[math-fun] generating function

Hello - Suppose that a(0), a(1), a(2), ... is a linear recurrence sequence with generating function p(x)/q(x), and that b(0), b(1), b(2), ... is a linear recurrence sequence with generating function u(x)/v(x). Does someone know a generating function for the product sequence a(0)*b(0), a(1)*b(1), a(2)*b*2), ... ? It appears that a denominator may be a polynomial whose roots are the reciprocal-products 1/(r(i)*s(j)), where r(I) ranges through all the roots of q(x) and s(j) ranges through all the roots of v(x). Clark Kimberling