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