Abstract
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants. We provide an algorithimic machinery to prove identities which so far could only be proved using classical hypergeometric approaches. These methods are implemented in the computer algebra package HarmonicSums.
1. Infinite nested Pochhammer sums
The goal of this article is to find and prove identities of the following form: where denotes the Pochhammer symbol, and C denotes Catalan’s constant.
Note that similar identities were given in [CitationAblinger 17, CitationLiu and Wang 19]. Such identities are of interest in physics: In particular, such sums have been studied in order to perform calculations of higher-order corrections to scattering processes in particle physics [CitationAblinger et al. 14, CitationDavydychev and Kalmykov 01, Citation04, CitationFleischer et al. 99, CitationJegerlehner et al. 03, CitationKalmykov and Veretin 00, CitationKalmykov et al. 07, CitationOgreid and Osland 98, CitationWeinzierl 04]. Moreover, similar identities were also considered [CitationBorwein et al. 01, CitationBorwein and Lisoněk 00, CitationLehmer 85, CitationZhi-Wei 11, CitationZucker 85], and there is a connection to Apéry’s proof of the irrationality of (see [CitationBorwein and Borwein 87]) [CitationWeinzierl 04, CitationZhi-Wei 11, CitationZucker 85].
While [CitationAblinger 17] basically deals with sums of the form we are going to consider a much wider class of sums in the frame of this paper. In addition, we will state a general computer algebra method to evaluate a large class of sums in terms of nested integrals. Moreover, we will be able to prove a structural theorem, about when such a sum can be expressed in terms of the so-called cyclotomic polylogarithms.
The main purpose of this article is to present methods which can be automated; hence, not all identities presented in this paper are new identities. To make more precise which class of sums we are considering, some definitions are in place. Let and let and for then we call defined as (1–1) (1–1) a cyclotomic harmonic sum (compare [CitationAblinger and Blümlein 13, CitationAblinger et al. 11, Citation14, CitationAblinger 13]) of depth r. Note that if ai = 1 and bi = 0 for we write (1–2) (1–2) and we call a multiple harmonic sum (see, e.g., [CitationAblinger et al. 13, CitationAblinger 13, CitationBlümlein and Kurth 99, CitationBlümlein 00, CitationVermaseren 99]).
The sums we are considering take the form (1–3) (1–3) where and f(n) is a cyclotomic harmonic sum. We will refer to Equation(1–3)(1–3) (1–3) as Pochhammer sum.
We are going to find representations of these Pochhammer sums in terms of special classes of integrals (that are similar to the iterated integrals in [CitationAblinger et al. 14] and correspond to the iterated integrals in [CitationAblinger 17]). These classes of integrals are iterated integrals over hyperexponential functions. More precisely a function f(x) is called hyperexponential if where q(x) is a rational function in x.
Then, an iterated integral over the hyperexponential functions is defined recursively by with the special case Since some letters might have a non-integrable singularity at the base point x= 0 we consistently define where c takes the unique value such that the integrand on the right hand side is integrable at It is important to note that this definition preserves the derivative In general, we set where k and are chosen to remove any non-integrable singularity. Again the result is unique and retains
In the following, we will define a subclass of iterated integrals (compare [CitationAblinger et al. 11]). For and where denotes Euler’s totient function, we define where denotes the ath cyclotomic polynomial, e.g., the first cyclotomic polynomials are given by
Now, let for we define cyclotomic polylogarithms recursively as follows (compare, e.g., [CitationAblinger et al. 11]):
We call k the weight of a cyclotomic polylogarithm and in case the limit exists we extend the definition to x = 1 and write
Note that restricting the alphabet to the letters and (2,0) leads to harmonic polylogarithms [CitationRemiddi and Vermaseren 00].
The proposed strategy to prove and find Pochhammer sum identities reads as follows and follows the method proposed in [CitationAblinger 17]):
Step 1: Rewrite the sums in terms of nested integrals.
Step 2: Rewrite the integrals in terms of cyclotomic polylogarithms (see [CitationAblinger 17, Section 4]).
Step 3: Provide a sufficiently strong database to eliminate relations among these cyclotomic polylogarithms and find reduced expressions (see Section 4).
This article focuses on Step 1, and we will present three different possibilities to find integral representations of Pochhammer sums. In order to accomplish this task, we view infinite sums as specializations of generating functions [CitationAblinger 17, CitationAblinger et al. 14]. Namely, if we are given an integral representation of the generating function of a sequence, then we can obtain an integral representation for the infinite sum over that sequence if the limit can be carried out. This approach to infinite sums can be summarized by the following formula:
For details on Step 2 (implemented in the command SpecialGLToH in HarmonicSums) and on Step 3 we refer to [CitationAblinger 17]. It has to be mentioned that we computed and used relation tables of harmonic polylogarithms at one up to weight 12, for cyclotomic polylogarithms of cyclotomy 4 and 6 we computed and used relation tables of cyclotomic polylogarithms at 1 up to weight 6. The size of these tables amounts to several gigabytes. Note that the full strategy has been implemented in the Mathematica package HarmonicSumsFootnote1 [CitationAblinger 14].
To complete this introduction we define a number of constants that will appear throughout this article:
Table
Here, we extend the definition (Equation1–Equation2) to negative indices by
Note that these constants do not possess any further relations induced by the algebraic properties given in Ablinger [CitationAblinger 17, Section 4], namely shuffle, stuffle, multiple argument, and duality relations, but it is presently not known, whether these constants obey further algebraic equations or not. In few of this question corresponding to logarithms of integers, we refer to Baker’s theorem [CitationBaker 66]. Note that for the subclass of multiple zeta values and Euler sums, we use the slightly different, but equivalent, set of constants compared to [CitationBlümlein et al. 10]. For details on how the constants of [CitationBlümlein et al. 10] can be rewritten in terms of the constants given above, we refer to the accompanying Mathematica notebook.
In the following sections, we will use different methods to compute integral representations of the generating function. In Section 2, we will use holonomic closure properties, while in sections Citation3 and Citation4 we will use rewrite rules. In Section 4, we will consider a subclass of Pochhammer sums, for which we can directly find representations in terms of cyclotomic polylogarithms, i.e., we do not have to deal with Step 2 of the proposed strategy.
2. Using closure properties of holonomic functions to derive generating functions
In the following, we repeat important definitions and properties (compare [CitationAblinger 16, CitationAblinger et al. 14, CitationKauers and Paule 11]). Let be a field of characteristic 0. A function is called holonomic (or D-finite) if there exist polynomials (not all pi being 0) such that the following holonomic differential equation holds: (2–1) (2–1)
We emphasize that the class of holonomic functions is rather large due to its closure properties. Namely, if we are given two such differential equations that contain holonomic functions f(x) and g(x) as solutions, one can compute holonomic differential equations that contain or as solutions. In other words, any composition of these operations over known holonomic functions f(x) and g(x) is again a holonomic function h(x). In particular, if for the inner building blocks f(x) and g(x), the holonomic differential equations are given, also the holonomic differential equation of h(x) can be computed.
Of special importance is the connection to recurrences. A sequence with is called holonomic (or P-finite) if there exist polynomials (not all pi being 0) such that the holonomic recurrence (2–2) (2–2) holds for all (from a certain point on). In the following, we utilize the fact that holonomic functions are precisely the generating functions of holonomic sequences: if f(x) is holonomic, then the coefficients fn of the formal power series expansion form a holonomic sequence. Conversely, for a given holonomic sequence the function defined by the above sum (i.e., its generating function) is holonomic (this is true in the sense of formal power series, even if the sum has a zero radius of convergence). Note that given a holonomic differential equation for a holonomic function f(x), it is straightforward to construct a holonomic recurrence for the coefficients of its power series expansion. For a recent overview of this holonomic machinery and further literature, we refer to [CitationKauers and Paule 11].
Since cyclotomic sums are holonomic sequences with respect to n and the iterated integrals we consider are holonomic functions with respect to x, we can use holonomic closure properties to derive integral representations of Pochhammer sums: Given a Pochhammer sum where g(n) is a cyclotomic sum. We proceed as proposed in on page 3: define and try to find an iterated integral representation of using the following steps:
Compute a holonomic recurrence equation for
Compute a holonomic differential equation for
Compute initial values for the differential equation.
Solve the differential equation to get a closed form representation for
This procedure is implemented in the packages HarmonicSums and can be called by
We will succeed in finding a closed form representation for f(x) in terms of iterated integrals, if we can find a full solution set of the derived differential equation. The command ComputeGeneratingFunction internally uses the differential solver implemented in HarmonicSums, which finds all solutions of holonomic differential equations that can be expressed in terms of iterated integrals over hyperexponential alphabets [CitationAblinger 16, CitationAblinger et al. 14, CitationBronstein 92, CitationPetkovšek 92, CitationHendriks and Singer 99]; these solutions are called d’Alembertian solutions [CitationAbramov and Petkovšek 94], in addition for differential equations of order two it finds all solutions that are Liouvillian [CitationAblinger 17a, CitationKovacic 86, CitationHendriks and Singer 99].
If we succeed in finding a closed form representation for f(x) in terms of iterated integrals, we proceed with Step 2 and Step 3 of the proposed strategy. Hence, we send and try to transform these iterated integrals to expression in terms of cyclotomic polylogarithms and finally we use relations between cyclotomic polylogarithms at one to derive an expression in terms of known constants.
The Pochhammer sum (2–3) (2–3) will deal as a representative example to illustrate all three different methods that are presented in this article. First, we work out the sum using the method presented above.
Example 1.
We consider the sum Equation(2–3)(2–3) (2–3) and start to derive a recurrence for we find:
Using the closure properties of holonomic functions, we find the following differential equation satisfied by
We can solve this differential equation for example using the differential equation solver implemented in HarmonicSums:
By checking initial values we find (2–4) (2–4)
At this point we send and use the command SpecialGLToH in HarmonicSums to derive an expression in terms of cyclotomic polylogarithms (compare [CitationAblinger 17, Section 3]). This leads to
Finally, we can use relations between cyclotomic polylogarithms at one (compare [Ablinger Citation17, Section 4]) to derive (2–5) (2–5)
Note that in the last step of this example we are actually only dealing with harmonic polylogarithms (see [CitationRemiddi and Vermaseren 00]).
Let us now list several identities that could be computed using this method:
Several formulas that can be found in [CitationLiu and Wang 19] can be also discovered and proved using the described method. Here, we are going to list some of them:
Note that this method can also be used to compute integral representations of sums of the form
Here, we find and sending for instance we get:
Finally, we consider (2–6) (2–6) proceeding as proposed, we find a differential equation of order 16:
Solving this differential equation is possible but takes quite some time, so this indicates that we might look for more feasible methods to find generating function representations for Pochhammer sums of that kind. In the following sections, we will introduce rewrite rules, which will allow to compute generating function representations of Pochhammer sums without having to solve differential equations.
3. Using rewrite rules to derive generating functions
In this section, we are going to state rewrite rules which will allow us to find integral representations of the generating functions of Pochhammer sums without having to solve differential equations. We will summarize these rewrite rules in the following lemmas. We start with the base cases where there is no inner sum present:
Lemma 2.
Let be a field of characteristic0. Then, the following identities hold in the ring of formal power series with and (3–1) (3–1) (3–2) (3–2) (3–3) (3–3) (3–4) (3–4)
In case an inner sum is present, we will make use of the following three lemmas.
Lemma 3.
Let be a field of characteristic0 and let Then the following identity holds in the ring of formal power series withd < 0: (3–5) (3–5)
Proof.
Both sides satisfy the following initial value problem for which has a unique solution near : □
Lemma 4.
Let be a field of characteristic0 and let Then the following identity holds in the ring of formal power series with: (3–6) (3–6)
Proof.
Both sides satisfy the following initial value problem for which has a unique solution near : □
Lemma 5.
Let be a field of characteristic 0 and let Then the following identity holds in the ring of formal power series with and : (3–7) (3–7)
Proof.
Both sides satisfy the following initial value problem for which has a unique solution near :
□
Note that formulas related to the previous lemmas concerning binomial sums can be found in [CitationAblinger et al. 14].
Let us now, for the second time, consider Equation(2–3)(2–3) (2–3) and illustrate how the previous lemmas can be used as rewrite rules to find integral representations of Pochhammer sums.
Example 6.
We again look for a closed form representation in terms of iterated integrals of
We start by using Lemma 5 twice:
Now we apply Lemma 3 followed by applying Equation(3–4)(3–4) (3–4) and Equation(3–1)(3–1) (3–1)
At this point, we rewrite the expression in terms of iterated integrals (this can be done by hand or by using the command GLIntegrate of HarmonicSums) and arrive again at Equation(2–4)(2–4) (2–4) and hence we can proceed as in Example 1 to arrive at
Note that this method is implemented in the package HarmonicSums using the command PochhammerSumToGL. Calling will immediately (after regrouping) give Equation(2–4)(2–4) (2–4) .
Reconsidering Equation(2–6)(2–6) (2–6) we find
Note that all the identities listed in Section 2 can also be computed using rewrite rules. But using these rewrite rules turns out to be much more efficient. We are now going to list several additional identities that could be computed with the help of this command: (3–8) (3–8) (3–9) (3–9) (3–10) (3–10) (3–11) (3–11) (3–12) (3–12) (3–13) (3–13)
To conclude this section we consider the sum
We find that it equals (3–14) (3–14)
Here, we fail to transform the iterated integrals in terms of cyclotomic polylogarithms; however, since the integrals are simple enough, we are able to perform the integrals in Equation(3–14)(3–14) (3–14) for example by using Mathematica and find the result
Another example where we fail to transform the iterated integrals in terms of cyclotomic polylogarithms but still can do the integrals is
In the following section, we will consider a subclass of Pochhammer sums, for which we will always be able to derive a representation in terms of cyclotomic polylogarithms.
4. Using rewrite rules to directly derive generating functions in terms of cyclotomic polylogarithms
In this section, we will deal with a subclass of the Pochhammer sums, namely we restrict the inner sum to be a multiple harmonic sum and we set with and a = 1 in (1–3), i.e., we are considering sums of the form (4–1) (4–1) where and with Considering a Pochhammer sum in this subclass we could again use the rewrite rules presented in Section 3 to find an integral representation, however we can also use the following lemmas. These new rewrite rules will directly lead to cyclotomic polylogarithms. We again start with the base cases where no inner sum is present (compare Lemma 2):
Lemma 7.
Let be a field of characteristic0. Then, the following identities hold in the ring of formal power series with and (4–2) (4–2) (4–3) (4–3)
In the cases where there is an inner multiple harmonic sum present we can refine the Lemmas 3, 4, and 5 and get the following result.
Lemma 8.
Let be a field of characteristic 0 and let Then the following identities hold in the ring of formal power series with and a multiple harmonic sum: (4–4) (4–4) (4–5) (4–5) (4–6) (4–6)
Here, we use the abbreviations and
Proof.
For all these equalities, it is possible to find an initial value problem, which has a unique solution near x= 0 and is satisfied by both sides of the respective equation. □
Note that the polynomials arising in the left hand sides of the equations in Lemmas 7 and Citation8 are of the form ti or for hence integrating over these integrands will lead to cyclotomic polylogarithms. Therefore, the Pochhammer sums of the form Equation(4–1)(4–1) (4–1) will be expressible in terms of cyclotomic polylogarithms, and we can state the following structural theorem.
Theorem 9.
Any sum of the form (4–7) (4–7) where and can be expressed in terms of cyclotomic polylogarithms at one.
Let us now, for the third time, consider Equation(2–3)(2–3) (2–3) and illustrate how the previous lemmas can be used as rewrite rules to directly find a representation in terms of cyclotomic polylogarithms.
Example 10.
We seek a closed form representation in terms of cyclotomic polylogarithms of so we use Equation(4–6)(4–6) (4–6) twice:
Now, we apply Equation(4–4)(4–4) (4–4) followed by applying Equation(4–3)(4–3) (4–3) and Equation(3–1)(3–1) (3–1) :
Now, we can send and rewrite this expression directly in terms of cyclotomic harmonic polylogarithms (again this can be done by hand or by using the command GLIntegrate of HarmonicSums) and arrive again at (4–8) (4–8)
Finally, we can again use relations between cyclotomic polylogarithms at one to derive Equation(2–5)(2–5) (2–5) .
Note that this is implemented in the command PochhammerSumToH, so calling will immediately give (after pretty printing)
Sending will give Equation(4–8)(4–8) (4–8) .
To conclude we are going to list several identities that could be computed with the help of this command (note that these identities could have also be computed using the methods presented in the previous sections): (4–9) (4–9) (4–10) (4–10) (4–11) (4–11) (4–12) (4–12) (4–13) (4–13) (4–14) (4–14)
Acknowledgments
The author would like to thank C. Schneider for useful discussions.
Additional information
Funding
Notes
1 The package HarmonicSums (Version 1.0 16/05/19) together with a Mathematica notebook containing a list of illustrative examples can be downloaded at http://www.risc.jku.at/research/combinat/software/HarmonicSums.
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