## 7.33 Combinations of the polynomials $Cnv(x)$ and Bessel functions. Integration of Gegenbauer functions with respect to the index

7.331

1.

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

ET II 44(10)a

$[y>0,2n−12

7.332

1.

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

ET II 59(23) ...

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