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Journal of the London Mathematical Society 1986 s2-33(1):133-148; doi:10.1112/jlms/s2-33.1.133
© 1986 by London Mathematical Society
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© Oxford University Press

An Integral of Products of Ultraspherical Functions and a q-Extension

Richard Askey, Tom H. Koornwinder and Mizan Rahman

Department of Mathematics, University of Wisconsin Madison 53706, USA
Centre for Mathematics and Computer Science PO Box 4079, 1009 AB, Amsterdam, Netherlands
Department of Mathematics and Statistics, Carleton University Ottawa, Ontario, Canada K1S 5B6

Let Pn(x) and Qn{x) denote the Legendre polynomial of degree n and the usual second solution to the differential equation, respectively. Din showed that Formula vanishes when |l–m| < n < l+m, and Askey evaluated the integral for arbitrary integral values ofl, m and n. We extend this to the evaluation of Formula, where Formula is the ultraspherical polynomial and C{lambda}n(x) is the appropriate second solution to the ultraspherical differential equation. A q-extension is found using the continuous q-ultraspherical polynomials of Rogers. Again the integral vanishes when |l–m| < n < l+m. It is shown that this vanishing phenomenon holds for quite general orthogonal polynomials. A related integral of the product of three Bessel functions is also evaluated.


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