Some Weighted Sum Formulas for Multiple Zeta, Hurwitz Zeta, and Alternating Multiple Zeta Values

He, Yuan; Chen, Zhuoyu

doi:https://doi.org/10.1155/2021/6672532

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Volume 2021 | Article ID 6672532 | https://doi.org/10.1155/2021/6672532

Some Weighted Sum Formulas for Multiple Zeta, Hurwitz Zeta, and Alternating Multiple Zeta Values

Yuan He¹and Zhuoyu Chen²

Academic Editor: Li Guo

Received30 Dec 2020

Accepted30 Apr 2021

Published13 May 2021

Abstract

We perform a further investigation for the multiple zeta values and their variations and generalizations in this paper. By making use of the method of the generating functions and some connections between the higher-order trigonometric functions and the Lerch zeta function, we explicitly evaluate some weighted sums of the multiple zeta, Hurwitz zeta, and alternating multiple zeta values in terms of the Bernoulli and Euler polynomials and numbers. It turns out that various known results are deduced as special cases.

1. Introduction

Let be all positive integers with . The multiple zeta values of the depth (sometimes called Euler sums) are usually defined by the following series:

These series appear in a variety of fields in mathematics, such as number theory, combinatorics, knot theory, the theory of mixed Tate motives, and quantum field theory; see, for example [1–4].

It is well known that one of the central problems on the multiple zeta values is to determine all the possible -linear relations among them. Goncharov’s ([5], Conjecture 4.2) conjecture implies that it suffices to study the relations among the multiple zeta values of the same weight . Perhaps the earliest result in this direction is Euler’s [6] sum formula, namely,where is the Riemann zeta function. In fact, there exists a general form of (2), as follows:which is referred to as the “sum conjecture” in [7], and was proved by Granville [8] and Zagier [9] independently around the year 1995. Some new relations for the multiple zeta values and their different variations and generalizations have been found in recent years. For example, Ohno and Zudilin [10] in 2008 proved a weighted form of Euler’s sum formula:

It becomes obvious from (2) that Ohno and Zudilin’s weighed sum formula can be rewritten as

Based on the equivalence of (4) and (5), Guo and Xie [11] in 2009 extended the weighted sum formula of Ohno and Zudilin to arbitrary depth and discovered that for positive integers with ,where for . For another weighted sum formulas of the multiple zeta values, one is referred to [12], where the weight coefficients are given by (symmetric) polynomials of the arguments. On the contrary, let be positive integers with and , and let be the sums of all multiple zeta values of the depth and the weight given by

Gangl et al. [13] in 2004 proved that

Shen and Cai [14] in 2012 obtained the formulas

After that, Hoffman [15] used the theory of symmetric functions to establish the general sum formulawhere is the -th Bernoulli number. Furthermore, Zhao [16] used the ideas developed in [15] to evaluate the sums of all multiple Hurwitz zeta values of the depth and the weight in terms of the Euler numbers. Moreover, Zhao [17] used the theory of symmetric functions to consider the more complicated alternating multiple zeta values and depicted that the sums of all alternating multiple zeta values of the depth and the weight can be evaluated in terms of the Riemann zeta function and the Euler numbers. More recently, Chen et al. [18] used the method of the generating functions to express by constructing a combinatorial identity of products of the multiple zeta values and the so-called multiple zeta-star values at the repetitions of , and then used Muneta’s [19] and Nakamura’s [20] results to reobtain Hoffman’s sum formula (10) and confirm Genčev’s ([21], Conjecture 4.1) conjecture on the evaluation of .

Subsequently, Shen and Jia [22] extended the sums of the multiple Hurwitz zeta values previously considered in [23] and showed that the sums of all multiple Hurwitz zeta values of the depth and the weight can be expressed by a combinatorial identity of products of the multiple Hurwitz zeta values and the so-called multiple Hurwitz zeta-star values at the repetitions of . In particular, Shen and Jia [22] obtained Zhao’s [16] sum formula with a slight different notation and evaluated the sums of all multiple Hurwitz zeta values of the depth and the weight in terms of the Euler numbers.

Motivated and inspired by the work of the above authors, we explicitly evaluate some weighted sums of the multiple zeta, Hurwitz zeta, and alternating multiple zeta values in terms of the Bernoulli and Euler polynomials and numbers by using the method of the generating functions and some connections between the higher-order trigonometric functions and the Lerch zeta function established by the first author [24]. The results presented here are the corresponding extensions of various known sum formulas.

This paper is organized as follows: In Section 2, we give several weighted sum formulas for the multiple zeta values, some of which generalize the sum formulas (10) and (11), and improve Eie and Ong’s [25] weighted sum formulas. In Section 3, we present some similar weighted sum formulas for the multiple Hurwitz zeta values and deduce Shen and Jia’s [22] sum formulas as special cases. Section 4 concentrates on the features that have contributed to the weighted sum formulas for the alternating multiple zeta values, and it then turns out that Zhao’s [17] sum formula is obtained in a rather simple way.

2. Sum Formulas for Multiple Zeta Values

For convenience, in the following, we always denote by the square root of such that , the Stirling numbers of the first kind, the Bernoulli polynomials, and the Euler polynomials. It is clear that taking and in the Bernoulli and Euler polynomials gives the Bernoulli numbers and the Euler numbers , respectively. We refer the reader to two standard books [26, 27] on basic properties for these special sequences and polynomials. We also write as the coefficients of in for nonnegative integer . We now state our first result as follows.

Theorem 1. Let be positive integers. Thenwhere, and in what follows, is the linear combination of the Stirling numbers of the first kind satisfying that for positive integer and nonnegative integer ,and is the linear combination of the Bernoulli polynomials given for positive integer and nonnegative integers by

Proof. Recall that Euler’s infinite product formula of the sine function iswhich holds true for arbitrary complex number (see [26], p. 75 or [28], pp. 12–18). The binomial series asserts that for complex number (see [27], p. 37),where are the binomial coefficients given for nonnegative integer bySo from (15) and (16), we discover that for complex number with ,Comparing the coefficients of on both sides of (18), it then follows that for complex number with ,We now evaluate the right-hand side of (19) from another view. Let be positive integers, and let be a real function defined on positive integer . If , then (see [24], Theorem 3.2)where is the Lerch zeta function given for real number , negative integer or zero, and complex number byNote that the series is an entire function of when is not an integer. Obviously, replacing by and by in (20) gives that for real number ,It follows from (19) and (22) thatIt is easily seen from the Taylor series expansion for the complex exponential function and the familiar binomial theorem thatSince for nonnegative integer (see [29], Theorem 12.13),where is the Hurwitz zeta function given for real number and complex number byso by applying (25) to (24), we arrive atNow (12) follows from (23) and (27). This completes the proof of Theorem 1.

Corollary 1. Let be a positive integer. Then

Proof. Since for nonnegative integer (see [26], p. 805),so by taking in Theorem 1, in view of , we get the desired result.
Corollary 1 is usually attributed to Hoffman ([15], Corollary 2) and was previously obtained by Aoki, Kombu, and Ohno ([30], Equation (4.6)), who stated it in the language of the multiple zeta-star values. We are in a good position to use Theorem 1 to yield the following result.

Theorem 2. Let be positive integers with . Thenwhere is a rational number given for positive integer and nonnegative integer by

Proof. Clearly, for real or complex parameter ,Just as a polynomial function of in the order of ascending power is divided by another polynomial function of in the order of ascending power, we discover thatApplying (33) to the right-hand side of (32), it then follows from (18) thatNoticing that from (19) and Theorem 1, we haveWe now evaluate the coefficients of in the infinite product of the right-hand side of (34). Let be all complex numbers determined by the factorization of the polynomial function over the complex number field satisfying thatThe famous Vieta’s theorem implies thatSo from (36), (37), and the remarkable formula see [31], Equation (36), or [7], Corollary 2.3,where denotes the repetitions of , we get thatInserting (35) and (39) into (34), it follows thatThus (30) follows immediately after making -times derivative with respect to and then taking on both sides of (40). This concludes the proof of Theorem 2.
It is easy to check that taking in Theorem 2 and then applying (29) and leads to Hoffman’s formula (10). It is worth noticing that the formula (40) can also be regarded as an extension of Theorem 1. In a similar consideration to Theorem 2, we have the following result.

Theorem 3. Let be the positive integers with . Thenwhere is a rational number given for positive integer and nonnegative integer byand are the higher-order Bernoulli polynomials defined by the generating function (see [32]):

Proof. We know from (16) and (33) thatIf we replace by in (15), then we haveIt follows from (15), (35), and (45) thatOn the contrary, from (36), (37), and the well known formula (see [31], Equation (37)),we obtain thatInserting (46) and (48) into (44), we haveTherefore, we get (41) and finish the proof of Theorem 3 when making -times derivative with respect to and then taking on both sides of (49).
It is trivial to check that Genčev’s conjecture (11) holds true when taking in Theorem 3. We here remark that Theorems 2 and 3, as well as the generalizations of the sum formulas (10) and (11), are the improvements of the results recently obtained by Eie and Ong ([25], Theorems 2.1 and 2.2), where they expressed the left-hand side of (30) by a combinatorial identity involving the higher derivative of one function and the sums of products of Bernoulli polynomials, and the left-hand side of (41) by a combinatorial identity involving the higher derivative of another function and the sums of products of Bernoulli polynomials.

3. Sum Formulas for Multiple Hurwitz Zeta Values

In this section, we shall study the multiple Hurwitz zeta values defined by the series (see [23, 33])where are all positive integers with and present some weighted sum formulas for them. The results showed here are very analogous to the ones in Section 2.

Theorem 4. Let be positive integers. Thenwhere is the linear combination of the Euler polynomials given for positive integer and nonnegative integers by

Proof. It is well known that Euler’s infinite product formula of the cosine function iswhich holds true for arbitrary complex number (see [26], p. 75 or [28], pp. 12–18). Hence, we obtain from (16) and (53) that for complex number with ,If we replace by in (22), we get that for real number ,which can be converted by the expression of the higher-order secant function stated in ([24], Theorem 3.2). It then follows thatObserve thatSince for nonnegative integer (see [34], Corollary 3),where is the alternating Hurwitz zeta function (also called Hurwitz Euler-eta function) given for real number and complex number byso by applying (58) to (57), we find thatInserting (60) into (56), we haveThus, we complete the proof of Theorem 4 by equating (54) and (61).
In what follows, we denote by the sums of all multiple Hurwitz zeta values of the depth and the weight given for positive integers with and byIt follows that we state the following result.

Corollary 2. Let be a positive integer. Then

Proof. By taking in Theorem 4, in view of and for nonnegative integer , the desired result follows immediately.
We mention that Corollary 2 was obtained by Shen and Jia ([22], p. 265) in the language of the multiple Hurwitz zeta-star values. We now give another weighted sum formula for the multiple Hurwitz zeta values as follows.

Theorem 5. Let be positive integers with . Thenwhere is a rational number given for positive integer and nonnegative integer by

Proof. We obtain from (33) and (54) thatNoticing that from (61), we haveand from (36), (37), and the formula (see [33] or [22], p. 265),we deduce thatInserting (67) and (69) into (66), we get thatThus the desired result follows by making -times derivative with respect to and then taking on both sides of (70). This completes the proof of Theorem 5.

Corollary 3. Let be positive integers with . Then

Proof. Taking in Theorem 5, in light of and for nonnegative integer , we get the desired result.
Corollary 3 is a general form of Shen and Cai’s [23] results for the cases in and was also found by Shen and Jia ([22], p. 265). For an equivalent version of Corollary 3, see ([16], Theorem 1.3) for details. Similarly, we state the following result.

Theorem 6. Let be positive integers with . Thenwhere is a rational number given for positive integer and nonnegative integer byand are the higher-order Euler polynomials defined by the generating function (see [35])

Proof. With the help of (16) and (33), we discover thatBy replacing by in (53), we find thatIt follows from (53), (67), and (76) thatIf we apply (36), (37), and the formula (see [22], p. 265),to the second infinite product in the right-hand side of (75), we haveInserting (77) and (79) into (75), it then follows thatThus we prove (72) immediately by making -times derivative with respect to and then taking on both sides of (80). This concludes the proof of Theorem 6.
In particular, we discover Shen and Jia’s ([22], p. 266) result as follows.

Corollary 4. Let be positive integers with . Then

Proof. Since for nonnegative integer , so by taking in Theorem 6, the desired result follows from and for nonnegative integer .

4. Sum Formulas for Alternating Multiple Zeta Values

As shown in [36, 37], the alternating multiple zeta values of depth (also called alternating Euler sums) are defined by the serieswhere are all positive integers and with . We next use the methods and results described in the second and third sections to obtain some weighted sum formulas for the alternating multiple zeta values.

Theorem 7. Let be positive integers. Then

Proof. By substituting for in (15) and for in (53), we find thatIt follows from (16) and (84) thatTrivially, the left-hand side of (85) can be written asIt is easy to see that from the generating function of the higher-order Euler polynomials, we haveand from (22) and (27), we can writeInserting (87) and (88) into (86), in view of (85), we get (83) immediately and thus finish the proof of Theorem 7.
If we denote by the sums of all alternating multiple zeta values of the depth and the weight given for positive integers with and bythen we obtain Zhao’s ([17], Equation (7)) result as follows.

Corollary 5. Let be a positive integer. Then

Proof. By taking in Theorem 7 and then using , the desired result follows from (29) and for nonnegative integer .

Theorem 8. Let be positive integers with . Thenwhere is a rational number given for positive integer and nonnegative integer byand denotes the greatest integer less than or equal to real number .

Proof. It is easily seen from (33) and (85) that the following identities are complete:Noticing that from (85) and Theorem 7, we haveand from (36), (37), and the formula (see [17], Lemma 2.1),we obtainInserting (94) and (96) into (93), it then follows thatTherefore (91) follows immediately when making -times derivative with respect to and then taking on both sides of (97). This concludes the proof of Theorem 8.

Corollary 6. Let be positive integers with . Then

Proof. By taking in Theorem 8, in light of , we get the desired result after using (29) and for nonnegative integer .
It is well known that the values of the Riemann zeta function at even positive integers can be computed by the Bernoulli numbers (see [29], Theorem 12.17):Hence, by applying (99) to Corollary 6, one can easily get Zhao’s ([17], Theorem 1.3) sum formula:where for nonnegative integer .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11771351).

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Copyright © 2021 Yuan He and Zhuoyu Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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