  This paper presents a new method to solve functional equations of
multivariate generating functions, such as
$$F(r,s)=e(r,s)+xf(r,s)F(1,1)+xg(r,s)F(qr,1)+xh(r,s)F(qr,qs),$$ giving a
formula for $F(r,s)$ in terms of a sum over finite sequences. We use this
method to show how one would calculate the coefficients of the generating
function for parallelogram polyominoes, which is impractical using other
methods. We also apply this method to answer a question from fully commutative
affine permutations.
