\[\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}M_{\kappa,\mu}%
\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=\frac{\Gamma\left(1+%
2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu+\kappa+\nu\right)}\*e^{-\frac{1}{2}x}%
x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*M_{\frac{1}{2}(\kappa+3\mu-\nu+\frac{%
1}{2}),\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}\left(x\right),\]