Interpolant of truncated multiple zeta functions

TL;DR

This paper introduces an analytic interpolant for truncated multiple zeta functions, representing it via Mellin transforms, establishing harmonic product relations, and exploring its properties.

Contribution

It presents a novel interpolant function for truncated multiple zeta functions, with Mellin transform representation and harmonic product relations, advancing understanding of their analytic structure.

Findings

Defined the interpolant $$ for truncated multiple zeta functions

Derived Mellin transform representation of the interpolant

Established harmonic product relations for the interpolant and related functions

Abstract

We introduce an analytic function $\Psi(s_1,\ldots,s_r;w)$ that interpolates truncated multiple zeta functions $\zeta_N(s_1,\ldots,s_r)$. We represent this interpolant as a Mellin transform of a function $G(q_1,\ldots,q_r;w)$ and, using this expression, give the analytic continuation. Further, the harmonic product relations for $\Psi$ and $G$ are established via relevant Hopf algebra structures, and some properties of the function $G$ are provided.