 # For n in NN and x>0 we define P_n(x) = (1)/(2 pi i) oint_Sigma (Gamma(t-n))/(Gamma(t+1)^2)x^tdt where Σ is a closed contour in the t-plane that encircles the points 0,1,…,n once in the positive region. Lisa Hardin 2022-07-16 Answered
For $n\in \mathbb{N}$ and $x>0$ we define
${P}_{n}\left(x\right)=\frac{1}{2\pi i}{\oint }_{\mathrm{\Sigma }}\frac{\mathrm{\Gamma }\left(t-n\right)}{\mathrm{\Gamma }\left(t+1{\right)}^{2}}{x}^{t}dt$
where Σ is a closed contour in the t-plane that encircles the points 0,1,…,n once in the positive region.
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All you need to do is to multiply the integral representation by $\left(-1{\right)}^{n}n!$, so you will have the right integral representation
${P}_{n}\left(x\right)=\frac{\left(-1{\right)}^{n}n!}{2\pi i}{\oint }_{\mathrm{\Sigma }}\frac{\mathrm{\Gamma }\left(t-n\right)}{\mathrm{\Gamma }\left(t+1{\right)}^{2}}{x}^{t}dt.$
For instance,
${P}_{5}\left(x\right)=1-5\phantom{\rule{thinmathspace}{0ex}}x+5\phantom{\rule{thinmathspace}{0ex}}{x}^{2}-\frac{5}{3}\phantom{\rule{thinmathspace}{0ex}}{x}^{3}+\frac{5}{24}\phantom{\rule{thinmathspace}{0ex}}{x}^{4}-\frac{1}{120}\phantom{\rule{thinmathspace}{0ex}}{x}^{5},$
which agrees with the Laguerre polynomial ${L}_{5}\left(x\right)$

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