Rényi entropies, L q norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions P. Sánchez-Moreno a,d,⇑ , J.S. Dehesa b,d , A. Zarzo c,d , A. Guerrero b,d a Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain b Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain c Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Universidad Politécnica de Madrid, Madrid, Spain d Instituto ‘‘Carlos I’’ de Física Teórica y Computacional, Universidad de Granada, Granada, Spain article info Keywords: Orthogonal poynomials Renyi entropy Hypergeometric functions Lauricella function Srivastava–Daoust function abstract The quantification of the spreading of the orthogonal polynomials p n ðxÞ can be investigated by means of the Rényi entropies R q ½q; q being a positive integer number, of the associated Rakhmanov probability densities, qðxÞ¼ xðxÞp 2 n ðxÞ, where xðxÞ is the corresponding weight function. The Rényi entropies are closely related to the L q -norms of the polynomi- als. In this manuscript, the L q -norms and the associated Rényi entropies of the real hyper- geometric orthogonal polynomials (i.e., Hermite, Laguerre, and Jacobi polynomials) and the generalized Hermite polynomials are expressed in an explicit way in terms of some gener- alized multivariate special functions of Lauricella and Srivastava–Daoust types which are evaluated at some specific values of 2q variables. These functions depend on 4q þ 1 and 6q þ 2 parameters, respectively, which are determined by the order q, the degree n of the polynomial, and the parameters of the orthogonality weight function xðxÞ. 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. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction The theoretical determination of integrals containing products of an arbitrary number of hypergeometric orthogonal polynomials is a long standing, yet generally unsolved, mathematical problem from the times of Erdélyi, Feldheim, Bus- bridge, Bailey, Carlitz and Titchmarsh (see e.g. [25,17,27,18]) up until now [4,8,1,2,9–11,14,62,33,34,39,42,44–46,52– 55,61,65,68,70,71,74,36]. It appears frequently in many scientific and technological domains, ranging from combinatorics [62] and weighted permutation problems [30,26,12,36,29,35], to the theory of angular momentum and mathematical phys- ics for the evaluation of matrix elements of Hermitian operators which describe physical observables of quantum systems (see e.g. [52,68,51] and references therein). Specially relevant from both fundamental and applied standpoints is the calcu- lation of the integrals containing arbitrary (i.e., non-necessarily integer) powers of a certain hypergeometric orthogonal poly- nomial, because (a) they are closely related to its L q norm (see e.g. [5,43,6]), to different quantifiers of its spreading all over its orthogonality interval, the information-theoretic lengths [22,32,63,64], and various information-theoretic quantities (Rényi and Tsallis entropies and lengths) of the probability density associated to the polynomial, (b) they admit combinatorial [62,29,42,39,36,35] and entropic [22,32,63,64] interpretations, and (c) they describe the expectation values of some 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.076 ⇑ Corresponding author at: Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain. E-mail addresses: pablos@ugr.es (P. Sánchez-Moreno), dehesa@ugr.es (J.S. Dehesa), alejandro.zarzo@upm.es (A. Zarzo), agmartinez@ugr.es (A. Guerrero). Applied Mathematics and Computation 223 (2013) 25–33 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc