Little q-Legendre polynomials and irrationality of certain Lambert series

TL;DR

This paper develops rational approximations for q-extensions of harmonic series and logarithms using little q-Legendre polynomials, establishing irrationality proofs with sharp bounds on the measure of irrationality.

Contribution

It introduces a novel application of little q-Legendre polynomials for irrationality proofs of q-extensions of classical constants.

Findings

Proves irrationality of certain q-extensions of harmonic series and logarithms.

Provides sharp upper bounds for the measure of irrationality.

Improves existing bounds for the irrationality measure of these quantities.

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 Pad\'e 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 V\"a\"an\"anen for the harmonic series and a better upper bound than the one given by Matala-aho and V\"a\"an\"anen for the logarithm.