Given two series $f(x) = \sum_{n=0}^{+\infty} f_n (x) x^n$, $g(y) = \sum_{m=0}^{+\infty} g_m y^m$, and a number series $K(m,n)$, is there a natural way to produce the series $\sum_{m,n}K(m,n)f_n g_m y^m x^n $?
For my problem, I have $$K(m,n) = \frac{1}{\sin(\pi(m \lambda_1 - n \lambda_2 - \lambda_2))\sin(\pi(m \lambda_1 - n \lambda_2 + \lambda_1))}$$
Ok, this is a dumb question indeed. Probably the solution is (correct me if I'm wrong):
use $K(x, y) \equiv \sum_{m,n}K(m,n)x^m y^n$,
and do a double Fourier transformation
$\int f(z_1)K(x-z_1, y-z_2) g(z_2) dz_1 dz_2$
and then substitute in the definitions, and use (up to some constants)
$\int d \xi e^{i m \xi} \sim \delta(m)$.