Complexity analysis of hypergeometric orthogonal polynomials

TL;DR

This paper analyzes the complexity measures of hypergeometric orthogonal polynomials, providing explicit and asymptotic results for Hermite, Laguerre, and Jacobi systems, and explores open problems in related special functions.

Contribution

It offers explicit and asymptotic complexity measures for classical orthogonal polynomials and investigates their behavior with respect to polynomial degree and parameters.

Findings

Explicit complexity measures for Hermite, Laguerre, Jacobi polynomials.

Asymptotic analysis of complexity measures as degree n increases.

Numerical examination of complexity measures in relation to polynomial parameters.

Abstract

The complexity measures of the Cr\'amer-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density $\rho_n(x)=\omega(x) p_n^2(x)$ of the polynomials $p_n(x)$ orthogonal with respect to the weight function $\omega(x)$, $x\in (a,b)$, are used to quantify various two-fold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways. Their explicit (Cr\'amer-Rao) and asymptotical (Fisher-Shannon, LMC) values are given for the three systems of orthogonal polynomials. Then, these complexity-type mathematical quantities are numerically examined in terms of the polynomial's degree $n$ and the parameters which characterize the weight function. Finally, several open problems about the generalised hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted $L_q$-norms of Laguerre and Jacobi polynomials are pointed out.