A Generating Function to Generalize the Sum Formula for Quadruple Zeta Values

Abstract

In the present paper, we prove an identity for the generating function of the quadruple zeta values with the action of the matrix ring $\mathrm{M}_{4}(\mathbb{Z})$. 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).