A $q$-analogue of the Koecher-Leshchiner generating function of odd zeta values

TL;DR

This paper derives a $q$-analogue of Koecher and Leshchiner's generating function for odd zeta values, connecting classical series with $q$-series and providing insights related to Apéry's irrationality proof.

Contribution

It introduces a novel $q$-analogue of a classical generating function for odd zeta values, extending the theory of special functions and series.

Findings

Derived a $q$-analogue of the Koecher-Leshchiner generating function

Connected the $q$-analogue to Apéry's series for $$

Provides a new perspective on $q$-series and zeta values

Abstract

In the 1980s, Koecher and, independently, Leshchiner found an elegant formula for the generating function of odd zeta values. In this short note, we derive a $q$-analogue of this formula, which provides a $q$-version of the accelerated series for $\zeta(3)$ used by Ap\'ery in his famous proof of irrationality.