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Chill2Macht
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# Is the following infinite product of fractions of linear factors equal to an exponential function or not?

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).

Is there a way to transform the infinite product so that it becomes an exponential function, e.g. by modifying each term individually?

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.