- Let $n$ be a natural number ( $0$ included ) and let $x > 0$ be real.
- Let $\gamma$ be a closed curve which encircles the points $0, 1, 2, \ldots, n$ in positive direction ( anti-clockwise ). $$ \mbox{Define}\quad\operatorname{P}_{n}\left(x\right) = \frac{1}{2\pi\mathrm{i}} \oint_{\gamma }\frac{\Gamma\left(t - n\right)} {\Gamma^{2}\left(t + 1\right)}\,x^{t}\,\mathrm{d}t. $$
Show that $P_{n}(x)$ is a polynomial of degree $n$.
My work so far: I tried to rewrite $\Gamma(t-n)$ by using the identity $\Gamma(z) = \frac{\Gamma(z+1)}{z}.$ Doing this I obtain that:
$P_{n}(x) = \frac{1}{2\pi i} \int_{\gamma}\frac{x^{t}}{\Gamma(t+1)(t-n)(t-n+1)...(t)}dt$.
From there, I can't see why that expression would be a polynomial. Any hints would be appreciated.
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\on}[1]{\operatorname{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ $\ds{\bbox[5px,#ffd]{}}$
The integrand numerator has poles at $\ds{1, 2, 3, \ldots, n}$: $\color{red}{The\ residue}$ at a pole $\ds{k \in \braces{1,2,3,\ldots,n}}$ is given by: \begin{align} &\lim_{t \to k}\bracks{\pars{t - k} {\Gamma\left(t - n\right) \over \Gamma^{2}\pars{t + 1}}\,x^{t}} \\[5mm] = &\ {x^{k} \over \Gamma^{2}\pars{k + 1}} \lim_{t \to k}\bracks{\pars{t - k} \Gamma\pars{t - n}} \\[5mm] = &\ {x^{k} \over\pars{k!}^{2}} \lim_{t \to k}\bracks{\pars{t - k} {\pi \over \Gamma\pars{1 - t + n} \sin\pars{\pi\bracks{t - n}}}} \\[5mm] = &\ \pi\,{x^{k} \over\pars{k!}^{2}\pars{n - k}!} \lim_{t \to k}\ {t - k \over \sin\pars{\pi\bracks{t - n}}} \\[5mm] = &\ \pi\,{x^{k} \over\pars{k!}^{2}\pars{n - k}!} \lim_{t \to k}\ {1 \over \cos\pars{\pi\bracks{t - n}}\pi} \\[5mm] = &\ \pars{-1}^{k - n}\, {x^{k} \over\pars{k!}^{2}\pars{n - k}!} \end{align}
Then, \begin{align} \operatorname{P}_{n}\pars{x} & \equiv \bbox[5px,#ffd]{{1 \over 2\pi\ic} \oint_{\gamma }{\Gamma\pars{t - n} \over \Gamma^{2}\pars{t + 1}}\,x^{t}\,\dd t} \\[5mm] = & \bbx{\pars{-1}^{n}\sum_{k = 1}^{n} {\pars{-1}^{k} \over\pars{k!}^{2}\pars{n - k}!}\,x^{k}} \\ & \end{align}