$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$
I found this equality through maple and it is also equal to $\frac{b}{a}x+1$ but what I'm interested in is how can one derive this equality and what is the relation between products like these and the Gamma function?
I need to simplify a similar expression that maple can't simplify, so if I can understand how to derive this, I may be able to simplify the more difficult expressions.
This expression came up in solving for the hypergeometric polynomials from it's difference equation. Multiplying by $\frac{a}{b}$ gives the first polynomial in the sequence. However I don't think it is related to simplifying the product in this way