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

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.