A $q$-Caputo Fractional Generalization of Tsallis Entropy: Series Representation and Non-Negativity Domains

TL;DR

This paper introduces a fractional generalization of Tsallis entropy using a $q$-Caputo operator, providing a series representation and analyzing the conditions for non-negativity, thus extending the entropy's mathematical framework.

Contribution

It presents a novel fractional generalization of Tsallis entropy via the $q$-Caputo differintegral, including a series representation and analysis of non-negativity domains.

Findings

Derived a closed series representation of the fractional entropy.

Showed the limits where the fractional entropy converges to standard Tsallis entropy.

Numerically identified regions of non-negativity depending on parameters.

Abstract

We introduce a fractional generalization of Tsallis entropy by acting with a $q$-Caputo operator on the generating family $\sum_i p_i^{\,x}$ evaluated at $x=1$. Concretely, we define $S_{q}^{\alpha}$ through the $q$-Caputo differintegral of order $0<\alpha<1$ and derive a closed series representation in terms of the $q$-Gamma function. The construction is anchored at the evaluation point, which ensures well-behaved limits: as $\alpha\!\to\!1$ we recover the standard Tsallis entropy $S_q$. Finally we perform a numerical calculation to show the regions where the obtained $q$-fractional entropy $S^{\alpha}_q$ can be non-negative (or negative) through the fractional parameter $\alpha$ and the non extensive index $q$.