I am trying to evaluate this integral over complex plain:
$$\int_\mathcal{C}e^{-\gamma|z|^2-zt^*+z^*t}\mathcal{L}_n(\lambda|z|^2)d^2z$$
where $z$ and $t$ are complex numbers, $\gamma$ and $\lambda$ are positive reals, $\mathcal{L}_n(a)$ is the $n$-th Laguerre polynomial, $x^*$ denotes complex conjugation and $|x|^2=x^*x$.
I'll be happy with a solution involving other Laguerre polynomials...
My usual approach of substituting the polar form for complex numbers doesn't seem to work here. Done anybody have other ideas?