Sum Formula of Multiple Hurwitz-Zeta Values

TL;DR

This paper generalizes formulas for sums of multiple Hurwitz-zeta values with even arguments to arbitrary depth, expanding previous results limited to small depths using symmetric function theory.

Contribution

It extends Shen and Cai's formulas for sums of multiple Hurwitz-zeta values to any depth through symmetric function theory.

Findings

Generalized formulas for T(2n,d) to arbitrary depth.

Connected multiple Hurwitz-zeta sums with symmetric functions.

Built on Hoffman’s symmetric function theory.

Abstract

Let s_1,...,s_d be d positive integers and consider the multiple Hurwitz-zeta value zeta(s_1,...,s_d;-1/2,...,-1/2)/2^w where w=s_1+...+s_d is called the weight. For d<n+1, let T(2n,d) be the sum of all these values with even arguments whose weight is 2n and whose depth is d. Recently Shen and Cai gave formulas for T(2n,d) for d<6 in terms of t(2n), t(2)t(2n-2) and t(4)t(2n-4). In this short note we generalize Shen-Cai's results to arbitrary depth by using the theory of symmetric functions established by Hoffman.