Does there exist a nicer form for $\beta(x + a, y + b) / \beta(a, b)$? 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 $0.$

You can also write this as a ratio of rising factorials.   The rising factorial is defined as $(a)_k = a(a+1)\cdots(a+k-1)$, with $(a)_0 = 1$.  Then use the recurrence relationship for the Gamma function to reduce the ratio of the beta functions to
$$\frac{\beta(a+x,b+y)}{\beta(a,b)} = \frac{(a)_x(b)_y}{(a+b)_{(x+y)}}.$$
I think that's as concise as it gets.  Of course, some folks aren't too fond of rising factorial notation...
