Irrationality proof of a $q$-extension of $\zeta(2)$ using little $q$-Jacobi polynomials

TL;DR

This paper constructs rational approximations for a $q$-analogue of $oldsymbol{ ext{zeta}(2)}$, proving its irrationality and providing a sharper measure of irrationality than previous bounds using little $q$-Jacobi polynomials.

Contribution

It introduces a novel method using Hermite-Padé approximation and little $q$-Jacobi polynomials to prove the irrationality of $oldsymbol{ ext{zeta}_q(2)}$ and improves the upper bound on its measure of irrationality.

Findings

Proves the irrationality of $ ext{zeta}_q(2)$.

Provides an upper bound for the measure of irrationality: approximately 3.8936.

Achieves a sharper bound than previous results by Zudilin.

Abstract

We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $\zeta_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $\mu(\zeta_q(2))\leq 10\pi^2/(5\pi^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $\zeta(2)$}, Mat. Sb. \textbf{193} (2002), no. 8, 49--70).