Heun functions related to entropies

TL;DR

This paper explores the relationship between indices of coincidence for common distributions and their connection to entropies, providing explicit formulas and representations involving Heun and hypergeometric functions.

Contribution

It introduces new connections between Heun functions and entropy measures, offering explicit solutions and representations for indices of coincidence.

Findings

Closed-form expressions for Heun functions related to entropies

Explicit representations of indices of coincidence

Connections between distributions and entropy measures

Abstract

We consider the indices of coincidence for the binomial, Poisson, and negative binomial distributions. They are related in a simple manner to the R\'{e}nyi entropy and Tsallis entropy. We investigate some families of Heun functions containing these indices of coincidence. For the involved Heun functions we obtain closed forms, explicit expressions, or representations in terms of hypergeometric functions.