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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
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