Sum of interpolated multiple $q$-zeta values

TL;DR

This paper introduces generating functions for interpolated multiple $q$-zeta values, expressing them via hypergeometric functions, and generalizes previous results to include height, $q$-deformation, and $t$-interpolation, with applications to sum formulas.

Contribution

It provides a systematic expression of generating functions for interpolated multiple $q$-zeta values, extending prior work with new generalizations.

Findings

Generated functions expressed in basic hypergeometric functions

Includes general height, $q$-deformation, and $t$-interpolation

Proves sum formulas for interpolated multiple $q$-zeta values

Abstract

Interpolated multiple $q$-zeta values are deformation of multiple $q$-zeta values with one parameter, $t$, and restore classical multiple zeta values as $t = 0$ and $q \to 1$. In this paper, we discuss generating functions for sum of interpolated multiple $q$-zeta values with fixed weight, depth and $i$-height. The functions are systematically expressed in terms of the basic hypergeometric functions. Compared with the result of Ohno and Zagier, our result includes three generalizations: general height, $q$-deformation and $t$-interpolation. As an application, we prove some expected relations for interpolated multiple $q$-zeta values including sum formulas.