I have the following PDF:
\begin{align}
p(x)=\frac{1}{m} \sum_{k=1}^{m-1} \frac{k!}{(k+n-m)!} [L_k^{n-m}(x)]^2 x^{n-m} e^{-x}, \quad \quad x \ge 0,
\end{align}
where $L_k^{n-m}(x)$ is the associated Laguerre polynomial of order $k$ and $n \ge m$
I need to compute the corresponding CCDF $\left( \text{i.e.} \, p(x \ge a) \right)$.
My attempt:
$p(x \ge a)=\int_a^\infty p(x) dx=\frac{1}{m} \sum_{k=1}^{m-1} \frac{k!}{(k+n-m)!} \int_a^\infty [L_k^{n-m}(x)]^2 x^{n-m} e^{-x} dx $. I am not sure if I have the right to switch between the sum and the integral.
I am not able to compute this integral, but I found that $\int_0^\infty x^{\alpha} e^{-x} [L_n^\alpha (x)]^2 dx=\frac{\Gamma(\alpha+n+1) }{n! }$, with $\alpha>0$.
How to continue the calculations ?