Infinite sum = product puzzle
                        Michael Somos
                    somos@cis.csuohio.edu
                        12 May 2004
                    (21 Sep 2007 tweak)

    Suppose you have two rational functions

         f(x)=x*(1-x)^2*(1-x^2)/(1-x^6)  and
         g(x)=(1-x)*(1-x^6)^6/(1-x^2)^2/(1-x^3)^3 .

Then

        f(x)+f(x^2)+f(x^3)+... = x*g(x)*g(x^2)*g(x^3)*... .

The puzzle is to find all such function pairs. The rules are
that f(x) is a product with one factor x and the rest are all
of the form (1-x^k) or 1/(1-x^k), and similarly for g(x) but
it has no factors of x. I know of just eleven solutions of
this puzzle including the one listed above. Probably there are
no others. It would be nice to generalize this in an interesting
way by weakening the rules to allow more solutions. For example,
one could allow f(x) to have a polynomial factor not of the form
(1-x^k). Another possibilty is to require that the power series
expansion coefficients of f(x) to be a periodic sequence. These
are just two examples that I thought of.