An intergral of products of ultraspherical functions and a q-extension. (English) Zbl 0564.33008

Askey, Richard; Koornwinder, Tom H.; Rahman, Mizan

J. Lond. Math. Soc., II. Ser. 33, 133-148 (1986).

If \(\{p_ n(x)\}\) are polynomials orthogonal with respect to a positive measure da(x) then \(\int^{\infty}_{-\infty}p_ n(x)p_ m(x)p_ k(x)d\alpha (x)=0\) 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 \(\int^{b}_{a}q_ n(x)p_ m(x)p_ k(x)w(x)dx\) vanishes when there is a triangle with sides k,m,n. Here \[ q_ n(z)=\int^{b}_{a}p_ n(t)[z-t]^{-1}d\alpha (t),\quad x\not\in [a,b], \] and \(q_ n(x)=[q_ n(x+io)+q_ n(x-io)/2]\) 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:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C05	Classical hypergeometric functions, \({}_2F_1\)
42C10	Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Keywords:

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