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arXiv · 1607.01978

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

Shuji Yamamoto
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Abstract

Kaneko and Tsumura introduced a new kind of multiple zeta functions $\eta(k_1,\ldots,k_r;s_1,\ldots,s_r)$. This is an analytic function of complex variables $s_1,\ldots,s_r$, while $k_1,\ldots,k_r$ are non-positive integer parameters. In this paper, we first extend this function to an analytic function $\eta(s'_1,\ldots,s'_r;s_1,\ldots,s_r)$ 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 $\zeta(k_1,\ldots,k_r)$ of Euler-Zagier type.

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Shuji Yamamoto. 2016-07-07. Multiple zeta functions of Kaneko-Tsumura type and their values at positive integers. https://arxiv.org/abs/1607.01978

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