# Is this known yet? (an expression for the Beta Function I have not yet seen)

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.

Mentor
2022 Award
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.