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On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials

N. M. Atakishiyev·A. U. Klimyk
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Abstract

This paper studies properties of q-Jacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra U_q(su_{1,1}). Spectrum and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthogonal basis in the representation space. The initial U_q(su_{1,1})-basis and the bases of these eigenfunctions are interconnected by matrices, whose entries are expressed in terms of little and big q-Jacobi polynomials. The orthogonality by rows in these unitary connection matrices leads to the orthogonality relations for little and big q-Jacobi polynomials. The orthogonality by columns in the connection matrices leads to an explicit form of orthogonality relations on the countable set of points for {}_3ϕ_2 and {}_3ϕ_1 polynomials, which are dual to big and little q-Jacobi polynomials, respectively. The orthogonality measure for the dual little q-Jacobi polynomials proves to be extremal, whereas the measure for the dual big q-Jacobi polynomials is not extremal.

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BibTeXRIS

N. M. Atakishiyev, A. U. Klimyk. 2003-10-06. On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials. https://arxiv.org/abs/math/0307250

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