Is the following infinite product: $$ \prod_{\substack{(a,b) \in \mathbb{Z}^2 \\ a > b}} \frac{x+b}{x+a} $$ defined? If so, does it simplify to an exponential function of $x$?
The subscript is over all integer pairs $a$ and $b$ such that $a>b $
Motivation/suggestions:
When plotting a graph of a finite truncation of the product :
it resembles a negative exponential function. However such an approximation appears to not be accurate for small values of $x$ (where it has vertical asymptotes).
Can it be transformed in such a way that it becomes an exponential function, by modifying each term individually so that the whole function is finite and the asymptotes lie at infinite distance in the limit?
Or can we show that the derivative of this infinite product with respect to $x$ is a constant factor of itself or not, thus proving that it is an exponential?
I believe it should be exponential because it was obtained by raising $e$ to the power a modified version of an integral of a constant function.
This infinite product definitely doesn't converge for any $x$. A necessary condition for an infinite product over $a,b$ to converge is that the individual factors tend to $1$ as $|a|,|b|\to\infty$, but that's not the case here (since $|a|$ and $|b|$ can tend to $\infty$ together in lots of different ways, for example with $b>0$ and $a=2b$).
If the intended expression is a limit of a specific sequence of finite products, that might be a different story due to cancellation of some sort; it would depend on the specific sequence.