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Rényi entropies, \(L_q\) norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions. (English) Zbl 1329.33013
Summary: The quantification of the spreading of the orthogonal polynomials \(p_n(x)\) can be investigated by means of the Rényi entropies \(R_q[{\rho}]\), \(q\) being a positive integer number, of the associated Rakhmanov probability densities, \({\rho}(x)={\omega}(x)p_n^2(x)\), where \({\omega}(x)\) is the corresponding weight function. The Rényi entropies are closely related to the \(L_q\)-norms of the polynomials. In this manuscript, the \(L_q\)-norms and the associated Rényi entropies of the real hypergeometric orthogonal polynomials (i.e., Hermite, Laguerre, and Jacobi polynomials) and the generalized Hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of Lauricella and Srivastava-Daoust types which are evaluated at some specific values of \(2q\) variables. These functions depend on \(4q+1\) and \(6q+2\) parameters, respectively, which are determined by the order \(q\), the degree \( n\) of the polynomial, and the parameters of the orthogonality weight function \({\omega}(x)\). The key idea is based on some extended linearization formulas for these polynomials. These results open the way to determine the Rényi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials.

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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