Fisher information of orthogonal hypergeometric polynomials

Abstract

The probability densities of the position and momentum of many quantum systems have the form $\rho(x)\propto p_n\sp 2(x)\omega(x)$, where $\{p_n(x)\}$ denotes a sequence of hypergeometric-type polynomials orthogonal with respect to the weight function $\omega(x)$. Here we derive an explicit expression for the Fisher information $I=\int {\rm d}x[\rho'(x)]\sp 2/\rho(x)$ corresponding to this kind of distribution, in terms of the coefficients of the second-order differential equation satisfied by the polynomials $p_n(x)$. 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.