I have the expression
$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$
where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$.
I have a feeling this should have a closed-form which is intuitive and makes less heavy use of the Beta function. Can someone describe to me whether this is true?
Here, $x$ and $y$ are integers larger than $0.$