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Frequency moments, $$L_q$$ norms and Rényi entropies of general hypergeometric polynomials. (English) Zbl 1312.33034
Summary: The basic variables of the information theory of quantum systems (e.g., frequency or entropic moments, Rényi and Tsallis entropies) can be expressed in terms of $$L_q$$ norms of general hypergeometrical polynomials. These polynomials are known to control the radial and angular parts of the wavefunctions of the quantum-mechanically allowed states of numerous physical and chemical systems. The computation of the $$L_q$$ norms of these polynomials is presently an interesting issue per se in the theory of special functions; moreover, these quantities are closely related to the frequency moments and other information-theoretic properties of the associated Rakhmanov probability density. In this paper we calculate the unweighted and weighted $$L_q$$-norms $$(q=2k,k\in\mathbb N)$$ of general hypergeometric real orthogonal polynomials (Hermite, Laguerre and Jacobi) and some entropy-like integrals of Bessel polynomials, in terms of $$q$$ and the parameters of the corresponding weight function by using their explicit expression and second order differential equation. In addition, the asymptotics $$(q\rightarrow\infty)$$ of the unweighted $$L_q$$ norms of the Jacobi polynomials is determined by the Laplace method.
##### 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
DLMF
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