Decades ago in an MIT talk, I gave a method to enumerate a family of telescoping products akin to Product[(1/2)*(1 + x^2^(-n)), {n, Infinity}] == (-1 + x)/Log[x] . (Which telescopes because you can shift the index by changing the x variable.) Recently, Mourad Ismail sent a paper reminding me to carry out the enumeration, on which Neil and I started yesterday. Neil found Product[(1/2)*(z^(-2)^(-k) + 1), {k, 1, Infinity}] == (z - 1)/(z^(2/3)*Log[z]) and fears it to be centuries old, after getting severely scooped on a continued fraction identity last week. I found Product[-(-1)^(2/3) + (-1)^(1/3)*z^(-2)^(-k), {k, Infinity}] == (-1 + (-1)^(2/3)*z)/((-1 + (-1)^(2/3))*z^(2/3)) which looks oscillatory and nonconvergent, but is really just equivalent to Product[2*Sin[Pi/6 + (-1/2)^k*t], {k, Infinity}]==(2*Cos[Pi/6 + t])/Sqrt[3] from http://www.tweedledum.com/rwg/idents.htm . (Nice(?) exercise.) Two of the above identities come from the scheme (a*x^4 + b*x^2 + c)/(c*x^2 + b*x + a) reducing to a polynomial, which Reduce promises contains three more. And there are other schemes... --rwg