MULTIPLE ZETA FUNCTIONS OF KANEKO-TSUMURA TYPE AND THEIR VALUES AT POSITIVE INTEGERS

Kaneko and Tsumura introduced a new kind of multiple zeta function η(k1, . . . , kr ; s1, . . . , sr). This is an analytic function of complex variables s1, . . . , sr, while k1, . . . , kr are nonpositive integer parameters. In this paper, we first extend this function to an analytic function η(s'1, . . . , s'r ; s1, . . . , sr) 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, . . . , kr) of Euler-Zagier type.