Table of Integrals, Series, and Products, 8th Edition

7.33 Combinations of the polynomials $C_n^v(x)$ and Bessel functions. Integration of Gegenbauer functions with respect to the index

7.331

1. 

${\displaystyle {\int }_1^{\infty }{x}^{2n+1-v}{(x^2-1)}^{v-2n-\frac{1}{2}}{C}_{2n}^{v-2n}\left(\frac{1}{x}\right)J_v(xy)\text{d}x={(-1)}^n{2}^{2n-v+1}{y}^{-v+2n-1}{[(2n)!]}^{-1}\mathrm{\Gamma }(2v-2n){[\mathrm{\Gamma }(v-2n)]}^{-1}\cos y}$

  ET II 44(10)a

$\left[y>0,2n-\frac{1}{2}<\mathrm{Re}v<2n+\frac{1}{2}\right]$

7.332

1. 

$\begin{array}{lll}{\displaystyle {\int }_0^{\infty }{x}^{v+1}\text{?}{(x^2+{\beta }^2)}^{-\frac{1}{2}v-\frac{3}{4}}{C}_{2n+1}^{v+\frac{1}{2}}\left[{(x^2+{\beta }^2)}^{-1/2}\beta \right]\text{?}{J}_{v+\frac{3}{2}+2n}\left[{(x^2+{\beta }^2)}^{1/2}a\right]\text{?}{J}_v(xy)\text{d}x}\hfill & ={(-1)}^n{2}^{1/2}{\pi }^{-1/2}{a}^{\frac{1}{2}-v}{y}^v{(a^2-y^2)}^{-1/2}\sin [\beta {(a^2-y^2)}^{1/2}]C_{2n+1}^{v+\frac{1}{2}}\left[{\left(1-\frac{y^2}{a^2}\right)}^{1/2}\right]\hfill & [0<y<a]\hfill \\ =0\hfill \end{array}$

  ET II 59(23) ...