Extended double shuffle relations and the generating function of triple zeta values of any fixed weight

TL;DR

This paper derives formulas for the generating function of triple zeta values of fixed weight using extended double shuffle relations, leading to new sum formulas and insights into multiple zeta values.

Contribution

It introduces new formulas for the generating function of triple zeta values based on extended double shuffle relations involving multi-fold products.

Findings

Derived formulas for the generating function of triple zeta values.

Obtained parameterized, weighted, and restricted sum formulas for triple zeta values.

Enhanced understanding of the algebraic structure of multiple zeta values.

Abstract

Extended double shuffle relations for multiple zeta values are obtained by the fact that any product of regularized multiple zeta values has two different representations, and the case of two-fold product is considered in general. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained by not only two-fold products of double and single zeta values but also three-fold products of single zeta values. As applications of the formulas, we also obtain some parameterized, weighted and restricted sum formulas for triple zeta values.