Summation with factorial terms (involving Laguerre polynomials)

Question

As part of an exercise including gamma functions and Laguerre polynomials, I need to show that for a Laguerre polynomial $L_n(x)$, $$\int\limits_0^\infty L_n(x)x^ke^{-x}dx = 0 \textrm{ with } n \in \mathbb{N} \textrm{ and }k \in \{0,1,\dots,n-1\}$$ I've managed to narrow it down to proving that $$\sum\limits_{i=0}^n\frac{(-1)^{n-i}(k+i)!}{(i!)^2(n-i)!} = 0\textrm{ with } n \in \mathbb{N} \textrm{ and }k \in \{0,1,\dots,n-1\}$$ But this is where I'm stuck. I used Maple and WolframAlpha to turn this sum into $$\frac{k!(-1)^n\Gamma(n-k)}{\Gamma(-k)\Gamma(n+1)^2}$$ Which is $0$ since $\Gamma(-k)$ would be infinity, due to the Gamma function having simple poles in the negative integers. However, neither Maple or Wolfram can tell me how I convert the sum into this fraction of gamma functions. Am I missing something obvious or are there any tricks to get there?

Answer 1

With the definition you have given, \begin{align}\int_0^{\infty}L_n(x)x^ke^{-x}dx&=\frac{(-1)^n n!}{2\pi i}\oint_{\Sigma}\frac{\Gamma(t-n)}{\Gamma^2(t+1)}\left(\int_0^{\infty}x^{t+k}e^{-x}dx\right)dt=\\ &=\frac{(-1)^n n!}{2\pi i}\oint_{\Sigma}\frac{\Gamma(t-n)\Gamma(t+k+1)}{\Gamma^2(t+1)}dt \end{align} What are the singular points of the integrand?

• $\Gamma(t-n)$ has simple poles at $t=n,n-1,n-2,\ldots$. However, almost all of these poles, namely $t=-1,-2,\ldots$ are compensated by the poles of one of the $\Gamma(t+1)$ in the denominator.

• Also, $\Gamma(t+k+1)$ has simple poles at $-k-1,-k-2,\ldots$ but all of them are compensated by the poles of the second $\Gamma(t+1)$.

Thus we have simple poles at $t=n,n-1,\ldots,0$ - precisely those inside of $\Sigma$. But now, instead of shrinking the integration contour, let us expand it to infinity. On the circle $|t|=R$, we have (think why) $$\frac{\Gamma(t-n)\Gamma(t+k+1)}{\Gamma^2(t+1)}=O\left(R^{k-n-1}\right)\qquad \text{as}\;R\rightarrow\infty.$$ Therefore, the integral over this circle will be (at most) $O\left(R^{k-n}\right)$, but this goes to zero as $R\rightarrow\infty$ for $k<n$.