Is This a New Expression for the Beta Function?

AI Thread Summary
A new expression for the Beta Function has been proposed, defined as B(x,y) = (Γ(x)/x) * (∑(k=1 to y) (Γ(x+y-k)/Γ(y-k+1)))⁻¹, applicable for non-negative integer pairs of x and y. The formula bears resemblance to the traditional Beta Function representation, B(x,y) = (x-1)!(y-1)!/(x+y-1)!. The discussion confirms that y is indeed the upper bound of the summation in the new expression. The community is invited to verify if this expression is previously known or truly novel. The exploration of this potential new formulation of the Beta Function is deemed interesting.
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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.
 
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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)!}##
 
fresh_42 said:
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.
 
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