On the sum relation of multiple Hurwitz zeta functions

TL;DR

This paper introduces special multiple Hurwitz zeta functions called multiple t-values and star t-values, and evaluates sums of these functions at even arguments in terms of Euler numbers, expanding understanding of their structure.

Contribution

It defines multiple t-values and star t-values, and provides explicit evaluations of sums of these functions at even arguments in terms of Euler numbers.

Findings

Sum of fixed-depth and weight multiple t-values can be explicitly evaluated.

Evaluations are expressed via classical Euler numbers.

Results extend the understanding of multiple Hurwitz zeta functions.

Abstract

In this paper we shall define a special-valued multiple Hurwitz zeta functions, namely the multiple $t$-values $t(\boldsymbol{\alpha})$ and define similarly the multiple star $t$-values as $t^{\star}(\boldsymbol{\alpha})$. 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 $t^{\star}(\{2m\}^n)$ are clear. We give the evaluations of them in terms of the classical Euler numbers through their generating functions.