Irrationality of $ζ_q(1)$ and $ζ_q(2)$

Details Irrationality of $ζ_q(1)$ and $ζ_q(2)$ Irrationality of $ζ_q(1)$ and $ζ_q(2)$

2006-04-13

arxiv.org/abs/math/0604312v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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