Irrationality of $\zeta_q(1)$ and $\zeta_q(2)$

Abstract

In this paper we show how one can obtain simultaneous rational approximants for $\zeta_q(1)$ and $\zeta_q(2)$ with a common denominator by means of Hermite-Pade approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, $\zeta_q(1)$, $\zeta_q(2)$ are linearly independent over the rationals. In particular this implies that $\zeta_q(1)$ and $\zeta_q(2)$ are irrational. Furthermore we give an upper bound for the measure of irrationality.