\frac{(-1)^{n-k}}{(k!)^{2}}\frac{(k+1)}{(k+1)}L_k(x)L_{k+1}(x)+{k}^2\frac{(-1)^{n-k}}{k(k-1)!k!}L_k(x)L_{k-1}(x)=(-1)^{n-k}\frac{2k+1-x}{(k!)^{2}}L^{2}_{k}(x)