Use of the generating function to generalize the sum formula for quadruple zeta values

TL;DR

This paper proves a generating function identity for quadruple zeta values, leading to new sum formulas and weighted analogues, extending previous results in number theory.

Contribution

It introduces a generating function approach to derive sum formulas for quadruple zeta values, including weighted versions that generalize prior findings.

Findings

Derived a new sum formula for quadruple zeta values

Established weighted analogues of the sum formula

Extended previous results by Guo, Xie, Ong, Eie, and Liaw

Abstract

In the present paper, we prove an identity for the generating function of the quadruple zeta values. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for quadruple zeta values. We also obtain its weighted analogues, which include the formulas for this case proved by Guo and Xie (2009, J. Number Theory 129, 2747--2765) and by Ong, Eie, and Liaw (2013, Int. J. Number Theory 9, 1185--1198).