Duality of q-polynomials, orthogonal on countable sets of points

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We review properties of q-orthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (self-adjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally interpret the duality of families of q-polynomials, orthogonal on countable sets of points. In order to obtain orthogonality relations for dual sets of polynomials, it is proposed to use two symmetric (self-adjoint) operators, representable (in some distinct bases) by Jacobi matrices. This approach is applied to several pairs of dual families of q-polynomials, orthogonal on countable sets, from the q-Askey scheme. For each such pair, the corresponding operators, representable by Jacobi matrices, are explicitly given. These operators are employed in order to find explicitly sets, on which the polynomials are orthogonal, and orthogonality relations for them.
71 pages, corrected typos, added references

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Classical Analysis and ODEs, 33D80; 33D45; 17B37

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http://salesiana.dossiersoluciones.com/handle/123456789/73189

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0411249
 http://arxiv.org/abs/math/0411249

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arXiv