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- W2079706068 abstract "Let $C_n^lambda (x)$ and $D_n^lambda (x)$ be the ultraspherical polynomial and ultraspherical function of the second kind, respectively. Askey, Koornwinder and Rahman showed that the integral $int _{ - 1}^{1} D_n^lambda (x)C_m^lambda (x)C_l^lambda (x)(1 - x^2 )^{2lambda - 1} dx$ vanishes when $l + m + n$ is odd and $|l - m| < n < l + m$. We find a dual of this result, namely, that [sumlimits_{n = 0}^infty {(n + lambda )frac{{Gamma (n + 1)}}{{Gamma (n + 2lambda )}}C_n^lambda (cos beta )C_n^lambda (cos gamma )D_n^lambda (cos alpha )} ] vanishes when $|beta - gamma | < alpha < beta + gamma $. We evaluate this sum for $alpha < |beta - gamma |$ and $alpha > beta + gamma $, and apply Hsü’s limiting technique to give a derivation of the Bessel function integral $int _0^infty x^{1 + nu } Y_nu (ax)J_v (bx)J_nu (cx)dx$ that has recently been evaluated by Askey, Koornwinder and Rahman by a different method." @default.
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- W2079706068 date "1985-07-01" @default.
- W2079706068 modified "2023-09-23" @default.
- W2079706068 title "An Infinite Series with Products of Jacobi Polynomials and Jacobi Functions of the Second Kind" @default.
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- W2079706068 doi "https://doi.org/10.1137/0516065" @default.
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