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.