dear community.
I'm looking for books/guides on functional difference equations. Can you recommend some? Below I try to explain what kind of equations I have in mind.
As an example, one of the equations that I'm directly interested in is $$ \left(\frac{\sin{2\pi (x-p/2)}}{\sin{2\pi x}}e^{-\frac12\partial_x}+\frac{\sin{2\pi (x+p/2)}}{\sin{2\pi x}}e^{+\frac12\partial_x}\right)F(x,y)=2\cos{2\pi y}\,F(x,y) $$ where $p$ is a fixed parameter.
I wasn't able to find any detailed guide on such equations. So, I'm asking for help.
I would also like to share my own thoughts on the topic. As far as I can see there is no essential difference between such equations and simple recurrence equations until one requires a solution to have some additional properties. As a 'physicist' I'm interested in solutions as 'nice' as possible. For example, I would not treat a piece-wise constant function as a legitimate solution to the equation above.
Any help is appreciated. I will try to clarify my question if needed.