On classical orthogonal polynomials |
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Authors: | N. M. Atakishiyev M. Rahman S. K. Suslov |
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Affiliation: | 1. Physics Institute, 370143, Baku, Azerbaijan 2. Department of Mathematics and Statistics, Carleton University, K1S 5B6, Ottawa, Ontario, Canada 3. “Kurchatow Institute”, Russian Research Center, 123182, Moscow, Russia
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Abstract: | Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a functiony(x(s)) on a nonuniform latticex(s) is given. It is shown that the difference-derivatives ofy(x(s)) also satisfy similar equations, if and only ifx(s) is a linear,q-linear, quadratic, or aq-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented. |
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