On the non-quadraticity of values of the q-exponential function and related q-series

Abstract

We investigate arithmetic properties of values of the entire function $$F(z)=F_q(z;\lambda )=\sum _{n=0}^{\infty }\frac{z^n}{{\prod }_{j=1}^n(q^j-\lambda )},|q|>1,\lambda \notin q^{{\mathbb{Z}}_{>0}},$$ that includes as special cases the Tschakaloff function ($\lambda =0$) and the $q$-exponential function ($\lambda =1$). In particular, we prove the non-quadraticity of the numbers $F_q(\alpha ;\lambda )$ for integral $q$, rational $\lambda$ and $\alpha \notin -\lambda q^{{\mathbb{Z}}_{>0}}$, $\alpha \ne 0$.

27 pages