Basic hypergeometric functions and orthogonal Laurent polynomials

Costa, Marisa; Godoy, Eduardo; Lamblém, Regina; Ranga, A.

doi:10.1090/S0002-9939-2011-11066-9

Basic hypergeometric functions and orthogonal Laurent polynomials
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by Marisa S. Costa, Eduardo Godoy, Regina L. Lamblém and A. Sri Ranga PDF

Proc. Amer. Math. Soc. 140 (2012), 2075-2089 Request permission

Abstract:

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or $q$-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 $0 < q < 1$ and the complex parameters $b$, $c$ and $d$ 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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