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Zbl 0564.33008
Askey, Richard; Koornwinder, Tom H.; Rahman, Mizan
An intergral of products of ultraspherical functions and a q-extension.
(English)
[J] J. Lond. Math. Soc., II. Ser. 33, 133-148 (1986). ISSN 0024-6107; ISSN 1469-7750/e

If $\{p\sb n(x)\}$ are polynomials orthogonal with respect to a positive measure da(x) then $\int\sp{\infty}\sb{-\infty}p\sb n(x)p\sb m(x)p\sb k(x)d\alpha (x)=0$ if there is no triangle with sides k,m,n. When the polynomials are the continuous q-ultraspherical polynomials of L. J. Rogers, the integral can be evaluated as a product for all integer k,m,n. If $d\alpha$ (x) has compact support, say [a,b], and the measure is absolutely continuous, $d\alpha (x)=w(\alpha)dx$, then it is shown that $\int\sp{b}\sb{a}q\sb n(x)p\sb m(x)p\sb k(x)w(x)dx$ vanishes when there is a triangle with sides k,m,n. Here $$q\sb n(z)=\int\sp{b}\sb{a}p\sb n(t)[z-t]\sp{-1}d\alpha (t),\quad x\not\in [a,b],$$ and $q\sb n(x)=[q\sb n(x+io)+q\sb n(x-io)/2]$ is the usual function of the second kind. When the polynomials are the Rogers polynomials the above integral is evaluated as a product. Limiting cases are ultraspherical polynomials, Hermite polynomials, and Bessel functions.
MSC 2000:
*33C45 Orthogonal polynomials and functions of hypergeometric type
33C05 Classical hypergeometric functions
42C10 Fourier series in special orthogonal functions

Keywords: q-ultraspherical polynomials; Rogers polynomials; ultraspherical polynomials; Hermite polynomials; Bessel functions; linearization of products

Cited in: Zbl 0579.33003

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