Rényi entropies, Lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions

摘要

The quantification of the spreading of the orthogonal polynomials pn(x) can be investigated by means of the Rényi entropies Rq[ρ],q being a positive integer number, of the associated Rakhmanov probability densities, ρ(x)=ω(x)pn2(x), where ω(x) is the corresponding weight function. The Rényi entropies are closely related to the Lq-norms of the polynomials. In this manuscript, the Lq-norms and the associated Rényi entropies of the real hypergeometric orthogonal polynomials (i.e., Hermite, Laguerre, and Jacobi polynomials) and the generalized Hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of Lauricella and Srivastava–Daoust types which are evaluated at some specific values of 2q variables. These functions depend on 4q+1 and 6q+2 parameters, respectively, which are determined by the order q, the degree n of the polynomial, and the parameters of the orthogonality weight function ω(x). The key idea is based on some extended linearization formulas for these polynomials. These results open the way to determine the Rényi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials.