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., [49, 35], [32, 48] and [4, 5], respectively. In particular, these nested sums cover the class of d’Alembertian solutions , a sub-class of Liouvillian solutions , of linear recurrence relations; for further details see .
Numerous properties of such sum classes, like the harmonic sums [17, 51], cyclotomic harmonic sums , generalized harmonic sums [30, 9] or binomial sums [23, 22, 52, 7] 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 , the analytic continuation [13, 18] of nested sums or the calculation of asymptotic expansions of such sums [21, 15, 16] has been worked out. For further details and generalizations of these results we refer to [8, 9, 7]. 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, 46, 44, 47]. This means that the sequences with entries from a field , that are produced by the reduced sums, are algebraically independent. In order to accomplish this task, the arising sums and products are represented in a difference ring, i.e., the sum objects are represented in a ring and the shift behaviour of the sums is modelled by a ring automorphism . More precisely, the sums and products are represented in an -extension111For the corresponding difference field theory see [28, 29]. [46, 44] with the distinguished property that the set of constants is precisely the field , i.e.,
Exactly this property enables one to embed the ring into the ring of sequences. This technology has been used to show in  that the sequences of the generalized harmonic numbers are algebraically independent over the rational sequences. In particular, fast summation algorithms [41, 45, 39] 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  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  and continued in [8, 2, 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 . Summarizing, using the property of the underlying quasi-shuffle 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 non-alternating harmonic sums has been accomplished in  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 and non-zero integers and for their cyclotomic versions : for being a field containing the rational numbers the summand is of the kind222 denotes the positive integers and .
and denotes the summation variable.
In this article we will consider the so-called basis sums induced by the quasi-shuffle 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 which is defined by the set of sequences 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: they are algebraically independent over the sub-ring of sequences that is generated by all rational functions from and . We will derive this result by showing that the basis sums generate an -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 . Based on this particularly nice structure it will follow by difference ring theory [50, 47] 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 -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 -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.
2. A general framework for cyclotomic harmonic sums
Throughout this article we assume that is a field containing as a subfield. In particular, we assume that there is a linear ordering on . For a set , denotes the set of all finite words over (including the empty word), i.e.,
Furthermore, we define the alphabet
as a totally ordered, graded set. More precisely, the degree of is denoted by . This establishes the grading . Moreover, we define the linear order on in the following way:
Furthermore, we define the function
Note that . For arbitrary letters in the connection is more complicated but there is always a relation of the form
Moreover, we define the weight function on these nested sums: 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.,
Instead of we will also write or with .
A product of two nested sums with the same upper summation limit can be written in terms of single nested sums: for
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 with . 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
Note that for every finite subset of we have
Throughout this article we will assume that
holds. In particular, we call a sum with also -sum.
3. A basic difference ring construction for the expression of -sums
In the following we will define a difference ring in which we will represent the expressions of -sums.
An expression of -sums in over a field is built by
rational expressions in with coefficients from , i.e., elements from the rational function field ,
that occurs in the numerator,
the -sums that occur as polynomial expressions in the numerator.
If is such an expression we use for the shortcut
Sometimes we also use the notation to indicate that the expression depends on a symbolic variable . We say that an expression of -sums has no pole for all with for some , if the rational functions occurring in do not introduce poles at any evaluation for with . If this is the case, one can perform the evaluation for all with . For a more rigorous definition of indefinite nested product-sum expressions (containing as special case the -sums) in terms of term algebras, we refer to  which is inspired by .
These expressions will be represented in a commutative ring and the shift operator acting on the expressions in terms of -sums will be rephrased by a ring automorphism . Such a tuple of a ring equipped with a ring automorphism is also called difference ring; if is a field, is also called a difference field. In such a difference ring we call a constant if and denote the set of constants by
In general, is a subring of . But in most applications we take care that itself forms a field.
Our construction will be accomplished step by step. Namely, suppose that we are given already a difference ring in which we succeeded in representing parts of our -sums. In order to enrich this construction, we will extend the ring from to and will extend the ring automorphism to a ring automorphism , i.e., for any we have that . We say that such a difference ring is a difference ring extension of ; in short, we also write . Since and agree on , we usually do not distinguish anymore between and .
We start with the rational function field and define the field/ring automorphism with
It is easy to verify that . So far, we can model rational expressions in in the field and can shift these elements with .
Next, we want to model the object with the relations and . Therefore we take the ring subject to the relation . Then one can verify that there is a unique difference ring extension of with . In particular, we have that
Precisely, this difference ring enables one to represent all rational expressions in together with objects that are rephrased by .
Before we can continue with our construction for -sums, we observe the following easy, but important fact.
Let be a difference ring and let be a polynomial ring, i.e., is transcendental over , and let . Then there is a unique difference ring extension of with .
We will use this lemma iteratively in order to adjoin all -sums to the difference ring . This construction is done inductively on the weight of the sums. It is useful to define the following function (compare (3)):
The base case is the already constructed difference ring with . Now suppose that we constructed the difference ring for all -sums of weight . Then we will construct a difference ring extension which covers precisely the -sums of weight . Consider all -sums with weight say
To these sums we attach the variables
of weight , respectively. Now we define the polynomial ring . To this end, we extend from to Suppose that models the -sums with . Note that
where is a -sum of weight . Let which models this sum. Therefore we extend from to subject to the relations
by applying Lemma 1 iteratively. This means, we first adjoin to and extend the automorphism with (9) for , then we adjoin to this ring the variable and extend the automorphism with (9) for , etc. We remark that this construction implies that
By construction is a difference ring extension of .
Finally, we define the polynomial ring
with infinitely many variables, which represents all -sums. In particular, we define the ring automorphism as follows. For any , we can choose a such that . This defines333Note that any other choice with will deliver the same evaluation. with . By construction, where is the automorphism of . It is easy to see that is a difference ring and that it is a difference ring extension of . Again we do not distinguish anymore between and . To sum up, we get the chain of difference ring extensions
For convenience, we will also write for the variable . In this way, we may write e.g.,
To give a résumé, we can express every expression of -sums over in . Conversely, if we are given a ring element , we denote by the expression that is obtained when all occurrences of are replaced by and all variables are replaced by the attached -sums with upper summation range . This will lead to an expression of -sums in over . In this way, we can jump between the function and difference ring worlds.
Now let . Then
recall that for an expression of -sums, is used to emphasize the dependence on the symbolic variable . If and have no poles for all for some , then it follows that
for all with . Moreover observe that we model the shift-behaviour accordingly: For any and any we have that
|and for any and any we have that|
The main goal of this article is to construct a difference ring, which represents all -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 . The following example shows immediately, that is a too naive construction.
Then one can easily verify that Even more, we get that for all , i.e., there are algebraic relations among these sums.
4. Quasi-shuffle algebras and the linearization operator
In order to eliminate such relations as given in Example 1, we will equip the difference ring construction with the underlying quasi-shuffle algebra.
Definition 2 (Non-commutative Polynomial Algebra).
Let be a totally ordered, graded set. The degree of is denoted by Let denote the free monoid over , i.e.,
We extend the degree function to by for and Let be a commutative ring. The set of non-commutative polynomials over is defined as
Addition in is defined component wise and multiplication is defined by
We define a new multiplication on 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 -sums; compare [25, 26, 27].
Definition 3 (Quasi-shuffle product).
is called quasi-shuffle product, if it distributes with the addition and
where , is a function satisfying
We specialize the quasi-shuffle algebra from Definition 3 in order to model the -sums accordingly. We consider the alphabet and define the degree of a letter a by . Finally, we define
and for all . This function obviously fulfils (S0)-(S3). In other words, if we take our commutative ring , then forms a quasi-shuffle algebra.
Let . By using the expansion of (14), we can write
for some uniquely determined (compare ). In particular, we have that
This linearization will be carried over to . Consider the -module
Now we are in the position to define the linearization function as follows. For , we take the and from (16) and define
By (17) it follows that
Finally, we extend to by linearity.
Let and take such that has no poles for all with . Then for all ,
We have that
In particular, as already indicated in (1) we get
Clearly, we can consider as a subset of , i.e., we can equip with the linearization function . Observe that for any we have that . In addition, we obtain the following lemma.
For any , and .
We only give a proof for since follows analogously. It suffices to prove for a monomial since then we can extend the result by linearity. First consider the product of two nested sums (compare (15)): let
with and Then