I am curious to know if some kind of addition formula can be derived for the following cases,
\begin{equation} \exp{\left(x y \cos{\phi}\right)} L_n^\alpha\left(x^2 + y^2 - 2 x y \cos{\phi}\right) \end{equation} or \begin{equation} \exp{\left(-x^2 + x y \cos{\phi}\right)} L_n^\alpha\left(x^2 + y^2 - 2 x y \cos{\phi}\right) \left(x^2 - x y \cos{\phi}\right)^\gamma\,, \quad \gamma \ge 0 \end{equation}
I am aware of the Bateman's form, \begin{equation} \exp{\left(x y e^{i \phi}\right)} L_n\left(x^2 + y^2 - 2 x y \cos{\phi}\right) = \sum_{k = 0}^\infty \left(x y e^{i\phi}\right)^{(k - n)} \frac{n!}{k!} L_n^{(k - n)}\left(x^2\right) L_n^{(k - n)}\left(y^2\right) \end{equation} The idea is to exponentiate the $\phi$ dependence to analytically perform the angular integration.
I found the answer. The addition formula for associated Laguerre polynomial was derived by T. Koornwinder, SIAM J. MATH. ANAL. Vol. 8, No. 3, May 1977.