We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on a series expression for the mutual energy of two probability measures naturally connected with the polynomials. The particular case of Gegenbauer polynomials is analyzed in detail. These results are applied also to the computation of the entropy of spherical harmonics, important for the study of the entropic uncertainty relations as well as the spatial complexity of physical systems in central potentials.

MSC: 33C45 42C05 34B24Keywords: Classical orthogonal polynomials Inner product involving difference operators Non-standard orthogonality a b s t r a c t It is well-known that the family of Hahn polynomials {h α,β n (x; N)} n≥0 is orthogonal with respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for Hahn polynomials for a degree higher than N.We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials and give the limit relations among them for all n ∈ N 0 . Furthermore, in order to get these results for the Krawtchouk polynomials we will obtain a more general property of orthogonality for Meixner polynomials.

Equilibrium measures in the real axis in the presence of rational external fields are considered. These external fields are called rational since their derivatives are rational functions. We analyze the evolution of the equilibrium measure, and its support, when the size of the measure, t, or other parameters in the external field vary. Our analysis is illustrated by studying with detail the case of a generalized Gauss-Penner model, which, in addition to its mathematical relevance, has important physical applications (in the framework of random matrix models). This paper is a natural continuation of [18], where equilibrium measures in the presence of polynomial external fields are thoroughly studied.

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We discuss the asymptotic behavior (as n → ∞) of the entropic integrals E n = −