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

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. $\mathbf{\text{193}}$ (2002), no. 8, 49–70).