Some explicit values of a $q$-multiple zeta function whose denominator power is not uniform

TL;DR

This paper derives explicit formulas for a generalized $q$-multiple zeta function with non-uniform small denominator powers, expanding known results beyond the equal or small uniform cases.

Contribution

It provides the first explicit formulas for the $q$-multiple zeta function with unequal small denominator powers, broadening the understanding of these functions.

Findings

Explicit formulas for the case of unequal small powers

Extension of previous results from equal/small powers to unequal cases

Enhanced understanding of $q$-multiple zeta functions

Abstract

One of the generalizations of multiple zeta values is the $q$-version, and in the case of finite sums, they may be expressed explicitly in polynomial form. Several results have been found when the powers of the factors in the denominator are equal and when they are small. In this paper, we give explicit formulas for the case when the powers are unequal and are small.