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

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 ∫ − 1 1 Q n ( x ) P m ( x ) P l ( x ) d x 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 ∫ − 1 1 D n λ ( x ) C m λ ( x ) C l λ ( x ) ( 1 − x 2 ) 2 λ − 1 d x , where C n λ ( x ) is the ultraspherical polynomial and C n λ ( 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.