At school, they ask me to solve this question: For $n \in \mathbb{N}$ and $x > 0$ we define
$P_n(x) = \frac{1}{2\pi i}\int_{\Sigma}\frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^t dt$
where $\Sigma$ is a closed contour in the t-plane that encircles the points 0,1,...,n once in the positive direction (for example, $\Sigma$ could e the circle of radius $R>n$ around the origin).
(a) Show that $P_n$ is a polynomial of degree $n$.
-> I have already shown (using te residue theorem) that this polynomial is equal to $P_n(x) = \sum_{k=0}^n \frac{(-1)^{n-k}}{(n-k)!\Gamma(k+1)^2}x^k$
(b) Show that $\int_0^\infty P_n(x) x^k e^{-x} dx = 0$ for $k = 0,...,n-1$.
-> I have already shown that this integral is equal to $\sum_{h=0}^n \frac{(-1)^{n-h}(h+k)!}{(n-h)!(h!)^2}$, but from there I don't see that this will be equal to zero for the appropriate k.
Question c and d are related to this one I think. There they ask:
(c) What is the value of the integral above for $k = n$?
(d) Calculate $\int_0^\infty [P_n(x)]^2 e^{-x} dx$
Can someone please help me with finding/proving (one of) these? Thanks!
Update: For question (b), we can just use an approximation where we get that the integral $\left | \oint_\Sigma \frac{1}{2\pi i}\frac{\Gamma(t-n)}{\Gamma(t+1)^2} \Gamma(t+k+1)dt \right |$ will be less then $O(R^{k-n})$ for $k<n$. Now (c) is the biggest question for me. I really don't see how to get there.