DLMF:18.17.E17 (Q5758)

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DLMF:18.17.E17
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    0 1 ( 1 - x 2 ) λ - 1 2 C 2 n ( λ ) ( x ) cos ( x y ) d x = ( - 1 ) n π Γ ( 2 n + 2 λ ) J λ + 2 n ( y ) ( 2 n ) ! Γ ( λ ) ( 2 y ) λ , superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 𝜋 Euler-Gamma 2 𝑛 2 𝜆 Bessel-J 𝜆 2 𝑛 𝑦 2 𝑛 Euler-Gamma 𝜆 superscript 2 𝑦 𝜆 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n}\left(x\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda\right)J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(% \lambda\right)(2y)^{\lambda}},}}
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    DLMF:18.17.E17
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