Little q-Legendre polynomials and irrationality of certain Lambert series

Abstract

We show how one can obtain rational approximants for $q$-extensions of the harmonic series and the logarithm (and many other similar quantities) by Padé approximation using little $q$-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for the harmonic series and a better upper bound than the one given by Matala-aho and Väänänen for the logarithm.

15 pages

References in corpus (1)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Cited by in corpus (2)

• Some q-analogues of supercongruences of Rodriguez-Villegas
• Leonhard Euler and a q-analogue of the logarithm