\[\int_{0}^{\infty}{\mathrm{e}}^{-xz}L^{(\alpha)}_{m}\left(x\right)L^{(\alpha)}_%
{n}\left(x\right)\,{\mathrm{e}}^{-x}\,x^{\alpha}\,\mathrm{d}x=\frac{\Gamma%
\left(\alpha+m+1\right)\Gamma\left(\alpha+n+1\right)}{\Gamma\left(\alpha+1%
\right)\,m!\,n!}\frac{z^{m+n}}{(z+1)^{\alpha+m+n+1}}{{}_{2}F_{1}}\left({-m,-n%
\atop\alpha+1};z^{-2}\right),\]