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Jacobi polynomials. II: An analytic proof of the product formula. (English) Zbl 0242.33020
Summary: An analytic proof is given for the author’s product formula for Jacobi polynomials and a new integral representation is obtained for the product $J_\alpha (x)J_\beta (y)$ of two Bessel functions. Similarly, a product formula for Jacobi polynomials due to {\it A. Dijksma} and the author [Nederl. Akad. Wet., Proc., Ser. A 74, 191--196 (1971; Zbl 0209.09302)] is derived in an analytic way. The proofs are based on Bateman’s work on special solutions of the biaxially symmetric potential equation. The paper concludes with new proofs for Gasper’s evaluation of the convolution kernel for Jacobi series and for Watson’s evaluation of the integral $$\int_0^\infty J_\alpha(\lambda x) J_\beta (\lambda y) J_\beta (\lambda z) \lambda^{1-\alpha}\, d\lambda\, .$$

33C45Orthogonal polynomials and functions of hypergeometric type
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
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