Computation of the entropy of polynomials orthogonal on an interval

TL;DR

This paper introduces an efficient method to compute the entropy of orthogonal polynomials on an interval using recurrence coefficients, with applications to Gegenbauer polynomials, spherical harmonics, and physical systems.

Contribution

It presents a novel algorithm based on a series expression for mutual energy, enabling entropy computation solely from recurrence relation coefficients.

Findings

Effective entropy computation method for orthogonal polynomials.

Detailed analysis of Gegenbauer polynomials.

Applications to spherical harmonics and physical systems.

Abstract

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.