Irrationality proof of certain Lambert series using little q-Jacobi polynomials

Abstract

We apply the Padé technique to find rational approximations to

$$h^{\pm }(q_1,q_2)=\sum _{k=1}^{\infty }\frac{q_1^k}{1\pm q_2^k}\text{,}0<q_1,q_2<1,q_1\in \mathbb{Q},q_2=1/p_2,p_2\in \mathbb{N}\setminus \left\{1\right\}\text{.}$$

A separate section is dedicated to the special case $q_i=q^{r_i},r_i\in \mathbb{N},q=1/p,p\in \mathbb{N}\setminus \left\{1\right\}$. In this construction we make use of little $q$-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of $h^{\pm }(q_1,q_2)$ and give an upper bound for the irrationality measure.