Is there a way to approximate the function below $$\frac{\Gamma(\phi)}{\Gamma(\mu\phi)\Gamma((1-\mu)\phi)}$$ where $\phi$ and $\mu$ are real values?
I'm trying to fit some regression model in a package called R and I'm with some problems because in this model $\phi>600$ in some cases and gamma() function doesn't support such big values.
This looks equal to $\frac{1}{{\rm {B}}(\mu\phi,\phi-\mu\phi)}$. There is a sharp inequality proved by Alzer, which reads: \begin{align} 0 \le \frac{1}{xy}-{\rm {B}}(x,y) \le 0.08731, \qquad \forall x,y\ge1 \end{align} The constants $``0"$ and $``0.08731"$ are best possible.
For $0\le x,y <1$, there exists no upper bound. There are a lot of upper bounds But Alzer is the famous and explicit one. Hope this help.