Abstract
We study two functions ?k(s) and ?k1?kr(s) introduced by Arakawa and Kaneko [3] and relate them with (finite) multiple zeta values and multiple zeta star values using elementary methods. In particular, we give an alternative proof of a result of Ohno [14].
Highlights
INTRODUCTIONOhno [14] applied his generalization of the duality and sum formulas for multiple zeta values to the result obtained by Arakawa and Taneko for ξk(n) in order to express ξk(n) for positive integers n in terms of so-called multiple zeta star values or non-strict multiple zeta values
Let Lik1,...,kr (z) denote the multiple polylogarithm function defined by Lik1,...,kr (z) = n1>n2>···>nr ≥1zn1 nk11 nk22 . . nkr r, with k1 ∈ N \ {1} and ki ∈ N = {1, 2, . . . }, 2 ≤ i ≤ r, and |z| ≤ 1
For z = 1 the multiple polylogarithm function Lik1,...,kr (1) = ζ(k1, . . . , kr) simplifies to a multiple zeta value, sometimes called multiple zeta function, where ζ(k1, . . . , kr) and ζN (k1, . . . , kr) denote the multiple zeta value defined by ζ
Summary
Ohno [14] applied his generalization of the duality and sum formulas for multiple zeta values to the result obtained by Arakawa and Taneko for ξk(n) in order to express ξk(n) for positive integers n in terms of so-called multiple zeta star values or non-strict multiple zeta values. Posed several questions concerning the function ξk1,...,kr (s) They asked for evaluations of ξk1,...,kr (s) to multiple zeta values, for k1, . Kr ∈ N and positive integers n by providing evaluations of the function ξk1,...,kr (n) to multiple zeta (star) values. For the evaluation of the general case ξk1,...,kr (n) we use a finite version (see for example [9]) of the well known stuffle identity for multiple zeta values [7]. The main results of this work concerning the evaluation of ξk1,...,kr (n) into multiple zeta (star) values will be stated in Theorems 2, 3, 4
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