Alternating multiple zeta values, and explicit formulas of some Euler–Apéry-type series

On some explicit evaluations of multiple zeta-star values

In this paper, we give some explicit evaluations of multiple zeta-star values which are rational multiple of powers of π. 1 Main Results The multiple zeta value (MZV) is defined by the convergent series ζ(k1, k2, . . . , kn) := ∑ m1>m2>···>mn>0 1 m1 1 m k2 2 · · ·m kn n , where k1, k2, . . . , kn are positive integers and k1 ≥ 2. The integers k = k1+k2+ · · ·+kn and n are called weight and dept...

Shuffle Product Formulas of Multiple Zeta Values

Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two multiple zeta values, some restricted shuffle product formulas of the product of two multiple zeta values, and a restricted shuffle product formula of the product of n multiple zeta values.

Algorithms for Some Euler-Type Identities for Multiple Zeta Values

. . . , s k are positive integers with s 1 > 1. For k ≤ n, let E(2n, k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. The well-known result E(2n, 2) = 3ζ(2n)/4was extended to E(2n, 3) and E(2n, 4) by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers E(2n, k) and then ...

New Families of Weighted Sum Formulas for Multiple Zeta Values

Evaluation formulas for Tornheim's type of alternating double series

In this paper, we give some evaluation formulas for Tornheim’s type of alternating series by an elementary and combinatorial calculation of the uniformly convergent series. Indeed, we list several formulas for them by means of Riemann’s zeta values at positive integers.