A two-parameter generalization of Shannon-Khinchin Axioms and the uniqueness theorem

TL;DR

This paper extends the Shannon-Khinchin axioms to a two-parameter framework, establishing a new uniqueness theorem and connecting it to the Leibniz rule of a two-parameter difference operator.

Contribution

It introduces a two-parameter generalization of the Shannon-Khinchin axioms and proves a corresponding uniqueness theorem, expanding the theoretical foundation of entropy measures.

Findings

Established a two-parameter Shannon additivity property.

Proved the uniqueness theorem for the generalized axioms.

Connected the additivity to the Leibniz rule of a two-parameter difference operator.

Abstract

Based on the one-parameter generalization of Shannon-Khinchin (SK) axioms presented by one of the authors, and utilizing a tree-graphical representation, we have further developed the SK Axioms in accordance with the two-parameter entropy introduced by Sharma-Taneja, Mittal, Borges-Roditi, and Kaniadakis-Lissia-Scarfone. The corresponding unique theorem is proved. It is shown that the obtained two-parameter Shannon additivity is a natural consequence from the Leibniz rule of the two-parameter Chakrabarti-Jagannathan difference operator.