\[\int_{0}^{\infty}e^{-zt-t^{2}}t^{2b-2}{\mathbf{M}}\left(a,b,t^{2}\right)\,%
\mathrm{d}t=\tfrac{1}{2}\pi^{-\frac{1}{2}}\Gamma\left(b-\tfrac{1}{2}\right)U%
\left(b-\tfrac{1}{2},a+\tfrac{1}{2},\tfrac{1}{4}z^{2}\right),\]