Journal of Applied Mathematics

Volume 2013 (2013), Article ID 802791, 7 pages

http://dx.doi.org/10.1155/2013/802791

Research Article

Algorithms for Some Euler-Type Identities for Multiple Zeta Values

Shifeng Ding and Weijun Liu

Department of Mathematics, Central South University, Changsha, Hunan 410083, China

Received 21 December 2012; Accepted 11 January 2013

Academic Editor: C. Conca

Copyright © 2013 Shifeng Ding and Weijun Liu. 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.

Abstract.

Multiple zeta values are the numbers defined by the convergent series , where , , , are positive integers with . For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is . The well-known result was extended to and by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers and then gave a direct formula for for arbitrary . In this paper we apply a technique introduced by Granville to present an algorithm to calculate and prove that the direct formula can also be deduced from Eisenstein's double product.

1. Introduction

The multiple zeta sums,

are also called Euler-Zagier sums, where are positive integers with . Clearly, the Riemann zeta function ,

is the case in (1). The multiple zeta functions have attracted considerable interest in recent years.

For Riemann’s zeta function , Euler proved the following identity:

Recently, some identities similar to (3) have also been established. Given two positive integers and (suppose ), define a number by

Then, for , the value of is known [1–5].

Following [6], for , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is ; that is,

In [7], Gangl et al. proved the following identities:

Recently, using harmonic shuffle relations, Shen and Cai proved the following results in [8]:

In [6], applying the theory of symmetric functions, Hoffman established the generating function for the numbers . He proved that

Based on this generating function, some formulas for for arbitrary are given. For example, Hoffman obtained that

where is the th Bernoulli number.

In this paper we use a technique introduced by Granville [9] to present an elementary recursion algorithm to calculate , we also give some direct formula for for arbitrary . Our algorithm may be of some interest if we note that it is obtained through an elementary analytic method and that the statement of the algorithm is fairly simple.

2. Statements of the Theorems

Theorem 1. Let denote a positive integer. Let , be a series of numbers defined by

Then, for any two positive integers and with , one has

Theorem 2. Given a positive integer , we have

When is not large, we may use the following recursion algorithm to calculate then use Theorem 1 to get the formula for .

Theorem 3. The coefficients , can be calculated recursively by the following formulas:

where , are the numbers defined by

In [6], Hoffman established an interesting result [6, Lemma 1.3] to obtain his formula (10) for . This lemma might be deduced from the theory of Bessel functions. Using the expressions for the Bessel functions of the first kind with a half integer index, we may deduce from the generating function (13) a direct formula for .

Theorem 4. For , one has

To deduce (17) from (16), we only need to write the expression of , respectively, according to whether is odd or even, and use (if is odd) or (if is even) to replace . In the two cases, we will get the expression (17) for . By Theorem 1, we have

which reproduces Hoffman’s formula (10).

3. Proofs of the Theorems

Proof of Theorem 1. The left side of (12) is

The second sum in (19) is the coefficient of in the formal power series

It follows that the coefficient of earlier is

Hence, the sum (19) is

Now, consider the function

We partition into two parts. Let

Then, we have , , for all, and

Consider the sum (22). For , we treat each sum in (22) with respect to as follows:

In the last step, begins with 1 since for .

It follows that the sum (22) becomes that

Clearly, the sum in (27) is the coefficient of in the Cauchy product of

that is, it is the coefficient of in the power series

Therefore, the sum (27) is

The proof is completed.

Remark 5. If we take to be a complex variable, then the series

is absolutely and uniformly convergent for in any compact set in the complex plane; thus, the function

is analytic in the complex plane. Hence, it may be expanded as a Taylor series.

Proof of Theorem 2. First we recall Euler’s classical formula

Similar to Euler’s formula, Eisenstein studied a product of two variables and proved that for the following formula holds (see [10, page 17]):

Let be temporarily fixed. By (34), for we have

Now, let . We get

We write . Or equivalently, let . Then, we get

Proof of Theorem 3. Taking logarithms of both sides of (32), we get that

By Remark 5, the series may be differentiated term-by-term; hence, we have

where we denote

The order of the summation can be changed since the series is dominated by for some positive constant . From (39), we get that

or

Write out the Cauchy product in the right side of (42), then compare the coefficient of on both sides. We get that

Proof of Theorem 4. We now study the the generating function

We may use L’Hospital’s rule to verify that

Now we expand out . We have

By (11) and (13), we have

Consider the function

Clearly, the sum in (47) can be rewritten as

where means the th derivative of a function with respect to .

We denote . Then, we have

and, hence,

which implies that

Finally, from (47) (49) we get that

We may apply Hoffman’s result [6, Lemma 1.3] to get the direct formula for

Here, we use some simple properties of the Bessel functions of the first kind to give its direct expression.

Lemma 6. Let be an integer and let . Then one has

where denotes the Bessel function of the first kind of index .

The Bessel functions with a half-integer index can be represented by elementary functions. The following lemma is well known.

Lemma 7. Let be an integer, and let . Then, one has

From Lemmas 6 and 7, and (53), we get that

This completes the proof of Theorem 4.

4. Examples

The direct formula for can be found from Theorem 4. However, we would like to use Theorem 3 to present some concrete examples to show how to calculate for small . The difficult part of the recursion formula (14) is for to calculate the sum

where we denote and .

It follows from that

Generally, we can use induction on to prove that if for we have gotten some positive integers such that

then the expression for is

Note that if is an even integer, then we have

Similarly, if is an odd integer, then we have

From (58)–(63), we get that

From formula (14), we get that

For , using , , and in formula (12), respectively, we will get identities (6) and (8). Moreover, for , we have

Acknowledgment

This work is supported by the National Natural Science Foundation of China (1127208).

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