Linear independence of values of the $q$-exponential and related functions

TL;DR

This paper proves the linear independence of values of the $q$-exponential function and related functions at algebraic points when $q$ is a Pisot-Vijayraghavan number, advancing understanding in $q$-series and transcendence theory.

Contribution

It establishes new linear independence results for $q$-exponential and related functions at algebraic points, specifically for $q$ as a Pisot-Vijayraghavan number.

Findings

Linear independence of $E_q(x)$ values at algebraic points for $q$ Pisot-Vijayraghavan number

Results extend to Tschakaloff function and generalized $q$-series

Advances in transcendence theory for $q$-functions

Abstract

In this paper, we establish the linear independence of values of the $q$-analogue of the exponential function, $E_q(x)$ and its derivatives at specified algebraic arguments, when $q$ is a Pisot-Vijayraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalized $q$-series.