I am trying to figure out the Fourier transform of Legendre polynomial $P_\ell [\cos(\theta-a t )]$:
$Q(\omega)=\int_{-\infty}^\infty P_\ell [\sin\phi\cos(\theta-a t )] e^{i \omega t} dt,$
where $\theta,\phi,a,\omega$ are all real. I know how to do it if I plug in a concrete value of $\ell$ since $P_\ell$ is just polynomial. However I am wondering if there exists a general expression that is valid for all $\ell$, which allows me to program. Thank you.
You can have a look at this article it gives a solution for finite Fourier transform where you are interested in a single period of integration. For the continuous time Fourier transform I doubt that there exists a closed form expression for all $l$ without any iterative forms but iterative calculations might be possible.