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

Abstract

Kaneko and Tsumura introduced a new kind of multiple zeta functions $\eta (k_1,\dots ,k_r;s_1,\dots ,s_r)$. This is an analytic function of complex variables $s_1,\dots ,s_r$, while $k_1,\dots ,k_r$ are non-positive integer parameters. In this paper, we first extend this function to an analytic function $\eta (s_1^{\prime },\dots ,s_r^{\prime };s_1,\dots ,s_r)$ of $2r$ complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these $\eta$-values and the multiple zeta values $\zeta (k_1,\dots ,k_r)$ of Euler-Zagier type.

13 pages; the proof of Proposition 2.5 has been clarified by introducing Lemma 2.4; to appear in Kyushu Journal of Mathematics

References in corpus (1)

• Mixed Tate Motives and Multiple Zeta Values

Cited by in corpus (1)

• On Arakawa-Kaneko zeta-functions associated with $G{L}_2(\mathbb{C})$ and their functional relations