# A note on the connection between non-additive entropy and $h$-derivative

## TL;DR

This paper introduces a two-parameter non-extensive entropy based on the $h$-derivative, unifying various known entropy measures including Tsallis, Abe, Shafee, Kaniadakis, and Boltzmann-Gibbs, and analyzes its properties.

## Contribution

It proposes a new two-parameter entropy framework using the $h$-derivative that generalizes and unifies multiple existing entropy measures.

## Key findings

- Recovers several known non-extensive entropies as special cases
- Demonstrates the properties of the new entropy framework
- Provides a mathematical foundation connecting different entropy forms

## Abstract

In order to study as a whole a wide part of entropy measures, we introduce a two-parameter non-extensive entropic form with respect to the $h$-derivative, which generalizes the conventional Newton--Leibniz calculus. This new entropy, $S_{h,h'}$, is proved to describe the non-extensive systems and recover several types of well-known non-extensive entropic expressions, such as the Tsallis entropy, the Abe entropy, the Shafee entropy, the Kaniadakis entropy and even the classical Boltzmann--Gibbs one. As a generalized entropy, its corresponding properties are also analyzed.