We can use the property $\Gamma{(a + 1)} = a \Gamma{(a + 1)}$ to simply the expression below to $\frac{a}{a + b}$ if $k = 1$.
$ \frac{\Gamma{(a+b)}}{\Gamma{(a)}} \frac{\Gamma(a+1)}{\Gamma{(a + 1 + k)}} $
How does it simplify? Here's what I've tried.
$ \frac{\Gamma{(a+b)}}{\Gamma{(a)}} \frac{\Gamma(a+1)}{\Gamma{(a + 1 + k)}} \\ \frac{\Gamma{(a+b)}}{\Gamma{(a)}} \frac{a\Gamma(a)}{\Gamma{(a + 2)}} \\ \frac{\Gamma{(a+b)}}{\Gamma{(a)}} \frac{a\Gamma(a)}{a^2\Gamma{(a)}} \\ \frac{a^b\Gamma{(a)}}{\Gamma{(a)}} \frac{a\Gamma(a)}{a^2\Gamma{(a)}} \\ \frac{a^{b+1}}{a^2} \\ a^{b - 1} $
That doesn't look like $\frac{a}{a + b}$. Where did I err?