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Transactions of the American Mathematical Society

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Sieved ultraspherical polynomials

Authors: Waleed Al-Salam, W. R. Allaway and Richard Askey
Journal: Trans. Amer. Math. Soc. 284 (1984), 39-55
MSC: Primary 33A45; Secondary 33A65, 42C05
MathSciNet review: 742411
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Abstract: The continuous $ q$-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway's Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway's and showed that these polynomials arose from the $ q$-ultraspherical polynomials when $ q$ approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in $ (-1, 1)$. Generating functions and other formulas are also found.

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Keywords: Sieved ultraspherical polynomials of first and second kind, continuous $ q$-ultraspherical polynomials, Fejér-Legendre polynomials, orthogonal polynomials, ultraspherical polynomials, Tchebycheff polynomials, recurrence relation, weight function, generating function, product linearization, connection coefficient
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