Weighted sums with two parameters of multiple zeta values and their formulas

TL;DR

This paper introduces new formulas for weighted sums of multiple zeta values involving two parameters, leading to identities and polynomial expressions that deepen understanding of their structure.

Contribution

It provides two novel formulas for weighted sums of multiple zeta values with two parameters, expanding the toolkit for analyzing their relationships.

Findings

Derived formulas express weighted sums as polynomials in usual zeta values.

Identified linear combinations of multiple zeta values as parameter-dependent polynomials.

Obtained identities for small-depth weighted sums.

Abstract

A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coeffcients in the rational polynomial ring generated by the two parameters, and obtain some identities for weighted sums of multiple zeta values of small depths.