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

The quantification of the spreading of the orthogonal polynomials p(n)(x) can be investigated by means of the Renyi entropies R-q[rho], q being a positive integer number, of the associated Rakhmanov probability densities, rho(x) = omega(x)p(n)(2)(x), where omega(x) is the corresponding weight function. The Renyi entropies are closely related to the L-q-norms of the polynomials. In this manuscript, the L-q-norms and the associated Renyi 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 omega(x). The key idea is based on some extended linearization formulas for these polynomials. These results open the way to determine the Renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials. (C) 2013 Elsevier Inc. All rights reserved.