −
12 ¯ S , , −
12 ¯ S , , −
12 ¯ S , , + 24 ¯ S , , , − ¯ S − ¯ S ∈ A H . Then one can easily verify that σ ( f ) = f. Even more, we get that expr( f )( λ ) = 0 for all λ ∈ N , i.e., there are algebraic relations among these sums. Quasi-shuffle algebras and the linearization operator
In order to eliminate such relations as given in Example 1, we will equip the difference ringconstruction ( A H , σ ) with the underlying quasi-shuffle algebra. Definition 2 (Non-commutative Polynomial Algebra) . Let G be a totally ordered, graded set.The degree of a ∈ G is denoted by | a | . Let G ∗ denote the free monoid over G , i.e., G ∗ = { a · · · a n | a i ∈ G, n ≥ } ∪ { ǫ } . We extend the degree function to G ∗ by | a a · · · a n | = | a | + | a | + · · · + | a n | for a i ∈ G and | ǫ | = 0 . Let R ⊇ Q be a commutative ring. The set of non-commutative polynomials over R is defined as R h G i := ( X w ∈ G ∗ r w w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r w ∈ R, r w = 0 for almost all w ) . Addition in R h G i is defined component wise and multiplication is defined by X w a w w · X w b w w := X w ( X uv = w a u b v ) w . We define a new multiplication ∗ on R h G i which is a generalisation of the shuffle product ,by requiring that ∗ distributes with the addition. We will see that this product can be usedto describe properties of H -sums; compare [25, 26, 27]. Definition 3 (Quasi-shuffle product) . ∗ : R h G i × R h G i −→ R h G i is called quasi-shuffleproduct, if it distributes with the addition and ǫ ∗ w = w ∗ ǫ = w , for all w ∈ G ∗ ,a u ∗ b v = a ( u ∗ v ) + b ( u ∗ v ) − [ a, b ]( u ∗ v ) , for all a, b ∈ G ; u , v ∈ G ∗ , (14) where [ · , · ] : G × G → G , ( G = G ∪ { } ) is a function satisfying S0 . [ a,
0] = 0 for all a ∈ G ;S1 . [ a, b ] = [ b, a ] for all a, b ∈ G ;S2 . [[ a, b ] , c ] = [ a, [ b, c ]] for all a, b, c ∈ G ; andS3 . Either [ a, b ] = 0 or | [ a, b ] | = | a | + | b | for all a, b ∈ G. LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 9
We specialize the quasi-shuffle algebra from Definition 3 in order to model the H -sumsaccordingly. We consider the alphabet G = H and define the degree of a letter a by | a | = w ( a ).Finally, we define [ a, b ] = ¯ λ ( a )¯ λ ( b ) (15)and [ a,
0] = 0 for all a, b ∈ H . This function obviously fulfils (S0)-(S3). In other words, if wetake our commutative ring R = K ( n )[ x ], then K ( n )[ x ] hHi forms a quasi-shuffle algebra.Let a , . . . , a r ∈ H ∗ . By using the expansion of (14), we can write a ∗ · · · ∗ a k = c d + · · · + c m d m (16)for some uniquely determined d i ∈ H ∗ , c i ∈ K ∗ (compare [8]). In particular, we have that | a | + · · · + | a k | = max i =1 ,...,m | d i | . (17)This linearization will be carried over to A H . Consider the K ( n )[ x ]-module V := K ( n )[ x ] s (1)1 ⊕ · · · ⊕ K ( n )[ x ] s (1) n ⊕ K ( n )[ x ] s (2)1 ⊕ · · · ⊕ K ( n )[ x ] s (2) n ⊕ . . . . (18)Now we are in the position to define the linearization function L : A H → V as follows. For a , . . . , a k ∈ H , we take the c i ∈ K and d i ∈ H from (16) and define L ( ¯ S a . . . ¯ S a ) = c ¯ S d + · · · + c m ¯ S d m . By (17) it follows that w ( ¯ S a . . . ¯ S a ) = max i =1 ,...,m w ( ¯ S d i ) . Finally, we extend L to A H by linearity.Since (14) reflects precisely (6), we obtain the following lemma. Lemma 2.
Let f ∈ A H and take δ ∈ N such that expr( f )( λ ) has no poles for all λ ∈ N with λ ≥ δ . Then for all λ ≥ δ , expr( f )( λ ) = expr( L ( f ))( λ ) . Example 2.
We have that L ( ¯ S ) = 4 ¯ S , + 6 ¯ S , + 4 ¯ S , −
12 ¯ S , , −
12 ¯ S , , −
12 ¯ S , , + 24 ¯ S , , , − ¯ S . (19) In particular, as already indicated in (1) we get S ( n ) = expr( L ( S ))( n ) =4 S , ( n ) + 6 S , ( n ) + 4 S , ( n ) − S , , ( n ) − S , , ( n ) − S , , ( n ) + 24 S , , , ( n ) − S ( n ) . Clearly, we can consider V as a subset of A H , i.e., we can equip ( A H , σ ) with the linearizationfunction L . Observe that for any f ∈ V we have that σ ( f ) ∈ V . In addition, we obtain thefollowing lemma. Lemma 3.
For any f ∈ A H , σ ( L ( f )) = L ( σ ( f )) and σ − ( L ( f )) = L ( σ − ( f )) .Proof. We only give a proof for σ − ( L ( f )) = L ( σ − ( f )) since σ ( L ( f )) = L ( σ ( f )) followsanalogously. It suffices to prove σ − ( L ( f )) = L ( σ − ( f )) for a monomial f ∈ A H sincethen we can extend the result by linearity. First consider the product of two nested sums(compare (15)): let ¯ λ (ˆ a )¯ λ (ˆ a ) = m X j =1 r j ¯ λ ( b j ) with r j ∈ K and b j ∈ A. Then σ − ( ¯ S ˆ a a ¯ S ˆ a a ) = σ − ¯ S ˆ a ( a ∗ ˆ a a ) + ¯ S ˆ a (ˆ a a ∗ a ) − m X j =1 r j ¯ S b j ( a ∗ a ) = σ − (cid:0) ¯ S ˆ a ( a ∗ ˆ a a ) (cid:1) + σ − (cid:0) ¯ S ˆ a (ˆ a a ∗ a ) (cid:1) − m X j =1 r j σ − (cid:16) ¯ S b j ( a ∗ a ) (cid:17) = ¯ S ˆ a ( a ∗ ˆ a a ) − ¯ λ (ˆ a ) ¯ S a ∗ ˆ a a + ¯ S ˆ a (ˆ a a ∗ a ) − ¯ λ (ˆ a ) ¯ S ˆ a a ∗ a − m X j =1 r j (cid:16) ¯ S b j ( a ∗ a ) − ¯ λ ( b j ) ¯ S a ∗ a (cid:17) = ¯ S ˆ a a ¯ S ˆ a a − ¯ λ (ˆ a ) ¯ S a ∗ ˆ a a − ¯ λ (ˆ a ) ¯ S ˆ a a ∗ a + m X j =1 r j ¯ λ ( b j ) ¯ S a ∗ a = (cid:0) ¯ S ˆ a a − ¯ λ (ˆ a ) ¯ S a (cid:1) (cid:0) ¯ S ˆ a a − ¯ λ (ˆ a ) ¯ S a (cid:1) = σ − ( ¯ S ˆ a a ) σ − ( ¯ S ˆ a a ) . (20)Now proceed by induction on the number of factors. Assume that the statement holds for f being the product of k factors ¯ S ˆ a a ¯ S ˆ a a · · · ¯ S ˆ a k a k and let¯ S ˆ a a ¯ S ˆ a a · · · ¯ S ˆ a k a k = X i c i ¯ S d i . Now we get σ − ( ¯ S ˆ a a ¯ S ˆ a a · · · ¯ S ˆ a k a k ¯ S ˆ a k +1 a k +1 ) = σ − X i c i ¯ S d i ¯ S ˆ a k +1 a k +1 ! = X i c i σ − (cid:0) ¯ S d i ¯ S ˆ a k +1 a k +1 (cid:1) . Using (20) and the induction hypothesis we conclude σ − ( ¯ S ˆ a a · · · ¯ S ˆ a k a k ¯ S ˆ a k +1 a k +1 ) = X i c i σ − (cid:0) ¯ S d i (cid:1) σ − (cid:0) ¯ S ˆ a k +1 a k +1 (cid:1) = σ − X i c i ¯ S d i ! σ − (cid:0) ¯ S ˆ a k +1 a k +1 (cid:1) = σ − ( ¯ S ˆ a a ) · · · σ − ( ¯ S ˆ a k a k ) σ − (cid:0) ¯ S ˆ a k +1 a k +1 (cid:1) . (cid:3) We conclude with the following lemma which will be essential to prove our main resultstated in Theorem 3.
Lemma 4.
Let ¯ V := K s (1)1 ⊕ · · · ⊕ K s (1) n ⊕ K s (2)1 ⊕ · · · ⊕ K s (2) n ⊕ . . . . Then: (1) If f ∈ K [ s (1)1 , . . . , s (1) n ][ s (2)1 , . . . , s (2) n ] . . . with f / ∈ K , then L ( f ) ∈ ¯ V . (2) If f ∈ ¯ V with f = 0 , then σ ( f ) = f . LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 11
Proof.
The first part is immediate. Since σ ( f ) = f if and only if f = σ − ( f ) we are going toprove that for every f ∈ ¯ V with f = 0 it follows that σ − ( f ) = f. Note that ¯ V ∩ K = {} . Let f ∈ ¯ V be of the form f = c ¯ S ˆ a a + c ¯ S ˆ a a + · · · + c k ¯ S ˆ a k a k for some c i ∈ K \ { } . Let b be the minimal letter in { ˆ a , ˆ a , . . . , ˆ a k } and let { r , r , . . . , r j } ⊆{ , . . . k } such that ˆ a r i = b for i ∈ { , . . . , j } and let { u , u , . . . , u h } ⊆ { , . . . k } such thatˆ a u i = b for i ∈ { , . . . , h } . Then we get σ − ( f ) = f − (cid:0) c ¯ λ (ˆ a ) ¯ S a + · · · + c k ¯ λ (ˆ a k ) ¯ S a k (cid:1) = f − ¯ λ ( b ) j X i =1 c r i ¯ S a ri | {z } M := − h X i =1 ¯ λ (ˆ a u i ) c u i ¯ S a ui | {z } U := . Due to the definition of b there cannot be a cancellation between the summands of M and U .Moreover, due to the choice of b we have that M = 0 . Hence σ − ( f ) = f. (cid:3) The reduced difference ring
We define the reduced difference ring where all relations of ( A H , σ ) are factored out by thequasi-shuffle algebra, i.e., we define I := { f ∈ A H | L ( f ) = 0 } . It is immediate that I is an ideal of A H and we can define the quotient ring E H := A H /I. Even more, I is a reflexive difference ideal [20], i.e., the following property holds: for any f ∈ I and any z ∈ Z we have σ z ( f ) ∈ I . Namely, for any f ∈ I , we have that L ( σ ( f )) = σ ( L ( f )) = σ (0) = 0 by Lemma 3. Hence, σ ( f ) ∈ I ; similarly, it follows that σ − ( f ) ∈ I .From this follows that we can construct the ring automorphism σ : A H /I → A H /I with σ ( a + I ) = σ ( a ) + I . In particular, identifying the elements f ∈ K ( n )[ x ] with f + I we canconsider ( A H /I, σ ) as a difference ring extension of ( K ( n )[ x ] , σ ).In the following we will elaborate how this difference ring can be constructed explicitly inan iterative fashion. Using the set of all power products Π byΠ = { ¯ S a ¯ S a . . . ¯ S a r | r ∈ N and a i ∈ H ∗ for 1 ≤ i ≤ r } we obtain the following possibility to generate I . Lemma 5. I = n l X i =1 r i ( L ( p i ) − p i ) | l ∈ N , and r i ∈ K ( n )[ x ] , p i ∈ Π for ≤ i ≤ l o . (21) Proof.
Denote the set on the right hand side by J . First let f ∈ J i.e., f = P li =1 r i ( L ( p i ) − p i )for some l ∈ N and r i ∈ K ( n )[ x ] , p i ∈ Π . We have L ( f ) = L l X i =1 r i ( L ( p i ) − p i ) ! = l X i =1 L ( r i ( L ( p i ) − p i )) = l X i =1 r i ( L ( p i ) − L ( p i )) = 0 . Hence f ∈ I and thus J ⊆ I . Now let f ∈ I, i.e., f = P li =1 r i p i for some l ∈ N , r i ∈ K ( n )[ x ]and p i ∈ Π with L ( f ) = 0 . We define v i := L ( p i ) − p i , hence p i = L ( p i ) − v i . Using thisdefinition we get f = l X i =1 r i p i = l X i =1 r i ( L ( p i ) − v i ) = l X i =1 r i L ( p i ) − l X i =1 r i v i = l X i =1 L ( r i p i ) − l X i =1 r i v i = L l X i =1 r i p i ! − l X i =1 r i v i = L ( f ) − l X i =1 r i v i = 0 − l X i =1 r i ( L ( p i ) − p i ) . Thus f ∈ J and therefore I ⊆ J . This completes the proof. (cid:3) If we define I d = I ∩ A H d , it follows again that I d is a reflexive difference ideal of A H d andthat the quotient ring ( E H d , σ ) defined as E H d := A H d /I d is a difference ring. Note that E H = K ( n )[ x ] . Even more, we get the chain of difference ring extensions( K ( n )[ x ] , σ ) = ( E H , σ ) ≤ ( E H , σ ) ≤ ( E H , σ ) ≤ · · · ≤ ( E H , σ ) . (22)In the following we will work out further how the construction from ( E H d − , σ ) to ( E H d , σ )can be carried out explicitly. Since L is weight preserving it follows that L ( p ) with p ∈ Π and w ( p ) = d depends linearly on sums with weight less or equal to d . Hence by Lemma 5 we get I d = n l X i =1 r i ( L ( p i ) − p i ) | l ∈ N with r i ∈ K ( n )[ x ] , p i ∈ Π where w ( p i ) ≤ d for 1 ≤ i ≤ l o = n g h + l X i =1 r i ( L ( p i ) − p i ) | g ∈ A H d , h ∈ I d − and l ∈ N with r i ∈ K ( n )[ x ] , p i ∈ Π where w ( p i ) = d for 1 ≤ i ≤ l o . Exploiting this property we can perform the following construction. At the specified weight d we can set up a matrix containing all the relations L ( p ) − p (23)with p ∈ Π and w ( p ) = d where the columns (except for the last one) represent the sums ofweight d coming from L ( p ) and the last column represents the polynomial p which is builtby sums of weight less than d. Now we can transform the matrix to its reduced row-echelonform. The sums corresponding to the corner elements can be reduced while the other sumsremain as variables. Exactly this crucial observation has been strongly utilized for harmonicsums in [14] in order to derive all algebraic relations induced by the quasi-shuffle algebra. Forfurther results and heavy calculations using
HarmonicSums we refer to [1, 8, 2, 9].Now we will link this observation with our difference ring construction. Let a ( d )1 , . . . , a ( d ) m d (24) LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 13 be all these variables of (8) which cannot be reduced by the quasi-shuffle algebra, resp. by L .Set t ( d ) i := a ( d ) i + I d . Then we get the ring E H d = A H d /I d = E H d − [ t ( d )1 , . . . , t ( d ) m d ] . (25)By construction this ring is a polynomial ring, i.e., the elements t ( d ) i are algebraically inde-pendent among each other.Finally, by the above considerations we conclude that E H = K ( n )[ x ][ t (1)1 , . . . , t (1) m ][ t (2)1 , . . . , t (2) m ] . . . . The above construction will be illustrated by the following example.
Example 3.
First, we consider the variables of the harmonic sums (i.e., H = A h ) of weight 4: v = ( ¯ S , , , , ¯ S , , , ¯ S , , , ¯ S , , , ¯ S , , ¯ S , , ¯ S , , ¯ S ) . Here we consider all non-trivial polynomials (23) for p ∈ Π with w ( p ) = 4 : R = { S , + 6 ¯ S , + 4 ¯ S , −
12 ¯ S , , −
12 ¯ S , , −
12 ¯ S , , + 24 ¯ S , , , − ¯ S − ¯ S , − S , − S , − S , + 2 ¯ S , , + 2 ¯ S , , + 2 ¯ S , , + ¯ S − ¯ S ¯ S , ¯ S , + ¯ S , − ¯ S − ¯ S ¯ S , S , − ¯ S − ¯ S , ¯ S , − S , , − S , , − S , , + 6 ¯ S , , , − ¯ S , , − ¯ S , − ¯ S , + ¯ S , , + ¯ S , , + ¯ S , , − ¯ S ¯ S , , − ¯ S , − ¯ S , + 2 ¯ S , , + ¯ S , , − ¯ S ¯ S , , − ¯ S , − ¯ S , + ¯ S , , + 2 ¯ S , , − ¯ S ¯ S , } . E.g., p = ¯ S ∈ Π yields first entry of R . Note that by Lemma 2 the elements in R transformedto H -sum expressions evaluate to for all n ≥ . E.g., the first entry is precisely the equationworked out in Example 2. Note further that I = { f h + X r ∈ R k r r | f ∈ A H , h ∈ I and k r ∈ K ( n )[ x ] } . The relations in R lead to the matrix ( M | h ) = − − −
12 6 4 4 − − ¯ S − − − − ¯ S ¯ S − − ¯ S ¯ S − − ¯ S − − − − ¯ S , − − − ¯ S ¯ S , − − − ¯ S ¯ S , − − − ¯ S ¯ S , where ( M | h ) v t gives back a vector with the entries from R . Now we reduce this matrix usingGaussian elimination together with some relations of lower weight to get the reduced echelon form (cid:0) M ′ | h ′ (cid:1) = − − ¯ S − ¯ S ¯ S − ¯ S ¯ S − ¯ S − − ¯ S (cid:0) ¯ S , − ¯ S (cid:1) − ¯ S ¯ S − − − ¯ S − ¯ S ¯ S , − − ¯ S − − ¯ S ¯ S . Set R ′ := ( M ′ | h ′ ) v t . By construction the elements of R can be written in terms of theelements of R ′ plus some extra elements from I . Thus the following holds: I = { f h + X r ∈ R ′ k r r | f ∈ A H , h ∈ I and k r ∈ K ( n )[ x ] } . The advantage of this representation of I with R ′ (instead of R ) is that one can read offrewrite rules to eliminate sums: the variables corresponding to the corner points of the reducedmatrix, i.e., the variables ¯ S , , , , ¯ S , , , ¯ S , , , ¯ S , , ¯ S , (26) can be reduced with the substitution rules ¯ S , , , → ¯ S S + 14 ¯ S ¯ S + 13 ¯ S ¯ S + 18 ¯ S , ¯ S , , → ¯ S , , − ¯ S , + ¯ S − ¯ S (cid:0) ¯ S , − ¯ S (cid:1) + 12 ¯ S ¯ S , ¯ S , , → − S , , + ¯ S , + ¯ S S + ¯ S ¯ S , , ¯ S , → ¯ S S , ¯ S , → − ¯ S , + ¯ S + ¯ S ¯ S (27) while the ¯ S , , , ¯ S , , ¯ S remain as variables . In other words, within the ring E H = A H /I these elements cannot be eliminated further and we arrive at the polynomial ring E H = A H /I = E H [ ¯ S , , + I , ¯ S , + I , ¯ S + I ] with the transcendental generators ¯ S , , + I , ¯ S , + I , ¯ S + I .We remark that we can calculate with the representants ¯ S , , , ¯ S , , ¯ S instead of the equiva-lence classes ¯ S , , + I , ¯ S , + I , ¯ S + I , respectively. There is only a subtle detail. If wewant to perform σ ( f ) for some f ∈ A H /I , we apply first σ ( f ) where σ is taken from ( A H , σ ) .Since the sums (26) might be again introduced by σ , we have to eliminate them by the foundsubstitution rules stated in (27) .Similarly, one can construct the polynomial ring E H = E H [ ¯ S , + I , ¯ S + I ] Reinterpreting the variables as H -sum expressions, the left hand sides equal the right hand sides for all n ≥ Note that a different order of sums in v would lead to a different set of remaining variables; certain orderswould lead to a basis built by Lyndon words, compare [25, 26, 27]. LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 15 with the substitution rules ¯ S , → − ¯ S , + ¯ S ¯ S + ¯ S and ¯ S , , →
16 ¯ S + 12 ¯ S ¯ S + 13 ¯ S , (28) the polynomial ring E H = E H [ ¯ S + I ] with the substitution rule ¯ S , →
12 ¯ S + 12 ¯ S , (29) and the polynomial ring E H = K ( n )[ x ][ ¯ S + { } ] . To sum up, we obtain the polynomial ring E H = K ( n )[ x ][ ¯ S + { } ][ ¯ S + I ][ ¯ S , + I , ¯ S + I ][ ¯ S , , + I , ¯ S , + I , ¯ S + I ] . Now define the function ρ : A H → ¯ E H where ρ ( f ′ ) is obtained by applying the substitutionrules of (27) , (28) and (29) such that one obtains an element of the polynomial ring ¯ E H = K ( n )[ x ][ ¯ S ][ ¯ S ][ ¯ S , , ¯ S ][ ¯ S , , , ¯ S , , ¯ S ] being a subring of A E . Then the difference ring ( E H , σ ) can be represented by the differencering (¯ E H , ¯ σ ) with the ring automorphism ¯ σ : ¯ E H → ¯ E H defined by ¯ σ ( f ) = ρ ( σ ( f )) . Here weapply first the given automorphism σ : A H → A H with f ′ := σ ( f ) ∈ A H and apply afterwardsthe constructed substitution rules to get the element ρ ( f ′ ) ∈ ¯ E H . In general, the representants in (24) for the variables of weight d (plus the other represen-tants with weights < d from the previous construction steps) can be used to express all theelements of f ∈ E H d . In particular, all the ring operations can be simply performed in thispolynomial ring. Only if one applies σ ( f ), one has to apply the corresponding substitutionrules in order to rewrite the expression again in terms of the representants. Remark 1.
So far, we fixed
H ∈ { A h , A a , A c ( M ) } in (7) and performed the construction forthe chosen H -sums to get the difference rings ( E H d , σ ) and ( E H , σ ) . Now we fix H = A c ( M ) and look what happens if we vary the finite set M ∈ N . More precisely, if M ⊂ M ⊂ M ⊂ . . . is a chain of finite subsets of N then A c ( M ) ⊂ A c ( M ) ⊂ · · · ⊂ A c ( M k ) ⊂ · · · ⊂ A c . Furthermore, if B ( d ) i denotes a set of representants of weight d for A c ( M i ) as defined in (24) ,then by using our construction above we can find representants such that B ( d )1 ⊂ B ( d )2 ⊂ B ( d )3 . . . As a consequence, it follows that E A c ( Mr ) d is a polynomial ring extension of E A c ( Mr − d wherethe transcendental generators are the elements ∪ ≤ j ≤ d V j with V j := B ( j ) r \ B ( j ) r − ; here V j arethe generators that represent the additional H -sums of weight j . Moreover, ( E A c ( Mr ) d , σ ) is adifference ring extension of ( E A c ( Mr − d , σ ) . We remark that any element from the equivalence class a ( d ) i + I d delivers the same evaluationexpression. Lemma 6.
Let f, g ∈ t ( d ) i = a ( d ) i + I d and take δ ∈ N such that for all δ ≤ λ ∈ N , expr( f )( δ ) and expr( g )( δ ) do not have poles. Then for all λ ≥ δ , expr( f )( λ ) = expr( g )( λ ) .Proof. Let h , h ∈ I d such that f = a ( d ) i + h and g = a ( d ) i + h . Since expr( a ( d ) i ) and expr( h i )have no poles, expr( f )( λ ) and expr( g )( λ ) have no poles for λ ≥ δ . Thereforeexpr( f )( λ ) − expr( g )( λ ) = expr( f − g )( λ ) = expr( h − h )( λ ) = expr( h )( λ ) − expr( h )( λ )for all λ ≥ δ . By Lemma 2 we conclude that expr( h i )( λ ) = expr( L ( h i ))( λ ) = 0 and soexpr( f )( λ ) = expr( g )( λ ). (cid:3) In addition, we obtain the following connection with the linearization operator.
Lemma 7.
For any f ∈ a ( d ) i + I we have that L ( f ) = L ( a ( d ) i ) .Proof. Write f = a ( d ) i + h with h ∈ I . Then L ( f ) = L ( a ( d ) i + h ) = L ( a ( d ) i )+ L ( h ) = L ( a ( d ) i ). (cid:3) This implies that we can define the linearization map ¯ L : E H → V with¯ L ( g + I ) = L ( g )for g ∈ A H . In addition, we get the following result. Lemma 8.
For any f ∈ E H , σ ( ¯ L ( f )) = ¯ L ( σ ( f )) .Proof. Write f = g + h with g ∈ A H and h ∈ I . By definition, L ( h ) = 0. Since I is a differenceideal, σ ( h ) ∈ I . Together with Lemma 3 we get σ ( ¯ L ( f )) = σ ( ¯ L ( g + h )) = σ ( L ( g )) = L ( σ ( g )) = ¯ L ( σ ( g ) + σ ( h )) = ¯ L ( σ ( g + h )) = ¯ L ( σ ( f )) . (cid:3) Structural results in R ΠΣ ∗ -ring In the following we will bring in addition difference ring theory. Here we will refine thenaive construction given by Lemma 1 that we used to construct the difference ring ( A , σ ) inSection 3. Namely, we will improve this type of extensions by the so-called Σ ∗ -extensions. Definition 4.
Take the difference ring extension ( A [ t ] , σ ) of ( A , σ ) from Lemma 1. Then thisextension is called Σ ∗ -extension if const( A [ t ] , σ ) = const( A , σ ) .In particular, ( A [ t ] . . . [ t e ] , σ ) is called a (nested) Σ ∗ -extension of ( A , σ ) if ( A [ t ] . . . [ t i ] , σ ) isa Σ ∗ -extension of ( A [ t ] . . . [ t i − ] , σ ) for all ≤ i ≤ e . In order to verify that const( A [ t ] , σ ) = const( A , σ ) holds, there is the following equivalentcharacterization; see [46, Thm. 2.12] or [44, Thm. 3]. Note that this result is a generaliza-tion/refinement of Karr’s difference field theory [28, 29]. Theorem 1.
Take the difference ring extension ( A [ t ] , σ ) of ( A , σ ) from Lemma 1 with σ ( t ) = t + β ( β ∈ A ) where const( A , σ ) is a field. Then this is a Σ ∗ -extension (i.e., const( A [ t ] , σ ) =const( A , σ ) ) iff there is no g ∈ A with σ ( g ) = g + β. (30) LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 17
Exactly this result is the driving property within the summation package
Sigma [38, 43]. In-definite nested sums are represented not in the naive construction given in Lemma 1, but sumsare only adjoined if the constants remain untouched, i.e., if the telescoping problem cannot besolved. More precisely, in
Sigma indefinite summation algorithms are implemented [41, 45, 39]that decide efficiently, if there exists a g ∈ A such that (30) holds. If such an element doesnot exist, we can perform the construction as stated in Lemma 1 and we have constructed aΣ ∗ -extension without extending the constant field. Otherwise, if we find such a g , we can use g itself to represent the “sum” with the shift-behaviour (30). In this way, the occurring sumswithin an expression are rephrased step by step by a tower of Σ ∗ extensions of the form (11)with K = const( A , σ ) = const( A , σ ) = const( A , σ ) = · · · = const( A , σ ) . However, this construction gets more and more expensive, the more variables are adjoined. Incontrast to that, if we restrict to H -sums, the construction of the tower of ring extensions (22),which relies purely on a linear algebra engine, is by far more efficient; see Example 3. Example 4.
Given such a reduction in terms of basis sums (using linear algebra), wecould verify with the summation package
Sigma that the difference ring extension ( E H d , σ ) of ( K ( n )[ x ] , σ ) with H = H a and d = 1 , , , , , , is a Σ ∗ -extension. For d = 7 we repre-sented 507 basis sums, i.e., we constructed a tower of 507 Σ ∗ -extensions within 5 days. Thecase d = 8 is currently out of scope. Similarly, we could show for H = A c ( M ) with variousfinite subsets M ⊂ N and d ∈ N that ( E H d , σ ) forms a Σ ∗ -extension of ( K ( n )[ x ] , σ ) . Note that Theorem 1 can be supplemented by the following structural result; for the differencefield version see, e.g., [40, Thm. 2.4].
Proposition 1.
Let ( A [ t ] . . . [ t e ] , σ ) be a Σ ∗ -extension of ( A , σ ) with σ ( t i ) − t i ∈ A where K := const( A , σ ) is a field. Let β ∈ A and let g ∈ A [ t ] . . . [ t e ] be a solution of (30) . Then g = P ei =1 c i t i + w with c i ∈ K and w ∈ A .Proof. We will show the result by induction on e . Suppose that the proposition holds for e − e extensions as stated in the proposition. Set G := A [ t ] . . . [ t e − ] and define β e := σ ( t e ) − t e ∈ A . Let β ∈ A and g ∈ G [ t e ] such that (30)holds. By [44, Lemma 6] (resp. [46, Lemma 7.2]) it follows that g = c t e + ˜ g for some c, ˜ g ∈ G .Comparing coefficients w.r.t. t e in σ ( c t e + ˜ g ) − ( c t e + ˜ g ) = β implies that σ ( c ) − c = 0, i.e., c ∈ const( G [ t e ] , σ ) = K . Therefore we get σ (˜ g ) − ˜ g = β − ( σ ( c t e ) − c t e ) = β − c β e ∈ A . Bythe induction assumption we conclude that ˜ g = P e − i =1 c i t i + w with c i ∈ K and w ∈ A . Hence g = c t e + P e − i =1 c i t i + w and the proposition is proven. (cid:3) In the following we will work out further properties in such a tower of Σ ∗ -extensions. Thiswill finally enable us to show us in one stroke that the tower (22) is built by an infinite towerof Σ ∗ -extensions, i.e., that const( E H , σ ) = const( A H /I, σ ) = K for H ∈ { A h , A a , A c ( M ) } with a finite set M ∈ N . In other words, for the special class of H -sums, we can dispense the user from all the difference ring algorithms of Sigma . We remark that one can hunt for recurrences of definite sums and can solve recurrence in termsd’Alembertian solution in such difference rings; for details see, e.g., [43] and references therein.
In the following let ( K ( n )[ x ] , σ ) be our difference ring with σ ( n ) = n + 1, σ ( x ) = − x and x = 1 where const( K ( n )[ x ] , σ ) = K . Let f, g ∈ K [ n ] \ { } . Note that σ k ( g ) ∈ K [ n ]. Hence we can define the dispersion [10]disp( f, g ) = max { k ≥ | gcd( f, σ k ( g )) = 1 } . Let q , . . . , q r ∈ K [ n ] \ { } be polynomials. Then we define the set K [ n ] { q ,...,q r } := { aq m . . . q m r r | ( m , . . . , m r ) ∈ N r \ { } ; a ∈ K [ n ]; gcd( q · · · q r , a ) = 1 } . With these notions, we restrict the class of Σ ∗ -extensions as follows. Definition 5.
Let ( K ( n )[ x ][ s ] . . . [ s e ] , σ ) be a Σ ∗ -extension of ( K ( n )[ x ] , σ ) and let P = { p , . . . , p r } with p i ∈ K [ n ] \ { } . Then the extension is called P -normalized, if for any ≤ i ≤ e we have that σ ( s i ) − s i ∈ K [ n ] P [ x ][ s . . . , s i − ] . Example 5.
As worked out in Example 4 (compare also Theorem 3 below), the differencering ( E H , σ ) (resp. (¯ E H , ¯ σ ) ) of Example 3 is a Σ ∗ -extension of ( K ( n )[ x ] , σ ) . In particular, itis a { n + 1 } -normalized Σ ∗ -extension. In the remaining subsection we will show in Proposition 2 that the telescoping solution g of (30) will not depend on n provided that one restricts to certain normalized Σ ∗ -extensionsand takes f with a particular shape. Lemma 9.
Consider the difference field ( K ( n ) , σ ) with n being transcendental over K andwith σ ( n ) = n + 1 . Let q ∈ K [ n ] \ K with disp( q, q ) = 0 , let u ∈ K ∗ and let v ∈ K [ n ] with gcd( v, q ) = 1 . Then there is no g ∈ K ( n ) such that σ ( g ) + u g = vq .Proof. Suppose that there is such a g ∈ K ( n ). By Abramov’s universal denominator bound-ing [11] or Bronstein’s generalization [19] (see Theorem 8 and 10 therein) it follows that g ∈ K [ n ]. Hence σ ( g ) + u g ∈ K [ n ], a contradiction that q / ∈ K . (cid:3) Lemma 10.
Consider the difference ring ( K ( n )[ x ] , σ ) with n being transcendental over n and σ ( n ) = n + 1 and with σ ( x ) = − x and x = 1 . Let p ∈ K [ n ] \ K with disp( p, p ) = 0 and let f ∈ K [ n ] { p } [ x ] with f = 0 . Then there is no g ∈ K ( n )[ x ] with σ ( g ) − g = f .Proof. Let g = g + g x with g i ∈ K ( n ) and f = f + f x with f i ∈ K [ n ] { p } such that σ ( g ) − g = f . Since f / ∈ K [ x ], there is a j ∈ { , } such that f j K . By coefficient comparisonwe get σ ( g j ) − ( − j g j = ( − j f j . Write f j = vq with gcd( v, q ) = 1. Note that q = c p m forsome m ∈ N and c ∈ K ∗ . Since disp( p m ) = 0, the existence of the solution g j contradicts toLemma 9. (cid:3) Lemma 11.
Let ( K ( n )[ x ] , σ ) be the difference ring from Lemma 10. Let P = { p , . . . , p r } with p i ∈ K [ n ] \ { } and let ( E , σ ) be a P -normalized Σ ∗ -extension of ( K ( n )[ x ] , σ ) . Set H := K ( n ) P [ x ][ s ] . . . [ s e ] . Then the following holds. (1) For any f, g ∈ H , f + g ∈ H . (2) For any f ∈ K [ s , . . . , s e ] , σ ( f ) − f ∈ H . For a unified formulation see Theorem 2 in [37].
LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 19
Proof.
The first statement is immediate. Now let f = P ( n ,...,n e ) ∈ N e c ( n ,...,n e ) s n . . . s n e e with c ( n ,...,n e ) ∈ K and set β i := σ ( s i ) − s i ∈ H . Then by expanding σ ( f ) into its monomials weget σ ( f ) = X ( n ,...,n e ) ∈ N e c ( n ,...,n e ) ( s + β ) n . . . ( s e + β e ) n e = f + X ( n ,...,n e ) ∈ N e \{ } d ( n ,...,n e ) s n . . . s n e e where for each ( n , . . . , n e ) ∈ N \ { } we have that d ( n ,...,n e ) ∈ K ( n ) P [ x ]. This last statementfollows from the fact that at least one β i ∈ H is a factor of d ( n ,...,n e ) . Since a sum of elementsof H is again from H , we conclude that σ ( f ) − f ∈ H . (cid:3) Proposition 2.
Let ( K ( n )[ x ] , σ ) as given in Lemma 10. Let P = { p , . . . , p r } with p i ∈ K [ n ] \ { } and disp( p i , p j ) = 0 for all ≤ i, j ≤ r . Let ( K ( n )[ x ][ s ] . . . [ s e ] , σ ) be a P -normalized Σ ∗ -extension of ( K ( n )[ x ] , σ ) . If f ∈ K [ n ] P [ x ][ s . . . , s e ] and g ∈ K ( n )[ x ][ s ] . . . [ s e ] such that σ ( g ) − g = f holds, then g ∈ K [ s , . . . , s e ] .Proof. We prove the statement by induction on the number of extensions. If e = 0, thesituation f = 0 cannot occur by Lemma 10. If f = 0, g ∈ K and the base case is proven.Now suppose that the theorem holds for e extensions and consider the Σ ∗ -extension ( H [ s ] , σ )of ( K ( n )[ x ] , σ ) with H = K ( n )[ x ][ s ] . . . [ s e ] and σ ( s ) = s + β with β ∈ H . Denote H P = K [ n ] P [ x ][ s ] . . . [ s e ]. Now suppose that σ ( g ) − g = f holds where g ∈ H [ s ] and f ∈ H P [ s ].Again, if f = 0, g ∈ K and we are done.Finally, suppose that f = 0 holds. Write g = P di =0 g i s i with g i ∈ H . If g i ∈ K [ s , . . . , s e ]for all i , we are done. Otherwise, let j be maximal such that g j / ∈ K [ s , . . . , s e ]. Define g ′ := P ji =0 g i s i ∈ H [ s ] and γ := P di = j +1 g i s i ∈ K [ s ] . . . [ s e ][ s ], i.e., g = g ′ + γ . By Lemma 11it follows that h := σ ( γ ) − γ ∈ H P [ s ]. In particular, since f ∈ H P [ s ], f ′ := f − h ∈ H P [ s ] byLemma 11. By construction, σ ( g ′ ) − g ′ = f ′ with f ′ = P ji =0 f ′ j s i for some f ′ j ∈ H P . In particular, by leading coefficient comparison, σ ( g j ) − g j = f ′ j . By the induction assumption it follows that g j ∈ K [ s , . . . , s e ], a contradiction. (cid:3) Algebraic independence of (cyclotomic) harmonic sums
First, we define the ring of sequences and the notion of difference ring embeddings. Considerthe set of sequences K N with elements h a λ i λ ≥ = h a , a , a , . . . i , a i ∈ K . With component-wise addition and multiplication we obtain a commutative ring; the field K can be naturallyembedded by identifying k ∈ K with the sequence h k, k, k, . . . i ; we write = h , , , . . . i .We follow the construction from [33, Sec. 8.2] in order to turn the shift S : h a , a , a , . . . i 7→ h a , a , a , . . . i (31)into an automorphism: We define an equivalence relation ∼ on K N by h a λ i λ ≥ ∼ h b λ i λ ≥ ifthere exists a d ≥ a k = b k for all k ≥ d . The equivalence classes form a ringwhich is denoted by S ( K ); the elements of S ( K ) (also called germs) will be denoted, as above,by sequence notation. Now it is immediate that S : S ( K ) → S ( K ) with (31) forms a ringautomorphism. The difference ring ( S ( K ) , S ) is called the ring of sequences (over K ). A difference ring homomorphism τ : G → G between difference rings ( G , σ ) and ( G , σ )is a ring homomorphism such that τ ( σ ( f )) = σ ( τ ( f )) for all f ∈ G . If τ is injective, wecall τ a difference ring monomorphism or a difference ring embedding. In this regard notethat ( τ ( G ) , σ ) forms a difference ring which is the same as ( G , σ ) up to the renaming of theelements by τ . Moreover, ( G , σ ) is a difference ring extension of ( τ ( G ) , σ ). In a nutshell,( G , σ ) is contained in ( G , σ ) by means of the embedding τ .Consider our difference field ( K ( n ) , σ ) with σ ( n ) = n + 1. Now define the evaluationfunction ev : K ( n ) × N → K ( n ) as follows. For pq ∈ K ( n ) with p, q ∈ K ( n ) and gcd( p, q ) = 1,ev( pq , k ) = ( p ( k ) q ( k ) if q ( k ) = 00 if q ( k ) = 0 (pole case); (32)here p ( k ) , q ( k ) with k ∈ N denotes the evaluation of the polynomials at n = k . Finally, wedefine the map τ : K ( n ) → K N by τ ( f ) = h ev( f, k ) i k ≥ = h ev( f, , ev( f, , ev( f, , . . . i . (33)Then one can easily see that τ is a difference ring homomorphism from ( K ( n ) , σ ) to ( S ( K ) , S ).In particular, τ is injective. Namely, take f ∈ K ( n ) with τ ( f ) = i.e., ev( f, k ) = 0 for all k ∈ N . Write f = pq with p ∈ K [ x ] and q ∈ K [ x ] \ { } . Since q has only finitely manyroots, τ ( f ) = implies that p has infinitely many roots. Thus p = 0 and therefore f = 0.In summary, τ is a difference ring embedding. In particular, this implies that ( τ ( K ( n )) , S )forms a difference field also called the field of rational sequences.Now consider our difference ring ( K ( n )[ x ] , σ ) with σ ( x ) = − x and x = 1. We extend evfrom K ( n ) to K ( n )[ x ] as follows. For f = f + f x with f , f ∈ K ( n ) we defineev( f, k ) = ev( f , k ) + ( − k ev( f , k ) . Furthermore we extend τ to K ( n )[ x ] with (33) by our extended map ev. Again one canstraightforwardly verify that τ is a difference ring homomorphism. Moreover, τ is injective:Let f ∈ Q ( n )[ x ] with f = f + f x such that τ ( f ) = . Then 0 = ev( f , k ) + ev( f , k ) =ev( f + f , k ) for almost all k ≥
0. As above, we conclude that f + f = 0. Similarly,we get 0 = ev( f , k + 1) − ev( f , k + 1) = ev( f − f , k ) which implies that f − f = 0.Consequently, f = f = 0. Summarizing, τ : K ( n )[ x ] → S ( K ) is a difference ring embedding.In particular, ( τ ( K ( n )[ x ] , S ) is a difference ring which is the same as ( K ( n )[ x ] , σ ) up to therenaming of the elements by τ . ( τ ( K ( n )[ x ] , S ) is also called the ring of rational sequencesadjoined with the alternating sequence.Now let H ∈ { A h , A a , A c ( M ) } for a finite set M ∈ N . We will extend successively ev and thus τ for ( E H d − , σ ) ≤ ( E H d , σ ) with d = 1 , , . . . as follows. For E H = K ( n )[ x ] we have constructedev and τ . Now suppose that we obtained already the evaluation function ev : E H d − → K which yields a difference ring homomorphism τ : E H d − → S ( K ). In addition, assume that (25)holds. Then take the sum representants (24) such that t ( d ) i = a ( d ) i + I d . Let f = X ( e ,...,e md ) ∈ N md f ( e ,...,e md ) t ( d )1 e . . . t ( d ) m d e md ∈ E H d where only finitely many f ( e ,...,e md ) ∈ E H d − are non-zero. Then we defineev( f, k ) = X ( e ,...,e md ) ∈ N md ev( f ( e ,...,e md ) , k ) expr( a ( d )1 )( k ) e . . . expr( a ( d ) m d )( k ) e md , LGEBRAIC INDEPENDENCE OF SEQUENCES GENERATED BY (CYCLOTOMIC) HARMONIC SUMS 21 i.e., the sum variables are replaced by the corresponding H -sums depending on the variable k and the expression from K ( n )[ x ] are treated by ev as described above. Finally, considerthe function τ : E H d → S ( K ) as given in (33) with the extended ev. Again one can verify bysimple calculations that τ forms a difference ring homomorphism; for further details see [46,Lemma 4.4].What remains to show is that τ is also injective. Here we utilize the following result thatis implied immediately by Theorem 2.3 and Lemma 5.8 from [47]. Theorem 2.
Let ( G , σ ) with G = K ( n )[ x ][ s ] . . . [ s e ] be a Σ ∗ -extension of the difference ring ( K ( n )[ x ] , σ ) . If τ : G → S ( K ) is a difference ring homomorphism, then τ is injective.Proof. Alternatively to the proof in [47], one can derive this result as follows: By Cor. 4.3and Lemma 4.4 of [46] ( G , σ ) has no nilpotent elements. Hence by [50, Cor. 1.24] ( G , σ ) isa simple difference ring. I.e., any difference ideal of ( G , σ ) is either { } or G . Now consider I = { f ∈ G | τ ( f ) = 0 } . Then this forms a difference ideal: if h ∈ I then τ ( σ ( h )) = S ( τ ( h )) = S ( ) = and thus σ ( h ) ∈ I . Since 1 / ∈ I , I = G . Hence I = { } , i.e., τ is injective. (cid:3) Now we are ready to prove our main result.
Theorem 3.
Let
H ∈ { A h , A a , A c ( M ) } . Then for any d ∈ N we have that ( E H d , σ ) with E H d = K ( n )[ x ][ t (1)1 , . . . , t (1) m ] . . . [ t ( d )1 , . . . , t ( d ) m d ] is a Σ ∗ -extension of ( K ( n )[ x ] , σ ) . In particular,the map τ : E H d → S ( K ) constructed above is a difference ring embedding.Proof. Let P ⊂ Z [ n ] be the shifted denominators of the alphabet H . By definition P isfinite and we have that P ⊆ { n + 1 , n + 3 , n + 4 , n + 5 , n + 5 , n + 7 , . . . } . Clearly,for any a, b ∈ P we have that disp( a, b ) = disp( b, a ) = 0. Now suppose that the theoremholds for d −
1. I.e., ( E H d − , σ ) is a Σ ∗ -extension of ( K ( n )[ x ] , σ ). Note that this is a P -normalized Σ ∗ -extension. Consider the difference ring extension ( E H d , σ ) of ( E H d − , σ ). Bythe construction from above there is a difference ring homomorphism τ : E H d → S ( K ). Inparticular, by τ ′ := τ | E H d − is a difference ring homomorphism from E H d − to S ( K ) and thus τ ′ is injective by Theorem 2. Suppose that ( H , σ ) with H = E H d − [ t ( d )1 , . . . , t ( d ) r − ] is a Σ ∗ -extension of ( E H d − , σ ). Then ( H , σ ) is a P -normalized Σ ∗ -extension of ( K ( n )[ x ] , σ ). Notethat const( H , σ ) = const( K ( n )[ x ] , σ ) = K is a field. Now suppose that ( H [ t ( d ) r ] , σ ) is nota Σ ∗ -extension of ( H , σ ). Take β := σ ( t ( d ) r ) − t ( d ) r ∈ K [ n ] P [ t (1)1 , . . . , t ( d − m d − ]. By Theorem 1it follows that there is a g ∈ K ( n )[ x ][ t (1)1 , . . . , t ( d − m d − ][ t ( d )1 , . . . , t ( d ) r − ] such that (30) holds. ByProposition 2 we conclude that g ∈ K [ t (1)1 , . . . , t ( d − m d − ][ t ( d )1 , . . . , t ( d ) r − ]. Define p := t ( d ) r − g ∈ K [ t (1)1 , . . . , t ( d − m d − ][ t ( d )1 , . . . , t ( d ) r ] \ { } (34)and consider q := σ ( p ) − p . By Proposition 1 we conclude that q ∈ K ( n )[ x ][ t (1)1 , . . . , t ( d − m d − ].Let s ( n ) be the H -sum that is represented by a ( d ) r . This means that s ( n ) = P nk =1 F ( k − β )( n ) = F ( n ). Let G ( n ) = expr( g )( n ). Note that F ( k ) has no pole for all n with n ≥ H -sums. Furthermore g is free of n and thus also G ( n ) is free ofrational expression in n (only the arising H -sums depend on n in the outermost summationsign). Hence G ( n ) and G ( n + 1) have no poles for all n with n ≥
0. Therefore we get G ( k + 1) − G ( k ) = expr( g )( k + 1) − expr( g )( k ) = expr( σ ( g ))( k ) − expr( g )( k ) If E H is free of x , this statement has been proven in [40]. = expr( σ ( g ) − g )( k ) = expr( β )( k ) = F ( k )for all k ≥
0. Hence F ( k ) = G ( k + 1) − G ( k ) for all k ≥ k from 0 to n − n ≥ s ( n ) = n X k =1 F ( k −
1) = n − X k =0 F ( k ) = n − X k =0 (cid:0) G ( k + 1) − G ( k ) (cid:1) = G ( n ) − c with c := G (0) ∈ K . Consequently, τ ( p ) = τ (( a ( d ) r + I ) − g ) = τ ( a ( d ) r ) − τ ( g ) = h s ( λ ) i λ ≥ − h G ( λ ) i λ ≥ = h s ( λ ) − G ( λ ) i λ ≥ = h c, c, c, . . . i and therefore τ ′ ( q ) = τ ( q ) = τ ( σ ( p ) − p ) = σ ( τ ( p )) − τ ( p ) = . Since τ ′ is injective, q = 0. Hence Lemma 8 and the linearity of ¯ L yield σ ( ¯ L ( p + I )) − ¯ L ( p + I ) = ¯ L ( σ ( p + I )) − ¯ L ( p + I ) = ¯ L ( σ ( p + I ) − ( p + I )) = ¯ L ( q + I ) = ¯ L (0+ I ) = 0 . Therefore σ ( u ) = u with u := ¯ L ( p + I ). By (34) it follows that u ∈ ¯ V by Lemma 4. Write p = p ′ + I d ∈ E H d \ { } with p ′ ∈ A H d . Note that L ( p ′ ) = 0 since p = 0 (i.e., p = 0 + I d ). Thus u = ¯ L ( p + I ) = L ( p ′ ) = 0, a contradiction to Lemma 4. (cid:3) Example 6.
By Theorem 3 the difference ring ( E H , σ ) , or equivalently the difference ring (¯ E H , ¯ σ ) , of Example 3 with H = A h is a Σ ∗ -extension of ( K ( n )[ x ] , σ ) . In particular, we obtainthe difference ring embedding τ : ¯ E H → S ( K ) given by τ ( ¯ S ) = h S ( λ ) i λ ≥ , τ ( ¯ S ) = h S ( λ ) i λ ≥ ,τ ( ¯ S , ) = h S , ( λ ) i λ ≥ , τ ( ¯ S ) = h S ( λ ) i λ ≥ ,τ ( ¯ S , , ) = h S , , ( λ ) i λ ≥ , τ ( ¯ S , ) = h S , ( λ ) i λ ≥ ,τ ( ¯ S ) = h S ( λ ) i λ ≥ . Therefore we get the polynomial ring τ ( E H ) = τ ( K ( n )[ x ])[ τ ( ¯ S )][ τ ( ¯ S )][ τ ( ¯ S , ) , τ ( ¯ S )][ τ ( ¯ S , , ) , τ ( ¯ S , ) , τ ( ¯ S )] , i.e., the sequences generated by the basis sums S ( n ) , S ( n ) , S , ( n ) , S ( n ) , S , , ( n ) , S , ( n ) ,and S ( n ) induced by the quasi-shuffle algebra are algebraically independent over the rationalsequences adjoined with the alternating sequence. Obviously, these sums are also algebraicallyindependent over the field of rational sequences τ ( K ( n )) . By induction it follows by Theorem 3 that there is a difference ring embedding τ : E H →S ( K ). In particular, following the construction from above, it has the form as stated in thefollowing corollary. Corollary 1.
Let
H ∈ { A h , A a , A c ( M ) } . Then const( E H , σ ) = K and there is a differencering embedding τ : E H → S ( K ) . Furthermore, τ ( E H ) = τ ( K ( n )[ x ])[ τ ( t (1)1 ) , . . . , τ ( t (1) m )][ τ ( t (2)1 ) , . . . , τ ( t (2) m )] . . . forms a polynomial ring with the sequences τ ( t ( d ) j ) = h expr( a ( d ) j )( λ ) i λ ≥ where expr( a ( d )1 ) , . . . , expr( a ( d ) m d ) are the H -sums of weight d which cannot be reduced furtherby the quasi-shuffle algebra. In the following we define M k = { , , . . . , k } and consider the general case H = A c .Let ( E A c ( M k ) d , σ d,k ) for k ∈ N and d ∈ N be the reduced difference ring for the cyclotomicharmonic sums H = A c ( M k ) with weight ≤ d as constructed in Section 5. By constructionwe get the chain of difference ring extensions( E A c ( Mk ) , σ ,k ) ≤ ( E A c ( Mk ) , σ ,k ) ≤ ( E A c ( Mk ) , σ ,k ) ≤ . . . (35)where the cyclotomic alphabet remains unchanged and the weights of the extensions areincreased step-wise. In particular, these extensions form Σ ∗ -extensions by Theorem 3. Fur-thermore, by Remark 1 we get the chain of difference ring extensions( E A c ( M d , σ d, ) ≤ ( E A c ( M d , σ d, ) ≤ ( E A c ( M d , σ d, ) ≤ . . . (36)where the weights do not increase but the cyclotomic alphabets are increased step-wise. Again,these extensions form Σ ∗ -extensions by Theorem 3.By (35) and (36) we get the Σ ∗ -extensions( E A c ( Mk ) k , σ k,k ) ≤ ( E A c ( Mk +1) k , σ k,k +1 ) ≤ ( E A c ( Mk +1) k +1 , σ k +1 ,k +1 ) (37)for all k ∈ N . Here the cyclotomic alphabets and weights are increased simultaneously stepby step.Finally, define, E A c := [ k ∈ N E A c ( Mk ) k . Since E A c ( Mk ) k is a subring of E A c ( Mr ) r for any r, k ∈ N with k ≤ r , E A c forms a ring. Furthermoredefine the function σ : E A c → E A c as follows. For f ∈ E A c take k ∈ N minimal such that f ∈ E A c ( Mk ) k . Then we define σ ( f ) := σ k,k ( f ). Lemma 12. ( E A c , σ ) is a difference ring which is a difference ring extension of ( E A c ( M r ) r , σ r,r ) .Proof. Let f ∈ E A c ( Mr ) r be arbitrary but fixed. Take l ∈ N being minimal such that f ∈ E A c ( Ml ) l .Then σ ( f ) = σ l,l ( f ). By iterative application of (37) it follows that σ r,r ( f ) = σ l,l ( f ) = σ ( f ).Therefore σ ( f ) = σ r,r ( f ) for all f ∈ E A c ( Mr ) r . (38)Let f, g ∈ E A c . Now take r such that f, g, σ ( f ) ∈ E A c ( Mr ) r . Then by (38) and the fact that σ r,r is a difference ring automorphism, we conclude that σ ( f g ) = σ ( f ) σ ( g ) and σ ( f + g ) = σ ( f ) + σ ( g ). Similarly, one can show that σ is bijective. Hence σ is a difference ring automorphism,i.e., ( E A c , σ ) is a difference ring. Finally, by (38) we conclude that ( E A c , σ ) is a difference ringextension of ( E A c ( M r ) r , σ r,r ) for any r ∈ N . (cid:3) At the end, with Lemma 12 and (37) we obtain the following chain of difference ringextensions ( E A c ( M )1 , σ , ) ≤ ( E A c ( M )2 , σ , ) ≤ ( E A c ( M )3 , σ , ) ≤ · · · ≤ ( E A c , σ )where each ( E A c ( M k − ) k − , σ k − ,k − ) ≤ ( E A c ( M k ) k , σ k,k ) forms a nested Σ ∗ -extension.Summarizing, we obtain the following result. Corollary 2.
Let H = A c . Then const( E H , σ ) = K and there is a difference ring embedding τ : E H → S ( K ) . In particular, τ ( E H ) = τ ( K ( n )[ x ])[ τ ( s (1)1 ) , . . . , τ ( s (1) µ )][ τ ( s (2)1 ) , . . . , τ ( s (2) µ )] . . . (39) forms a polynomial ring with the sequences τ ( s ( k ) j ) = h expr( b ( k ) j )( n ) i n ≥ where expr( b ( k )1 ) , . . . , expr( b ( k ) µ k ) are all A c ( M ( k )) -sums of weight ≤ k which are not containedin A c ( M ( k − and which cannot be reduced further by the quasi-shuffle algebra. We remark that the construction of the difference ring ( E A c , σ ) can be accomplished inmany different ways yielding a different sorting of the sum-generators in (39). Here we chosea construction where each extension step is built only by finitely many Σ ∗ -extensions.8. Conclusion
We showed that the reduced representation (induced by the quasi-shuffle algebra) of theharmonic sums with the alphabet A h and A a and the cyclotomic harmonic sums with thealphabet A c constitute a tower of Σ ∗ -extensions. This means we can model these nested sumsin a difference ring extension where the constants are not extended. As a consequence thesenested sums can be embedded into the ring of the sequences. Furthermore, this shows thatthe found relations due to the quasi-shuffle algebra are complete, i.e., there are no furtherrelations that might occur between the sequences of the nested sums. We emphasize thatthese techniques are not restricted to cyclotomic harmonic sums. Most of the ideas presentedin this article can be carried over to the more general class of R ΠΣ ∗ -extensions [46, 44] (whichcontains the class of Σ ∗ -extensions). Since this class covers, e.g., q –hypergeometric sequences,one can also treat the generalized (cyclotomic) harmonic sums with the alphabet A g := A ∩ ( N × N × N × S )for certain finite sets S ⊂ K , the inverse binomial sums and also their q -versions in thissetting. We expect that the results and tools presented in this work will be helpful to showalgebraic independence in the ring of sequences also for these more general classes of nestedsums. References [1] J. Ablinger. A computer algebra toolbox for harmonic sums related to particle physics. Master’s thesis,Johannes Kepler University, 2009.[2] J. Ablinger.
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Research Institute for Symbolic Computation, J. Kepler University Linz, A-4040 Linz, Austria
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