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

TL;DR

This paper studies the arithmetic nature of values of a generalized q-exponential function, proving they are non-quadratic under certain conditions, which advances understanding of their algebraic independence.

Contribution

It establishes the non-quadraticity of values of a broad class of q-exponential functions, including special cases like the Tschakaloff function.

Findings

Proves non-quadraticity of F_q(α;λ) for specific q, λ, α.

Extends results to include classical q-exponential and Tschakaloff functions.

Provides new insights into the algebraic properties of q-series values.

Abstract

We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;\lambda)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-\lambda)}, \qquad |q|>1, \quad \lambda\notin q^{\mathbb Z_{>0}}, $$ 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 $\alpha\notin-\lambda q^{\mathbb Z_{>0}}$, $\alpha\ne0$.