Algorithms for Some Euler-Type Identities for Multiple Zeta Values
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.
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 , for , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is ; that is,
In , Gangl et al. proved the following identities:
Recently, using harmonic shuffle relations, Shen and Cai proved the following results in :
In , 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  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 , 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 haswhere 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.
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 formula (14), we get that
This work is supported by the National Natural Science Foundation of China (1127208).
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