On the irrationality of generalized $q$-logarithm

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 Padé approximations to ${\ell }_p(x,z)$.

12 pages

References in corpus (4)

• Diophantine problems for q-zeta values
• A double-sum Kronecker-type identity
• On the non-quadraticity of values of the q-exponential function and related q-series
• A determinantal approach to irrationality

Cited by in corpus (1)

• A determinantal approach to irrationality