In an engineering application, I've been trying to calculate the following integral involving Laguerre polynomials:

$\int_{-\infty}^\infty dx L_n(x^2+\beta^2) e^{-x^2/2+i x \alpha}$,

where $n=0,1,2,3,\ldots$ and $\alpha,\beta$ are real.

For $\beta=0$ the integral can be found in Gradshteyn: $\sqrt{2\pi}/(2^n n!)e^{-\alpha^2/2}[H_n(\alpha/\sqrt{2})]^2$, where $H_n$ is the Hermite polynomial.

However, I am not sure what to do in the case $\beta\neq 0$ -- any thoughts?

Note: Mathematica numerically evaluates the integral for small $n$ (and general $\beta\neq 0$), but I am looking for the solution in a closed form for any $n$.