Simplify $\Gamma(n + x)/\Gamma(x)$

Question

Is there a way to simplify $\Gamma(n + x)/\Gamma(x)$ for $n \in \mathbb{N}$ ? It's easy for the first few values of $n$ but I was wondering whether there is a known general formula (expressed as a polynomial of $n$ and $x$?).

Answer 1

Well, if by "simplifying" you mean get rid of the Gamma function, then, yes, you can simply start by using your definition of Gamma.

If $ n $ is an arbitrary integer, we have for any $ k\in\mathbb{N} $, $ x>0 $ : \begin{aligned} \Gamma\left(x+k+1\right)&=\left(x+k\right)\Gamma\left(x+k\right)\\ \Longrightarrow \prod_{k=0}^{n-1}{\frac{\Gamma\left(x+k+1\right)}{\Gamma\left(x+k\right)}}&=\prod_{k=0}^{n-1}{\left(x+k\right)}\\ \iff \ \ \ \ \ \ \ \ \ \ \ \frac{\Gamma\left(x+n\right)}{\Gamma\left(x\right)}&=\prod_{k=0}^{n-1}{\left(x+k\right)} \end{aligned}

Answer 2

Hint: start with $\,n=1\,$ and use this property.
Observe further that $\displaystyle\dfrac{\Gamma(2 + x)}{\Gamma(x)}=\dfrac{\Gamma(1+(1 + x))}{\Gamma(1+x)}\dfrac{\Gamma(1 + x)}{\Gamma(x)}$ and so on...