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

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742411-6

MathSciNet review:
742411

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Abstract | References | Similar Articles | Additional Information

Abstract: The continuous $q$-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allawayâ€™s Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allawayâ€™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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Additional Information

Keywords:
Sieved ultraspherical polynomials of first and second kind,
continuous <IMG WIDTH="15" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$q$">-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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© Copyright 1984
American Mathematical Society