Exploring an Unexplored Expression for the Beta Function

[PhysicsRock]

So, I've recently played around a little with the Gamma Function and eventually managed to find an expression for the Beta Function I have not yet seen. So I'm asking you guys, if you've ever seen this expression somewhere or if this is a new thing. Would be cool if it was, so here's the formula:
$$
B(x,y) = \frac{\Gamma(x)}{x} \cdot \left( \sum_{k=1}^{y} \frac{\Gamma(x+y-k)}{\Gamma(y-k+1)} \right)^{-1}
$$

Obviously, this only works for non-negative integer pairs of $x$ and $y$. Still pretty interesting I think.

 

[fresh_42]

Sure you have $y$ as upper bound of the sum?

Anyway, it looks very similar to $B(x,y)=\dfrac{(x-1)!(y-1)!}{(x+y-1)!}$

 

[PhysicsRock]

Sure you have $y$ as upper bound of the sum?

Anyway, it looks very similar to $B(x,y)=\dfrac{(x-1)!(y-1)!}{(x+y-1)!}$

Yes, $y$ is definitely the upper bound.