I am trying to derive a Stirling's approximation for the Multivariate Beta function. This does not seem very hard, but I cannot even derive the Stirling's approximation for the usual Beta function. From Wikipedia, we have:
$B(x,y) \sim \sqrt{2\pi} \frac{x^{x-\frac{1}{2}}y^{y-\frac{1}{2}}}{(x+y)^{x+y-\frac{1}{2}}}$
I thought I could just use the Stirling's formula for $\Gamma(x)$ and the relation between $\Gamma$ and $B$.
That is:
$\Gamma(x) \sim \sqrt{2\pi} (x-1)^{x-\frac{1}{2}} e^{1-x}$.
From here:
$B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \sim \sqrt{2\pi} \frac{(x-1)^{x-\frac{1}{2}}(y-1)^{y-\frac{1}{2}}}{(x+y-1)^{x+y-\frac{1}{2}}} $
This is close but different from the formula I find on Wikipedia (and other sources). Any help appreciated.