Deformed mathematical objects stemming from the $q$-logarithm function

TL;DR

This paper introduces new deformed mathematical objects based on the $q$-logarithm, exploring their algebraic properties and deriving entropic functionals, including a form of Tsallis entropy.

Contribution

It classifies and constructs novel deformed numbers, operators, and derivatives from the $q$-logarithm, expanding the mathematical framework related to nonextensive entropy.

Findings

Objects exhibit commutativity, associativity, and distributivity within classes

Two entropic functionals are derived, including Tsallis entropy

New algebraic structures generalize classical mathematical concepts

Abstract

Generalized numbers, arithmetic operators and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of $q$-logarithm/$q$-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity and distributivity, and also a pair of linear/nonlinear derivatives are observed within each class. Two entropic functionals emerge from the formalism, one of them is the nonadditive Tsallis entropy.