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

Kaneko and Tsumura introduced a new kind of multiple zeta functions $η(k_1,\ldots,k_r;s_1,\ldots,s_r)$. 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 $η(s'_1,\ldots,s'_r;s_1,\ldots,s_r)$ of $2r$ complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these $η$-values and the multiple zeta values $ζ(k_1,\ldots,k_r)$ of Euler-Zagier type.