quantum mechanics - Shift operator (integral calculus involving Hermite polynomials)

I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and a shift operator matrix, I thought it'd be better to post it here.

I have to calculate the integral

$$\frac{1}{2^nn!\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$

where $H_n(x)$ is the $n^{th}$ Hermite polynomial and prove that it equals

$$\sqrt{\frac{m_!}}\left(\frac{k}{\sqrt{2}}\right)^{|n-l|}L_{m_<}^{|n-l|}\left(-\frac{k^2}{2}\right)\exp\left(\frac{k^2}{4}\right)$$

where $m_<$ and $m_>$ denote the smaller and the larger respectively of the two indices $n$ and $l$ and where $L_n^m$ are the associated Laguerre polynomials.

The last term is $\exp(k^2/4)$, hence I suppose that I begin with

$$\frac{1}{2^nn!\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx-\frac{k^2}{4}}e^{\frac{k^2}{4}}H_l(x)\;\mathrm{d}x$$ $$\frac{1}{2^nn!\sqrt{\pi}}e^{\frac{k^2}{4}}\int_{-\infty}^{+\infty}H_n(x)e^{-(x-\frac{k}{2})^2}H_l(x)\;\mathrm{d}x$$

but here I'm stuck... No matter what or how I can't go further.

Thanks for your help!