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).

That's right for the denominator. I do not know if is easy to give a formula for the numerator. See Prop. 3.9 on page 9 (and the ensuing remark) of my paper: http://ajc.maths.uq.edu.au/pdf/40/ajc_v40_p115.pdf A standard reference for this sort of thing is Stanley's "Enumerative Combinatorics"; however, when I looked there, he only had the statement about the rationality of the generating function of the pointwise product. I needed the degree of the recurrence (and a little bit more) in my paper. Michael Reid