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.References
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Additional Information
- Marisa S. Costa
- Affiliation: Pós-Graduação em Matemática, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
- Email: isacosta.mat@bol.com.br
- Eduardo Godoy
- Affiliation: Departamento de Matemática Aplicada II, E.T.S.I. Industriales, Universidade de Vigo, Campus Lagoas-Marcosende, 36310 Vigo, Spain
- Email: egodoy@dma.uvigo.es
- Regina L. Lamblém
- Affiliation: Pós-Graduação em Matemática, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
- Email: parareginae@hotmail.com
- A. Sri Ranga
- Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, UNESP-Universi- dade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
- MR Author ID: 238837
- Email: ranga@ibilce.unesp.br
- Received by editor(s): September 7, 2010
- Received by editor(s) in revised form: February 11, 2011
- Published electronically: October 19, 2011
- Additional Notes: This work was partially support by the joint project CAPES(Brazil)/DGU(Spain)
The second author’s work was also supported by the European Community fund FEDER
The third and fourth authors have also received other funds from CNPq, CAPES and FAPESP of Brazil for this work - Communicated by: Walter Van Assche
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 2075-2089
- MSC (2010): Primary 33D15, 42C05; Secondary 33D45
- DOI: https://doi.org/10.1090/S0002-9939-2011-11066-9
- MathSciNet review: 2888195