On the sum relation of multiple Hurwitz zeta functions

Abstract

In this paper we shall define the special-valued multiple Hurwitz zeta functions, namely the multiple t-values t(α) and define similarly the multiple star t-values as t^⋆(α). Then we consider the sum of all such multiple (star) t-values of fixed depth and weight with even argument and prove that such a sum can be evaluated when the evaluations of t({2m}ⁿ) and t^⋆(f{m}ⁿ) are clear. We give the evaluations of them in terms of the classical Euler numbers through their generating functions.

Mathematics Subject Classifcation (2010): 40B05, 11M35, 05A15.

Keywords: Hurwitz zeta function, multiple zeta value, sum formula, generating functions, infinite series and products