[math-fun] Funny meromorphic function

This formula f(z) = Product_{1 <= n < oo} (1 + 1/(z-n)^2 - 1/n^2) converges where this sum does: Sum_{1 <= n < oo} (1/(z-n)^2 - 1/n^2) = Sum (n^2 - (z-n)^2) / (n^2 (z-n)^2) = Sum (-z^2 + 2 n z) / (n^2 (z-n)^2) , which clearly converges for z ≠ n in Z+. Does anyone recognize this function, or is it similar to other known functions? —Dan