Analytic properties of generalized Mordell-Tornheim type of multiple zeta-functions and $L$-functions

Abstract

Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced. In this paper, similarly we consider generalized multiple zeta-functions and $L$-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases. We prove the meromorphic continuation to the multi-dimensional complex space, and give the results on possible singularities.