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Cramer–Rao information plane of orthogonal hypergeometric polynomials. (English) Zbl 1086.33010
The classical hypergeometric polynomials $$\{p\,_n(x)\}^\infty_{n=0}$$, which are orthogonal with respect to a weight function $$\omega(x)$$ 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 $$\rho\,_n(x)=p\,_n(x)^2\,\omega(x)$$. The Rakhmanov density $$\rho_n(x)$$ 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.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 94A17 Measures of information, entropy 62B10 Statistical aspects of information-theoretic topics
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