$L_p^q(x)$ is associated Laguerre polynomials and defined as below
$$ L_p^q(x)=\frac{x^{-q}e^x}{p!} \frac{d^p}{dx^p}\left( x^{p+q} e^{-x}\right) $$
I want to show the orthogonality of it. I mean, how to show
$$ \int_0^\infty e^{-x}xL_p^1(x)L_r^1(x) dx = {}_{n+r}P_{k} \ \ \delta_{p,r} $$
Now, ${}_n P_k$ is defined as below
$$ {}_a P_b = \frac{a!}{(a-b)!} $$