Multiple zeta functions of Kaneko-Tsumura type and their values at positive integers

2016/07/07 by Yamamoto, Shuji
#11M32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1607.01978

Abstract

Kaneko and Tsumura introduced a new kind of multiple zeta functions η(k₁,…,k_r;s₁,…,s_r). This is an analytic function of complex variables s₁,…,s_r, while k₁,…,k_r are non-positive integer parameters. In this paper, we first extend this function to an analytic function η(s'₁,…,s'_r;s₁,…,s_r) of 2r complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these η-values and the multiple zeta values ζ(k₁,…,k_r) of Euler-Zagier type.

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