Cramer–Rao information plane of orthogonal hypergeometric polynomials

The classical hypergeometric polynomials {pn(x)}n=0∞, which are orthogonal with respect to a weight function ω(x)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 ρn(x)=pn(x)2ω(x)ρn(x)=pn(x)2ω(x). The Rakhmanov density ρn(x)ρn(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.