Entropies and Heun functions associated with positive linear operators

Abstract

We consider a parameterized probability distribution p(x)=(p0(x),p1(x),…) and denote by S(x) the squared l2-norm of p(x). The properties of S(x) are useful in studying the Rényi entropy, the Tsallis entropy, and the positive linear operator associated with p(x). We show that for a family of distributions (including the binomial and the negative binomial distributions), S(x) is a Heun function reducible to the Gauss hypergeometric function 2F1. Several properties of S(x) are derived, including integral representations and upper bounds. Examples and applications are given, concerning classical positive linear operators.