A combinatorial identity of multiple zeta values with even arguments ∗

Let ζ(s1, s2, · · · , sk;α) be the multiple Hurwitz zeta function. Given two positive integers k and n with k 6 n, let E(2n, k;α) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. In this note we present some generating series for the numbers E(2n, k;α).

There are various relations among multiple zeta values.One of the well-known Q-linear relations among multiple zeta values is the sum formula ( see [2,3,4,7,9]), which states that In [6], M. Igarashi proved a generalization of the sum formula.Here we quote Igarashi's result in a slightly different form.
Theorem 1. ( [6]) Let α be a complex number with Re α > 0 and let n, k be positive integers such that n k + 1.Then the following identity holds.
th derivative of a function with respect to its variable x, and (a) N denotes the Pochhammer symbol.
Given two positive integers n and k with n k, let E(2n, k; α) be the sum of all the multiple zeta values of even-integer arguments having weight 2n and depth k, i.e., (The notation E(2n, k; α) is similar to that in [5,12].In [5], Hoffman defined a number E(2n, k) which is equivalent to the number E(2n, k; 1) in our notation, and in [12], the number E(2n, k; 1/2)/2 2n in our notation is denoted by T (2n, k).) The result of Gangl, Kaneko and Zagier [1] that was extended by Shen and Cai [11] to Applying the theory of symmetric functions, Hoffman [5] and Zhao [12], respectively, established the generating functions for the numbers E(2n, k; 1) and T (2n, k), respectively.Hoffman [5] proved that And Zhao [12] proved that Based on these generating functions, some formulas for the numbers E(2n, k; 1) and T (2n, k) for arbitrary n k have been obtained.For example, where B 2j is the 2jth Bernoulli number.
In this note we use a method introduced by Granville [2] to establish an algorithm similar to that in Theorem 1 for the calculation of the numbers E(2n, k; α).We prove that the calculation of E(2n, k; α) involves the Euler Γ-function and the direct formulas for E(2n, k; 1) and E(2n, k; 1/2) can be deduced from the Bessel functions of the first kind.The main results of this paper are the following theorems.
Theorem 2. Let α be a complex number with α ∈ C\{0, −1, −2, • • • }.Let N 0 denote an integer and let (a Then for positive integers n and k (suppose n k), we have Theorem 3. Given an integer N 0 and a complex number where Γ denotes Euler's Gamma function.
the electronic journal of combinatorics 21(2) (2014), #P2.27 Corollary 4. Theorem 2 and Theorem 3 yield that: , where [ • ] (k) denotes the kth derivative of a function with respect to its variable x and Γ denotes the Gamma function.
Corollary 5.For positive integers n and k (suppose n k ), we have Corollary 6.The direct formulas for E(2n, k; 1) and E(2n, k; 1 2 ) are that: 2 Some Lemmas This paper uses some properties of Euler's Gamma function Γ(s) and the Bessel function J p (x) of the first kind.Here we list some needed Lemmas.There are several equivalent forms of the Gamma function Γ(s) developed by Euler, one by Weierstrass: where γ denotes the Euler-Mascheroni constant defined by It is easy to deduce from Weierstrass' definition of Γ(s) the following lemma.

Lemma 8. [10]
The following identities hold for s ∈ C: We turn to introduce some properties of the Bessel function of the first kind with a half-integer index.The Bessel function J p (x) is said to be of a half integer index if p = k ± 1 2 with k being an integer.It is well known that the Bessel function of the first kind with a half integer index can be represented by elementary functions.
Proof.The Bessel function J p (x) is defined by the series where the radius of convergence of the series is +∞.It follows from ( 7) that (using the well-known identity Γ(n + 1/2) = √ π(2n)!/(n!2 2n ), ∀n 0) On the other hand, by Taylor's expansion of cos √ x, we have Observing the right-hand sides of ( 8) and ( 9), the result follows immediately.
Lemma 10.Let k 0 be an integer and let x > 0. Then the Bessel function J k+1/2 (x) is represented by the electronic journal of combinatorics 21(2) (2014), #P2.27 Proof.We omit the detailed steps.It is shown that ( [8]) the Bessel function J k+1/2 (x) satisfies the relation It follows from ( 11) that and so on.Then using induction on k we can prove that ( 10) is equivalent to (11).
Lemma 11.Let k 1 be an integer.Then by Lemma 9 and Lemma 10, we have 3 Proofs Proof of Theorem 2. The left side of (2) is The second sum in (13) is the coefficient of x 2n in the formal power series It follows that the coefficient of x 2n above is Hence the sum (13) is Now we take each n j = N in turn.Then the sum (15) becomes that where the series {P (N, i)} ∞ i=0 and {Q(N, i)} ∞ i=0 are defined in the following manner.Define P (N, 0) = Q(N, 0) 1, and Q(N, j) = 0, ∀j > N ; If j > 1 then define P (N, j − 1) to be .
In other words, the series {P (N, i)} ∞ i=0 and {Q(N, i)} ∞ i=0 are defined by the following generating functions: the electronic journal of combinatorics 21(2) (2014), #P2.27 Now the second sum k j=1 P (N, j − 1)Q(N, k − j) in ( 16) is the coefficients of x k−1 in the power series where a Now on both sides of (18) we put y → N + α.Then we get the required identity in Theorem 3.