Fisher information of orthogonal hypergeometric polynomials

The probability densities of position and momentum of many quantum systems have the form p(x) proportional to P-n(2) (x)omega(x), where {p(n)(x)} denotes a sequence of hypergeometric-type polynomials orthogonal with respect to the weight function omega(x). Here we derive the explicit expression of the Fisher information I = integral dx[rho',(x)](2)/rho(x) corresponding to this kind of distributions, in terms of the coefficients of the second-order differential equation satisfied by the polynomials p, (x). We work out in detail the particular cases of the classical Hermite, Laguerre and Jacobi polynomials, for which we find the value of Fisher information in closed analytical form and study its asymptotic behaviour in the large n limit. (c) 2004 Elsevier B.V. All rights reserved.