Transcendental Numbers and the Lambert-Tsallis Function

TL;DR

This paper investigates the transcendental nature of generalized exponential and Lambert functions, specifically the Lambert-Tsallis W_q and q-exponential e_q, using number theory to identify conditions for transcendence.

Contribution

It introduces a framework to determine when the generalized Lambert-Tsallis and q-exponential functions are transcendental based on parameters q and z.

Findings

Conditions on q and z for W_q(z) to be transcendental

Conditions on q and z for exp_q(z) to be transcendental

Application of Gelfond-Schneider theorem to generalized functions

Abstract

To decide upon the arithmetic nature of some numbers may be a non-trivial problem. Some cases are well know, for example exp(1) and W(1), where W is the Lambert function, are transcendental numbers. The Tsallis q-exponential, e_q (z), and the Lambert-Tsallis W_q (z) function, where q is a real parameter, are, respectively, generalizations of the exponential and Lambert functions. In the present work we use the Gelfond-Schneider theorem in order to show the arithmetic conditions on q and z such that W_q (z) and exp_q (z) are transcendental.