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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 - p n (x)p m (x)p k (x)dα(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 dα (x) has compact support, say [a,b], and the measure is absolutely continuous, dα(x)=w(α)dx, then it is shown that a b 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)= a b p n (t)[z-t] -1 dα(t),x[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:
33C45Orthogonal polynomials and functions of hypergeometric type
33C05Classical hypergeometric functions, 2 F 1
42C10Fourier series in special orthogonal functions