Derivatives and Special Values of Higher-Order Tornheim Zeta Functions

Abstract

We study analytic properties of the higher-order Tornheim zeta function, defined by a certain $n$-fold series ($n\geq 2$) in $n+1$ complex variables. In particular, we consider the function $\omega_{n+1}(s)$, obtained by setting all variables equal to $s$. Using a free-parameter method due to Crandall, we first give an alternative proof of the trivial zeros of $\omega_{n+1}(s)$ and evaluate $\omega_{n+1}(0)$. Our main result, however, is the evaluation of $\omega_{n+1}'(0)$ for any $n\geq 2$. This is again achieved by using Crandall's method, and it generalizes recent results in the cases $n=2, 3$. Properties of Bernoulli numbers and of higher-order Bernoulli numbers and polynomials play an important role throughout this paper.