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

Abstract

We show how one can use Hermite-Padé 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: $\mu ({\zeta }_q(2))\le 10{\pi }^2/(5{\pi }^2-24)\approx 3.8936$. 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).

arXiv comment13 pages, one reference was corrected, two were added

Cited by in corpus (1)

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