Beta Function proof help

Question

I'm trying to prove that beta(x,y)=beta(x+1,y)+beta(x,y+1) but not sure where to start. Any help would be greatly appreciated! Thanks

Answer 1

$$B(x+1,y)+B(x,y+1)=\int_0^1\left(t^x(1-t)^{y-1}+t^{x-1}(1-t)^y\right) dt=$$

$$\int_0^1t^{x-1}(1-t)^{y-1}\left[\require{cancel}\cancel t+1-\cancel t\right]dt=B(x,y)$$

Answer 2

You have probably proven that $$ B(x,y) = \frac{ \Gamma(x)\Gamma(y) } {\Gamma(x+y)}$$ where $\Gamma$ is the gamma function and also that $$\Gamma(x+1) = x\Gamma(x).$$ Then we have $$B(x+1,y) + B(x,y+1) = \frac {\Gamma(x+1)\Gamma(y) + \Gamma(x)\Gamma(y+1) } {\Gamma(x+y+1)} = \frac{\Gamma(x)\Gamma(y)(x+y)}{\Gamma(x+y)(x+y)} = B(x,y).$$