Interpolation of $q$-analogue of multiple zeta and zeta-star values

TL;DR

This paper introduces a new two-parameter generalization of multiple zeta values that combines $q$-analogue and $t$-interpolation, and establishes several algebraic relations for these interpolated values.

Contribution

It develops the concept of $t$-$q$MZVs$, unifying existing $q$-analogue and $t$-interpolated multiple zeta values, and proves key algebraic relations for them.

Findings

Proved Kawashima type relation for $t$-$q$MZVs

Established cyclic sum formula for $t$-$q$MZVs

Derived Hoffman type relation for $t$-$q$MZVs

Abstract

We know at least two ways to generalize multiple zeta(-star) values, or MZ(S)Vs for short, which are $q$-analogue and $t$-interpolation. The $q$-analogue of MZ(S)Vs, or $q$MZ(S)Vs for short, was introduced by Bradley, Okuda and Takeyama, Zhao, etc. On the other hand, the polynomials interpolating MZVs and MZSVs using a parameter $t$ were introduced by Yamamoto. We call these $t$-MZVs. In this paper, we consider such two generalizations simultaneously, that is, we compose polynomials, called $t$-$q$MZVs, interpolating $q$MZVs and $q$MZSVs using a parameter $t$ which are reduced to $q$MZVs as $t=0$, to $q$MZSVs as $t=1$, and to $t$-MZVs as $q$ tends to $1$. Then we prove Kawashima type relation, cyclic sum formula and Hoffman type relation for $t$-$q$MZVs.