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Complexity analysis of hypergeometric orthogonal polynomials. (English) Zbl 1351.60036
Summary: The complexity measures of the Crámer-Rao, Fisher-Shannon and LMC (López-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 (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 \(n\) 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 \(L_q\)-norms of Laguerre and Jacobi polynomials are pointed out.
MSC:
60F99 Limit theorems in probability theory
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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