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Linearization and Krein-like functionals of hypergeometric orthogonal polynomials. (English) Zbl 1410.33027
This work deals with the computation of the generalized Krein-like \(r\)-integral functionals of hypergeometric orthogonal polynomials defined by \[ \mathcal{J}_{\{m_{r}\}} \left(s, \beta \right) = \int_{\Delta} \left(\omega(x)\right)^{\beta} x^{s} p_{m_{1}}(x)\cdots p_{m_{r}}(x)\, dx \tag{1}\] where \(s\) and \(\beta\) are real parameters and \(\omega(x)\) is the weight function on the real interval \(\Delta\) with respect to which the polynomials \(\{p_{m}(x) \}\) are orthogonal, that is, \[ \int_{\Delta} \omega(x) p_{n}(x)p_{m}(x)\, dx =d_{n}^2\, \delta_{n,m}. \] The authors explain how to compute the integrals (1) by means of a Lauricella-based approach and apply it to the three families of orthogonal polynomials: Hermite, Laguerre and Jacobi. The results obtained are then used in the development of power, Krein-like, exponential and logarithmic moments of the Rakhmanov probability density regarding each one the three families.
On the subject of the particular Krein-like \(2\)-functionals \[ \mathcal{J}_{m,n} \left(s, \beta \right) = \int_{\Delta} \left(\omega(x)\right)^{\beta} x^{s} p_{m}(x)p_{n}(x)\, dx \tag{2} \] with \(\Delta=[a,b]\), two different methods are exploited for the three classical orthogonal polynomial sequences (OPS). On one hand, the authors compute (2) taking into account the second order differential equation fulfilled by each family and on the other hand they provide expressions established through characteristics of each classical OPS, namely the explicit expansion of \(p_{n}(x)\) and the linearization formula for the product of two polynomials.
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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