Complexity analysis of hypergeometric orthogonal polynomials

The complexity measures of the Crámer–Rao, Fisher–Shannon and LMC (López-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density ρn(x)=ω(x)pn2(x) of the polynomials pn(x)pn(x) orthogonal with respect to the weight function ω(x)ω(x), x∈(a,b)x∈(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 (Crámer–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 nn and the parameters which characterize the weight function. Finally, several open problems about the generalized hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted LqLq-norms of Laguerre and Jacobi polynomials are pointed out.