Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too! 


The original motivation for this is the (finite) product $f(n)=\prod\_{i=1}^{n-1}(1-\frac{i}{i^2+n})$ that I had to bound some time ago. Applying some calculus (logarithm to convert into sum, relate to a series, bound with an integral) I could show that 
$f(n)>\frac{1}{\sqrt{n}}-\frac\pi{8n}$ if $n>9$ (and actually $f(n)\sim \frac{1}{\sqrt{n}}-\frac\pi{8n}$ for big $n$) but I was left with the question whether there is a closed form for "my" finite product, or for the corresponding infinite product $\phi(n)=\prod\_{i\ge 1}(1-\frac{i}{i^2+n})$(Any information on it would also make my day).


**EDIT May 17**: the infinite product $\phi(n)$ is zero (see Robert Israel's answer). Nevertheless, the square of $f(n)$ is $(1/n)\prod\_{i=1}^{n-1}(1-(\frac{i}{i^2+n})^2)$, and it is still possible that the infinite product $\prod\_{i\ge1}(1-(\frac{i}{i^2+n})^2)$ converges.


So, is there a place to look for techniques to deal with such products if the need arises?



Functions defined as infinite products
According to Maple, the finite product $f(n) = \frac{\Gamma(n - (1 + \sqrt{1-4n})/2) \Gamma(n-(1-\sqrt{1-4n})/2) \Gamma(1-\sqrt{-n}) \Gamma(1+\sqrt{-n})}
{\Gamma(n-\sqrt{-n}) \Gamma(n+\sqrt{-n}) \Gamma((1-\sqrt{1-4n})/2) \Gamma((1+\sqrt{1-4n})/2)}$.
The infinite product is 0, because $\sum\_i i/(i^2+n)$ diverges.


