On sieved orthogonal polynomials. III. Orthogonality on several intervals

Ismail, Mourad E. H.

doi:10.1090/S0002-9947-1986-0819937-1

On sieved orthogonal polynomials. III. Orthogonality on several intervals
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by Mourad E. H. Ismail

Trans. Amer. Math. Soc. 294 (1986), 89-111

DOI: https://doi.org/10.1090/S0002-9947-1986-0819937-1

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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 ${\{ 1 - ({(1 + c)^2}/4\operatorname {c} )T_k^2(x)\} ^{1/2}}/|{U_{k - 1}}(x)|$ 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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