Let $B(x,y)$ denote the Beta function.
I have $$\frac{B(x+2, y)}{B(x, y)}$$ Wikipedia says $$B(x+1, y) = B(x, y) \cdot \frac {x}{x+y}$$ Thus, $$\frac{B(x+2, y)}{B(x, y)}$$ should equal $$\frac {x^2}{(x+y)^2}$$ But I am told by a computer and my source material that I am wrong. Have I just made simple mistake?
For context, I am attempting to find the 2nd moment of the Beta Distribution. I am aware of the derivation involving the gamma function, but I just wanted to see where this went.
To apply the formula to $B(x+2,y)$, you have to replace $x$ with $x+1$. Hence
$$B(x+2,y)=B(x+1,y)\frac{x+1}{x+1+y}=B(x,y)\frac{x}{x+y}\frac{x+1}{x+1+y}$$