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I have the following problem

Show that $L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$ satisfies $x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$ where the contour is counterclockwise around the origin.

I have busted through with the x-derivatives, but I'm not sure where to go from there... or if that is even the way to tackle this problem... There was talk about Taylor expanding the integrand and using the properties of contour integration to whittle down the terms to a finite number of contributions inside the integrals. Is this nonsense, or legit?

any help would be great, Thanks.

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    This can help you http://mathworld.wolfram.com/LaguerreDifferentialEquation.html2012-04-20

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