A functional equation related to generalized entropies and the modular group

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.

Originally uploaded as an appendix to arXiv:1709.07807v1. Changes in v2: the introduction was extended to summarize in more detail previous results; there is a new lemma at the end of Section 4