Orthogonal polynomials derive normalization factor

Question

I have the following problem, and have literally no idea where to start! Even a hint to get me going would be apreciated. I believe the question is concerning Laguerre polynomials.

Let $\alpha$ > -1. Suppose that a monic orthogonal polynomial sequence $\{P_n\}_{n\geq0}$ (for $\mathscr{L}$) satisfies the orthogonality relation

$\mathscr{L}[x^m P_n(x)] = \frac{1}{{\Gamma(\alpha +1)}}\int_0^\infty P_n(x)x^me^{-x}x^\alpha dx = N_n\delta_{n,m},$ for $m\leq n$, and $m,n \geq 0$

with $N_n =\mathscr{L}[x^n P_n(x)] \neq 0$, for $n\geq 0$

Obtain an explicit expression for $N_n$.

Answer 1

Hint: $$\int_0^\infty x^{p-1}e^{-x} dx = \Gamma(p),$$ where $\Gamma$ is the Gamma function.