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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: 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 $ 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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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
Email: ranga@ibilce.unesp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11066-9
PII: S 0002-9939(2011)11066-9
Keywords: Basic hypergeometric functions, continued fractions, orthogonal Laurent polynomials, Szegő polynomials
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
Article copyright: © Copyright 2011 American Mathematical Society