A functional equation related to generalized entropies and the modular group

TL;DR

This paper solves a functional equation linked to generalized entropies, utilizing the modular group's properties to reveal symmetries and propose a broader connection between probabilities and arithmetic.

Contribution

It introduces a novel approach using modular group transformations to analyze functional equations related to generalized entropies.

Findings

Solution of a key functional equation involving generalized information functions

Identification of modular group symmetries in the functional system

Implication of a broader link between conditional probabilities and arithmetic

Abstract

We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.