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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On sieved orthogonal polynomials. III. Orthogonality on several intervals


Author: Mourad E. H. Ismail
Journal: Trans. Amer. Math. Soc. 294 (1986), 89-111
MSC: Primary 33A65
MathSciNet review: 819937
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Abstract: We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is $ \{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\} $, where $ c$ is a positive parameter. The weight function of the polynomials of the second kind is $ {\{ 1 - ({(1 + c)^2}/4\operatorname{c} )T_k^2(x)\} ^{1/2}}/\vert{U_{k - 1}}(x)\vert$ when $ c \geqslant 1$. When $ c < 1$ we pick up discrete masses located at the zeros of $ {U_{k - 1}}(x)$. The weight function of the polynomials of the first kind is also included. Sieved generalizations of the symmetric Pollaczek polynomials and their $ q$-analogues are also treated. Their continuous spectra are also the above mentioned set. The $ q$-analogues include a sieved version of the Rogers $ q$-ultraspherical polynomials and another set of $ q$-ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0819937-1
PII: S 0002-9947(1986)0819937-1
Keywords: Chebyshev polynomials, ultraspherical polynomials, polynomials orthogonal on several intervals
Article copyright: © Copyright 1986 American Mathematical Society