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

TL;DR

This paper constructs a $q$-analogue of the Koecher-Leshchiner generating function for the odd zeta values. The approach derives a finite telescoping identity using $q$-integers $[n]_q$, $q$-factorials, and multiple harmonic $q$-sums $H_{q,k}({2}^s)$ to express the $q$-analogue of the generating function. Letting $N\to\infty$ recovers the infinite series identity, connecting to Apéry-like accelerated series for $\\zeta(3)$. The results provide a $q$-deformation framework for generating functions of odd zeta values and extend Apéry-type irrationality techniques to the quantum setting.

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 $ζ(3)$ used by Apéry in his famous proof of irrationality.