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

7.54 Combinations of hypergeometric and Bessel functions

7.541

${\displaystyle {\int }_0^{\infty }{x}^{\alpha +\beta -2\nu -1}{(x+1)}^{-\nu }}{e}^{xz}{K}_{\nu }[(x+1)z]F(\alpha ,\beta ;\alpha +\beta -2\nu ;-x)\text{d}x={\pi }^{-\frac{1}{2}}\cos (\nu \pi )\mathrm{\Gamma }(\frac{1}{2}-\alpha +\nu )\mathrm{\Gamma }(\frac{1}{2}-\beta +\nu )\mathrm{\Gamma }(\gamma ){(2z)}^{-\frac{1}{2}-\frac{1}{2}\gamma }{W}_{\frac{1}{2}\lambda ,\frac{1}{2}(\beta -\alpha )(2z)}$

  ET II 401(16)

$\begin{array}{cc}\gamma =\alpha +\beta -2\nu & [\mathrm{Re}(\alpha +\beta -2\nu )>0,\mathrm{Re}(\frac{1}{2}-\alpha +\upsilon )>0,\mathrm{Re}(\frac{1}{2}-\beta +\upsilon )>0,|\arg z|<\frac{3}{2}\pi ]\end{array}$

7.542

1.¹² 

${\displaystyle {\int }_0^{\infty }{x}^{\sigma -1}}F_p{p-1}_{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_{p-1};-\lambda x^2)Y_{\nu }(xy)\text{d}x=\frac{\mathrm{\Gamma }(b_1)\mathrm{\dots }\mathrm{\Gamma }(b_{p-1})}{2{\lambda }^{\frac{1}{2}\sigma }\mathrm{\Gamma }(a_1)\mathrm{\dots }\mathrm{\Gamma }(a_p)}{G}_{p+2,p+3}^{p+2,1}(\frac{y^2}{4\lambda }|\begin{array}{c}{b}_0^{*},\mathrm{\dots },b_{p-1}^{*},l\\ h,k,a_1^{*},\mathrm{\dots },a_p^{*},l\end{array})$

  ET II 118(53)

$a_j^{*}=a_j-\frac{\sigma }{2},j=1,\mathrm{\dots },p;b_0^{*}=1-\frac{\sigma }{2};b_j^{*}=b_j-\frac{\sigma }{2},j=1,\mathrm{\dots },p$