On the irrationality of generalized $q$-logarithm

TL;DR

This paper proves the irrationality of a generalized $q$-logarithmic series for certain parameters, extending previous results on $q$-harmonic series, using Hankel determinants and Padé approximations.

Contribution

It establishes the irrationality of a broad class of generalized $q$-logarithmic series, a significant extension of known results.

Findings

Proves irrationality of $ ext{ell}_p(x,z)$ for specified parameters.

Introduces a novel proof technique using Hankel determinants and Padé approximations.

Generalizes known results on $q$-harmonic series and $q$-logarithms.

Abstract

For integer $p$, $|p|>1$, and generic rational $x$ and $z$, we establish the irrationality of the series $$\ell_p(x,z)=x\sum_{n=1}^\infty\frac{z^n}{p^n-x}.$$ It is a symmetric ($\ell_p(x,z)=\ell_p(z,x)$) generalization of the $q$-logarithmic function ($x=1$ and $p=1/q$ where $|q|<1$), which in turn generalizes the $q$-harmonic series ($x=z=1$). Our proof makes use of the Hankel determinants built on the Pad\'e approximations to $\ell_p(x,z)$.