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

If $\left\{{p}_{n}\left(x\right)\right\}$ are polynomials orthogonal with respect to a positive measure da(x) then ${\int }_{-\infty }^{\infty }{p}_{n}\left(x\right){p}_{m}\left(x\right){p}_{k}\left(x\right)d\alpha \left(x\right)=0$ 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, $d\alpha \left(x\right)=w\left(\alpha \right)dx$, then it is shown that ${\int }_{a}^{b}{q}_{n}\left(x\right){p}_{m}\left(x\right){p}_{k}\left(x\right)w\left(x\right)dx$ vanishes when there is a triangle with sides k,m,n. Here

${q}_{n}\left(z\right)={\int }_{a}^{b}{p}_{n}\left(t\right){\left[z-t\right]}^{-1}d\alpha \left(t\right),\phantom{\rule{1.em}{0ex}}x\notin \left[a,b\right],$

and ${q}_{n}\left(x\right)=\left[{q}_{n}\left(x+io\right)+{q}_{n}\left(x-io\right)/2\right]$ 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 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 42C10 Fourier series in special orthogonal functions