Definite integral of a Laguerre polynomial and exponentials

In our investigations on the effect of strong magnetic fields on the properties of elementary particles we have been faced with a definite integral of the form $$\int_0^{2\pi}d\theta\ L_{n}(s^2+t^2+2st\cos\theta)\ e^{-ik\theta}\, \exp{(-st\,e^{i\theta})}\ ,$$ where $L_n(x)$ is a Laguerre polynomial, $s$ and $t$ are real numbers and $n$ and $k$ are integers, with $n \geq 0$. In the present article we show that this integral can be solved analytically. The result can be used to get an alternative proof of an addition formula for Laguerre polynomials.