Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Basic hypergeometric functions and orthogonal Laurent polynomials

Authors: Marisa S. Costa, Eduardo Godoy, Regina L. Lamblém and A. Sri Ranga
Journal: Proc. Amer. Math. Soc. 140 (2012), 2075-2089
MSC (2010): Primary 33D15, 42C05; Secondary 33D45
Posted: October 19, 2011
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Abstract: A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or -hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials $\{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty }$ , where and the complex parameters , and are such that $b \neq -1, -2, \ldots$ , $c-b+1 \neq -1, -2, \ldots$ , $\mathcal {R}e(d) > 0$ and $\mathcal {R}e(c-d+2) > 0$ . Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.

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