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