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

Kelly Postelmans·Walter Van Assche
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Abstract

In this paper we show how one can obtain simultaneous rational approximants for $ζ_q(1)$ and $ζ_q(2)$ 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, $ζ_q(1)$, $ζ_q(2)$ are linearly independent over the rationals. In particular this implies that $ζ_q(1)$ and $ζ_q(2)$ are irrational. Furthermore we give an upper bound for the measure of irrationality.

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BibTeXRIS

Kelly Postelmans, Walter Van Assche. 2006-04-13. Irrationality of $ζ_q(1)$ and $ζ_q(2)$. https://arxiv.org/abs/math/0604312

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