\[\int_{0}^{\infty}\hat{L}^{(k)}_{n}\left(x\right)\hat{L}^{(k)}_{m}\left(x\right%
)\hat{W}_{k}(x)\,\mathrm{d}x=\frac{(n+k)\Gamma\left(n+k-1\right)}{(n-1)!}%
\delta_{n,m}.\]