Fisher information of orthogonal hypergeometric polynomials

  • Authors:

  • Jorge Sánchez-Ruiz;Jesús S. Dehesa

  • Affiliations:

  • Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain and Instituto Carlos I de Física Teórica y Computacional ...;Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain and Departamento de Física Moderna, Universidad de Granada, 18071 Granada, Spain

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2005

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Abstract

The probability densities of position and momentum of many quantum systems have the form ρ(x) ∞ pn2(x)ω(x), where {pn(x)} denotes a sequence of hypergeometric-type polynomials orthogonal with respect to the weight function ω(x). Here we derive the explicit expression of the Fisher information I = ∫ dx[ρ'(x)]2/ρ(x) corresponding to this kind of distributions, in terms of the coefficients of the second-order differential equation satisfied by the polynomials pn(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.