\[n\left(\left(x+k\right)^{2}(n+k-1)-k\right)\hat{L}^{(k)}_{n+1}\left(x\right)+(%
n+k-1)\left(\left(x+k\right)^{2}(x-2n-k+1)+2k\right)\hat{L}^{(k)}_{n}\left(x%
\right)+(n+k-2)\left(\left(x+k\right)^{2}(n+k)-k\right)\hat{L}^{(k)}_{n-1}%
\left(x\right)=0,\]