On the irrationality of generalized $q$-logarithm

For integer $p$, $\mathrm{\mid }p\mathrm{\mid }>1$, and generic rational $x$ and $z$, we establish the irrationality of the series $${\mathrm{\ell }}_p(x,z)=x\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{p^n-x}\mathrm{.}$$ It is a symmetric (${\mathrm{\ell }}_p(x,z)={\mathrm{\ell }}_p(z,x)$) generalization of the $q$-logarithmic function ($x=1$ and $p=1\mathrm{/}q$ where $\mathrm{\mid }q\mathrm{\mid }<1$), 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 ${\mathrm{\ell }}_p(x,z)$.