A note on the multiple zeta functions and their variants at identical arguments

TL;DR

This paper analyzes multiple zeta functions and their variants at identical arguments, expressing them as polynomials in the Riemann zeta function and establishing functional relations among these functions.

Contribution

It provides explicit polynomial descriptions of MZF coefficients using Bell polynomials and derives functional relations between MZF, multiple t-functions, and their star variants.

Findings

Coefficients of MZF polynomials described via Bell polynomials.

Singularities of these functions characterized explicitly.

Functional relations established among MZF, M$t$F, and star variants.

Abstract

In this article, we study the multiple zeta functions (MZF) and some of its variants at identical arguments. Using the harmonic product, these functions can be expressed as polynomials in the Riemann zeta function. Firstly, we note that an explicit description of the coefficients of these polynomials can be given in terms of the values of the complete Bell polynomials. This for instance, easily leads to the complete description of the singularities of these functions. But more importantly, this enables us to establish a functional relation between MZF, Multiple $t$-functions (M$t$F) and their star variants at identical arguments.