Cramer–Rao information plane of orthogonal hypergeometric polynomials

The classical hypergeometric polynomials { p n ( x ) } n = 0 ∞ , which are orthogonal with respect to a weight function ω ( x ) defined on a real interval, are analyzed in the Cramer–Rao information plane, that is the plane defined by both Fisher information and variance of the probability density ρ n ( x ) = p n ( x ) 2 ω ( x ) . The Rakhmanov density ρ n ( x ) of these polynomials, which describes the probability density of the quantum states for various physical prototypes in an exact manner and for numerous physical systems to a very good approximation, is discussed in detail.