Algebraic Independence of Sequences Generated by (Cyclotomic) Harmonic Sums

Indefinite nested sums are important building blocks to assemble closed forms for combinatorial counting problems or for problems that arise, e.g., in particle physics. Concerning the simplicity of such formulas an important subtask is to decide if the arising sums satisfy algebraic relations among each other. Interesting enough, algebraic relations of such formal sums can be derived from combinatorial quasi-shuffle algebras. We will focus on the following question: can one find more relations if one evaluates these sums to sequences and looks for relations within the ring of sequences. In this article we consider the sequences of the rather general class of (cyclotomic) harmonic sums and show that their relations coincide with the relations found by their underlying quasi-shuffle algebra. In order to derive this result, we utilize the quasi-shuffle algebra and construct a difference ring with the following property: (1) the generators of the difference ring represent (cyclotomic) harmonic sums, (2) they generate within the ring all (cyclotomic) harmonic sums, and (3) the sequences produced by the generators are algebraically independent among each other. This means that their sequences do not satisfy any polynomial relations. The proof of this latter property is obtained by difference ring theory and new symbolic summation results. As a consequence, any sequence produced by (cyclotomic) harmonic sums can be formulated within our difference ring in an optimal way: there does not exist a subset of the arising sums in which the sequence can be formulated as polynomial expression.


Introduction
Special functions like the harmonic numbers and more generally indefinite nested sums defined 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 in connection with symbolic summation see, e.g., [35,49], [32,48], and [4,5], respectively. In particular, these nested sums cover the class 214 J.AblingerandC. Schneider 2 J. Ablinger and C. Schneider of d'Alembertian solutions [12], a sub-class of Liouvillian solutions [24], of linear recurrence relations; for further details see [34]. Numerous properties of such sum classes, like the harmonic sums [17,51], cyclotomic harmonic sums [8], generalized harmonic sums [9,30], or binomial sums [7,22,23,52] have been explored. In particular, the connection of the nested sums to nested integrals (i.e., to multiple polylogarithms and generalizations of them) via the (inverse) Mellin transform [36], the analytic continuation [13,18] of nested sums or the calculation of asymptotic expansions of such sums [15,16,21] has been worked out. For further details and generalizations of these results we refer to [7][8][9]. The underlying algorithms are implemented in the Mathematica package HarmonicSums [2,3].
Among all these algorithmic constructions, a key technology is the elimination of algebraic dependencies of the arising nested sums within a given expression to gain compact representations. Here the Mathematica package Sigma [38,43] provides strong tools that can simplify, among many other features, an expression in terms of indefinite nested product-sums to an expression in terms of such sums that are all algebraically independent in the analysis sense [40,42,44,46,47]. This means that the symbolic sums in the simplified expression evaluate to sequences with entries from a field K, that are algebraically independent. Internally, these sums are represented as variables in a polynomial ring A and the shift behavior of the sums is modeled by a ring automorphism σ : A → A. More precisely, the sums and products are represented in an RΠΣ * -extension † [44,46] with the distinguished property that the set of constants is precisely the field K, i.e., {c ∈ A | σ (c) = c} = K.
Exactly this property enables one to embed injectively the ring A into the ring of sequences and the algebraic independence of the variables (which represent the sums) implies that also the sequences produced by the sums/variables are algebraically independent. This technology has been used to show in [46] that the sequences of the generalized harmonic numbers are algebraically independent over the rational sequences. In particular, fast summation algorithms [39,41,45] in the setting of difference rings and fields support this construction algorithmically and expressions with up to several hundred algebraically independent sums can 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 ring algorithms 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 classes of indefinite nested sums can be utilized: they obey quasi-shuffle algebras [25][26][27]. This enables one to rewrite any polynomial expression in terms of indefinite nested sums as a linear combination of indefinite nested sums. As worked out in [14] and continued in [2,8,9], this feature can be used to hunt for algebraic relations among the occurring indefinite nested sums and to express the compact result in terms of the so-called basis sums which cannot be eliminated further by the quasi-shuffle algebra. Using the HarmonicSums package expressions as mentioned above could be reduced to several MB in terms of about several thousand basis sums; for details see [6]. Summarizing, using the property of the underlying quasi-shuffle † For the corresponding difference field theory see [28,29].

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Algebraic Independence of (Cyclotomic) Harmonic Sums 3 algebra one obtains dramatic compactifications within the demanding calculations in particle physics. A natural question is if the obtained sums induced by the quasi-shuffle algebra are also algebraically independent in the sense of analysis, i.e., if the sequences produced by the nested sums are algebraically independent. A special variant for nonalternating harmonic sums has been accomplished in [21] using the knowledge of certain integral representations. In the following we will focus on the general case for the harmonic sums [17,51] ∑ with non-negative integers n and non-zero integers c i (1 ≤ i ≤ k) and for their cyclotomic versions [8]: for K being a field containing the rational numbers the summand and i j denotes the summation variable. In this article we will consider the so-called basis sums induced by the quasishuffle algebra. This means we consider a particular chosen set of nested sums that generate all other nested sums and that do not possess any further relations using the quasi-shuffle algebra operation. Our main result is that these basis sums are also algebraically independent as sequences. More precisely, consider the ring of sequences (or ring of germs [33]) which is defined by the set of sequences K N 0 = { a n n≥0 |a n ∈ K} equipped with component-wise addition and multiplication where two sequences are identified as equal if they differ only by finitely many entries. Then we will show that the basis sums evaluated to such elements of the ring of sequences are algebraically independent over the sub-ring of sequences that is generated by all rational functions from K(n) and (−1) n . In this regard, we use the following definition.
In this case, the smallest subring containing A and x 1 , . . . , x r is denoted by A[x 1 , . . . , x r ] and is called a polynomial ring with variables x 1 , . . . , x r and coefficients in A.
We will derive this result by showing that the basis sums generate an RΠΣ *extension in the difference ring sense. This means that the basis sums generate a polynomial ring equipped with a shift operator such that the set of constants is precisely K. Based on this particularly nice structure it will follow by difference ring ‡ N denotes the positive integers and N 0 = N ∪ {0}. 216 J.AblingerandC. Schneider 4 J. Ablinger and C. Schneider theory [47,50] that this difference ring can be embedded by an injective difference ring homomorphism into the ring of sequences. In other words, the algebraic properties of the polynomial ring (in particular, the algebraic independence of variables of the polynomial 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 framework for (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 link the quasi-shuffle algebra with our difference ring construction. In Section 5, we define the reduced difference ring for (cyclotomic) harmonic sums in which all algebraic relations are eliminated that are induced by the quasi-shuffle algebra. We will provide new structural results obtained by the difference ring theory of RΠΣ * -extensions in Section 6. In Section 7 we will combine all these results and will show that our reduced difference ring is built by a tower of RΠΣ * -extensions. As a consequence, we can conclude that this ring can be embedded into the ring of sequences. A conclusion is given in Section 8.

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 finite words over B (including the empty word), i.e., Furthermore, we define the alphabet 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 the linear order < on A in the following way: Furthermore, we define the function AlgebraicIndependenceof(Cyclotomic)HarmonicSums 217 Algebraic Independence of (Cyclotomic) Harmonic Sums 5 Note that λ ((1, 0, c 1 , z 1 ), i)λ ((1, 0, c 2 , z 2 ), i) = λ ((1, 0, c 1 + c 2 , z 1 z 2 ), i). For arbitrary letters in A the connection is more complicated but there is always a relation of the form with r j ∈ Q, (e j , f j , g j , h j ) ∈ A, and g j ≤ c 1 + c 2 , see, e.g., [2,8]. For n ∈ N 0 , k ∈ N, a i ∈ A with 1 ≤ i ≤ k we define nested sums as (compare [2,8]) Moreover, we define the weight function w on these nested sums: w(S a 1 , a 2 , a 3 ,..., a k (n)) = |a 1 | + · · · + |a k | and extend it to monomials such that the weight of a product of nested sums is the sum of the weights of the individual sums, i.e., w S a 1 (n)S a 2 (n) · · · S a k (n) = w (S a 1 (n)) + w(S a 2 (n)) + · · · + w S a k (n) .
Instead of S a 1 , a 2 ,..., a k (n) we will also write S a 1 a 2 ···a k (n) or S a (n) with a = a 1 a 2 · · · a k ∈ A * . A product of two nested sums with the same upper summation limit can be written in terms of single nested sums: for n ∈ N 0 , Note that the product of the two sums within the summands of the right side can be expanded further by using again this product formula. Applying this reduction recursively will lead to a linear combination of sums S b 1 ,..., b r (n) with b i ∈ A. In particular, the maximum of all the weights 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, 0, c, 1) with c ∈ N, i.e., we restrict to then we are dealing with harmonic sums, see, e.g., [14,51]. (2) If we consider only letters of the form (1, 0, c, ±1) with c ∈ N, i.e., we restrict to 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 then we are dealing with cyclotomic harmonic sums, see, e.g., [2,8].
Note that for every finite subset M of N we have Throughout this article we will assume that holds. In particular, we call a sum S a 1 a 2 ···a k (n) with a i ∈ H also H-sum.

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 expressions of H-sums.

Definition 3.1. 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 rational function field K(n), (2) (−1) n that occurs in the numerator, (3) the H-sums that occur as a polynomial expressions in the numerator.
If f is such an expression we use for λ ∈ K(n) the shortcut Sometimes we also use the notation f (n) to indicate that the expression depends on a symbolic variable n. We say that an expression e(n) of H-sums has no pole for all n ∈ N 0 with n ≥ λ for some λ ∈ N 0 , if the rational functions occurring in e(n) do not introduce poles at any evaluation n → ν for ν ∈ N 0 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 AlgebraicIndependenceof(Cyclotomic)HarmonicSums 219 Algebraic Independence of (Cyclotomic) Harmonic Sums 7 definition of indefinite nested product-sum expressions (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 acting on 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 a difference ring; 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 by In general, const (A, σ ) is a subring of (A, σ ). But in most applications we take care that const (A, σ ) itself forms a field.
Our construction will be accomplished step by step. Namely, suppose that we are given already 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 extend the ring automorphism σ to a ring automorphism 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 automor- It is easy to verify that const(K(n), σ ) = const (K, σ ) = K. So far, we can model rational expressions in n in the field K(n) and can shift these elements with σ .
Next, we want to model the object (−1) n with the relations ((−1) n ) 2 = 1 and (−1) n+1 = −(−1) n . Therefore, we take the ring K(n)[x] subject to the relation x 2 = 1. Then one can verify that there is a unique difference ring extension (K(n)[x], σ ) of (K(n), σ ) with σ (x) = −x. In particular, we have that Precisely, this difference ring (K(n)[x], σ ) enables one to represent all rational expressions in n together with objects (−1) 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 3.2. Let (A, σ ) be a difference ring and let A[t]
be a polynomial ring, i.e., t is transcendental 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 to define the following function (compare (2.1)): The base case is the already constructed difference ring . To these sums we attach the variables σ . Finally, we define the polynomial ring with infinitely many variables, which represents all H-sums. In particular, we define the ring automorphism σ : A H → A H as follows. For any f ∈ A H , we can choose AlgebraicIndependenceof(Cyclotomic)HarmonicSums 221 Algebraic Independence of (Cyclotomic) Harmonic Sums 9 It is easy to see that A H , σ is a difference ring and that it 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 For convenience, we will also writeS a 1 ,..., a k for the variable s In this way, we may write, e.g., To give a résumé, 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 obtained when all occurrences of x are replaced by (−1) n and all variables s are replaced by the attached H-sums with upper summation range n. This will lead to an expression of H-sums in n over K. In this way, we can jump between the function and difference ring worlds. Then and recall that for an expression e of H-sums, e(n) is used to emphasize the dependence on the symbolic variable n. If expr( f )(n) and expr(g)(n) have no poles for all n ≥ δ for some δ ∈ N 0 , then it follows that and for all λ ∈ N 0 with λ ≥ δ . Moreover observe that we model the shift-behaviour accordingly: for any k ∈ N and any λ ≥ δ we have that and for any k ∈ N and any λ ≥ δ + k we have that 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 the introduction, we will rely on the fact that the constants are precisely the elements K. The following example shows immediately that A H , σ is a too naive construction.
Then one can easily verify that σ ( f ) = f . Even more, we get that expr( f )(λ ) = 0 for all λ ∈ N 0 , 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 3.3, we will equip the difference ring construction A H , σ with the underlying quasi-shuffle algebra.
Definition 4.1. (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 1 · · · a n | a i ∈ G, n ≥ 1} ∪ {ε} .
We extend the degree function to G * by |a 1 a 2 · · · a n | = |a 1 |+ |a 2 |+· · · +|a n | for a i ∈ G and |ε| = 0. Let R ⊇ Q be a commutative ring. The set of non-commutative polynomials over R is defined as R G := ∑ w∈G * r w w r w ∈ R, r w = 0 for almost all w .

Addition in R G is defined component wise and multiplication is defined by
We define a new multiplication * on R G which is a generalisation of the shuffle product, by requiring that * distributes with the addition. We will see that this product can be used to describe properties of H-sums; compare [25][26][27].  AlgebraicIndependenceof(Cyclotomic)HarmonicSums 223 Algebraic Independence of (Cyclotomic) Harmonic Sums 11 We specialize the quasi-shuffle algebra from Definition 4.2 in order to model the H-sums accordingly. We consider the alphabet G = H and define the degree of a letter a by |a| = w(a). Finally, we define and [a, 0] = 0 for all a, b ∈ H. This function obviously fulfils (S0)-(S3). In other words, if we take our commutative ring R = K(n)[x], then K(n)[x] H forms a quasishuffle algebra. Let a 1 , . . . , a r ∈ H * . By using the expansion of (4.1), we can write for some uniquely determined d i ∈ H * , c i ∈ K * (compare [8]). In particular, we have that This linearization will be carried over to A H . Consider the K(n)[x]-module Now we are in a position to define the linearization function L : A H → V as follows.
For a 1 , . . . , a k ∈ H, we take the c i ∈ K and d i ∈ H from (4.3) and define L S a 1 · · ·S a 1 = c 1Sd 1 + · · · + c mSd m .

By (4.4) it follows that
Finally, we extend L to A H by linearity. Since (4.1) reflects precisely (2.4), we obtain the following lemma.   Clearly, we can consider V as a subset of A H , i.e., we can equip A H , σ with the linearization function L. Observe that for any f ∈ V we have that σ ( f ) ∈ V. In addition, we obtain the following lemma.
Proof. We only give a proof for σ −1 (L( f )) = L σ −1 ( f ) since σ (L( f )) = L(σ ( f )) follows analogously. It suffices to prove σ −1 (L( f )) = L σ −1 ( f ) for a monomial f ∈ A H since then we can extend the result by linearity. First consider the product of two nested sums (compare (4.2)): let Now proceed by induction on the number of factors. Assume that the statement holds for f being the product of k factorsSâ 1 a 1Sâ2 a 2 · · ·Sâ k a k and let

AlgebraicIndependenceof(Cyclotomic)HarmonicSums 225
Algebraic Independence of (Cyclotomic) Harmonic Sums 13 Using (4.7) and the induction hypothesis we conclude We conclude with the following lemma which will be essential to prove our main result stated in Theorem 7.2.
n 2 ⊕ · · ·. Then: ( 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 −1 ( f ) = f .

The Reduced Difference Ring
We define the reduced difference ring where all relations of A H , σ are factored out by the quasi-shuffle algebra, i.e., we define It is immediate that I is an ideal of A H and we can define the quotient ring 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 4.5. Hence, σ ( f ) ∈ I; similarly, it follows that σ −1 ( f ) ∈ I. From this it 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 can consider 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 in an iterative fashion. Using the set of all power products Π by Π = S a 1S a 2 · · ·S a r | r ∈ N and a i ∈ H * for 1 ≤ i ≤ r we obtain the following possibility to generate I.
Proof. Denote the set on the right hand side by J.

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Algebraic Independence of (Cyclotomic) Harmonic Sums 15 Thus f ∈ J and therefore I ⊆ J. This completes the proof.
Even more, we get the chain of difference ring extensions In the following we will work out further how the construction from E H d−1 , 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 than or equal to d. Hence, by Lemma 5.1 we get Exploiting this property we can perform the following construction. At the specified weight d we can set up a matrix containing all the relations with p ∈ Π and w(p) = d where the columns (except for the last one) represent the sums of weight d coming from L(p) and the last column represents the polynomial p which is built by sums of weight less than d. Now we can transform the matrix to its reduced row-echelon form. The sums corresponding to the corner elements can be reduced while the other sums remain as variables. Exactly this crucial observation has been strongly utilized for harmonic sums in [14] in order to derive all algebraic relations induced by the quasi-shuffle algebra. For further results and heavy calculations using HarmonicSums we refer to [1,2,8,9]. Now we will link this observation with our difference ring construction. Let By construction this ring is a polynomial ring, i.e., the elements t are algebraically independent among each other.
Finally, by the above considerations we conclude that The above construction will be illustrated by the following example. Here we consider all non-trivial polynomials (5.3) for p ∈ Π with w(p) = 4: For example, p =S 1 4 ∈ Π yields the first entry of R. Note that by Lemma 4.3 the elements in R transformed to H-sum expressions evaluate to 0 for all n ≥ 0. For example, the first entry is precisely the equation worked out in Example 4.4. Note further that The relations in R lead to the matrix where (M | h) v t gives back a vector with the entries from R. Now we reduce this matrix using Gaussian elimination together with some relations of lower weight to get the reduced echelon form By construction the elements of R can be written in terms of the elements of R plus some extra elements from I 3 . Thus the following holds: The advantage of this representation of I 4 with R (instead of R) is that one can read off rewrite rules to eliminate sums: the variables corresponding to the corner points of the reduced matrix, i.e., the variables S 1, 1, 1, 1 ,S 1, 1, 2 ,S 1, 2, 1 ,S 2, 2 ,S 1, 3 (5.6) can be reduced with the substitution rules ¶ S 1, 1, 1, 1 →S We remark that we can calculate with the representantsS 2, 1, 1 ,S 3, 1 ,S 4 instead of the equivalence classesS 2, 1, 1 + I 4 ,S 3, 1 + I 4 ,S 4 + I 4 , respectively. There is only a subtle detail. If we want 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 (5.6) might be again introduced by σ , we have to eliminate them by the found substitution rules stated in (5.7).
Similarly, one can construct the polynomial ring with the substitution rules where ρ( f ) is obtained by applying the substitution rules of (5.7), (5.8), and (5.9) such that one obtains an element of the polynomial ringĒ H 4 = K(n)[x] S 1 S 2 S 2, 1 ,S 3 S 2, 1, 1 ,S 3, 1 ,S 4 being a subring of A E . Then the difference ring E H 4 , σ can be represented by the difference ring Ē H 4 ,σ with the ring automorphismσ :Ē H 4 →Ē H 4 defined bȳ σ ( f ) = ρ(σ ( f )). Here we apply first the given automorphism σ : A H → A H with f := σ ( f ) ∈ A H and apply afterwards the constructed substitution rules to get the element ρ( f ) ∈Ē H 4 . In general, the representants in (5.4) for the variables of weight d (plus the other representants with weights < d from the previous construction steps) can be used to express all the elements of f ∈ E H d . In particular, all the ring operations can be simply performed in this polynomial ring. Only if one applies σ ( f ), one has to apply the corresponding substitution rules in order to rewrite the expression again in terms of the representants.
AlgebraicIndependenceof(Cyclotomic)HarmonicSums 231 Algebraic Independence of (Cyclotomic) Harmonic Sums 19 is a chain of finite subsets of N then denotes a set of representants of weight d for A c(M i ) as defined in (5.4), then by using our construction above we can find representants such that As a consequence, it follows that E here V j are the generators that represent the additional H-sums of weight j. Moreover, We remark that any element from the equivalence class a (d) i + I d delivers the same evaluation expression.
In addition, we obtain the following connection with the linearization operator.
This implies that we can define the linearization mapL : E H → V with L(g + I) = L(g) for g ∈ A H . In addition, we get the following result. Proof. Write f = g + h with g ∈ A H and h ∈ I. By definition, L(h) = 0. Since I is a difference ideal, σ (h) ∈ I. Together with Lemma 4.5 we get =L(σ ( f )).

Structural Results in RΠΣ * -Ring
In the following we will bring in in addition difference ring theory. Here we will refine the naive construction given by Lemma 3.2 that we used to construct the difference ring (A, σ ) in Section 3. Namely, we will improve this type of extensions by the so-called Σ * -extensions.
Exactly this result is the driving property within the summation package Sigma [38,43]. Indefinite nested sums are represented not in the naive construction given in Lemma 3.2, but sums are only adjoined if the constants remain untouched, i.e., if the telescoping problem cannot be solved. More precisely, in Sigma indefinite summation algorithms are implemented [39,41,45] that decide efficiently, if there exists a g ∈ A such that (6.1) holds. If such an element does not exist, we can perform the construction as stated in Lemma 3.2 and we have constructed a Σ * -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 (6.1). In this way, the occurring sums within an expression are rephrased step by step by a tower of Σ * extensions * of the form (3.4) with K = const(A 0 , σ ) = const (A 1 , σ ) = const(A 2 , σ ) = · · · = const (A, σ ) . * We remark that one can hunt for recurrences of definite sums and can solve recurrence in terms d'Alembertian solution in such difference rings; for details see, e.g., [43] and references therein.
However, this construction gets more and more expensive, the more variables are adjoined. In contrast to that, if we restrict to H-sums, the construction of the tower of ring extensions (5.2), which relies purely on a linear algebra engine, is by far more efficient; see Example 5.2. Example 6.3. Given such a reduction in terms of basis sums (using linear algebra), we could 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, 2, 3, 4, 5, 6, 7 is a Σ * -extension. For d = 7 we represented 507 basis sums, i.e., we constructed a tower of 507 Σ *extensions within 5 days. The case d = 8 is currently out of scope. Similarly, we could show for H = A c(M) with various finite subsets M ⊂ N and d ∈ N that E H d , σ forms a Σ * -extension of (K(n)[x], σ ).
Note that Theorem 6.2 can be supplemented by the following structural result; for the difference field version see, e.g., [40,Thm. 2.4].
Proof. We will show the result by induction on e. Suppose that the proposition holds for e − 1 such extensions. Now consider the case of e extensions as stated in the proposition. Set G := A[t 1 ] · · · [t e−1 ] and define β e := σ (t e ) − t e ∈ A. Let β ∈ A and g ∈ G[t e ] such that (6.1) holds. By [44, Lemma 6] ([46, Lemma 7.2], respectively) it follows that g = ct e +g for some c,g ∈ G. Comparing coefficients with respect to t e in σ (ct e +g) − (ct 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 (ct e ) − ct e ) = β − c β e ∈ A. By the induction assumption we conclude thatg = ∑ e−1 i=1 c i t i + w with c i ∈ K and w ∈ A. Hence, g = ct e + ∑ e−1 i=1 c i t i + w and the proposition is proven.
In the following, we will work out further properties in such a tower of Σ * -extensions. This will finally enable us to show us in one stroke that the tower (5.2) is built by an infinite tower of Σ * -extensions, i.e., that In other words, for the special class of H-sums, we can dispense the user from all the difference ring algorithms of Sigma.
With these notions, we restrict the class of Σ * -extensions as follows. In the remaining subsection we will show in Proposition 6.10 that the telescoping solution g of (6.1) will not depend on n provided that one restricts to certain normalized Σ * -extensions and takes f with a particular shape. Lemma 6.7. Consider the difference field (K(n), σ ) with n being transcendental over K and with 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 = v q .
Proof. Suppose that there is such a g ∈ K(n). By Abramov's universal denominator bounding [11] or Bronstein's generalization † [19] (see Theorems 8 and 10 therein) it follows that g ∈ K[n]. Hence, σ (g) + u g ∈ K[n], a contradiction that q / ∈ K. Proof. Let g = g 0 +g 1 x 1 with g i ∈ K(n) and , there is a j ∈ {0, 1} such that f j ∈ K. By coefficient comparison we get σ (g j ) − (−1) j g j = (−1) j f j . Write f j = v q with gcd(v, q) = 1. Note that q = c p m for some m ∈ N and c ∈ K * . Since disp(p m , p m ) = 0, the existence of the solution g j contradicts to Lemma 6.7.  We follow the construction from [33,Sec. 8.2] in order to turn the shift S : a 0 , a 1 , a 2 , . . . → a 1 , a 2 , a 3 , . . . (7.1) into an automorphism: we define an equivalence relation ∼ on K N 0 by a λ λ ≥0 ∼ b λ λ ≥0 if there exists a d ≥ 0 such that a k = b k for all k ≥ d. The equivalence classes form a ring which 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 (7.1) forms a ring automorphism. The difference ring (S(K), S) is called the ring of sequences (over K).
Consider our difference field (K(n), σ ) with σ (n) = n + 1. Now define the evaluation function ev : K(n) × N → K(n) as follows. For p q ∈ K(n) with p, q ∈ K[n] and gcd(p, q) = 1, 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 ) = 0, i.e., ev( f , k) = 0 for all k ∈ N 0 . Write f = p q with p ∈ K[x] and q ∈ K[x] \ {0}. Since q has only finitely many roots, τ( f ) = 0 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 2 = 1. We extend ev from K(n) to K(n)[x] as follows. For f = f 0 + f 1 x with f 0 , f 1 ∈ K(n) we define ev( f , k) = ev( f 0 , k) + (−1) k ev( f 1 , k).