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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.

Roland
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  • This is precisely a (inverted) Beta function, for which there exists good estimates. – davidlowryduda May 19 '17 at 23:35
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    R has the lbeta() and lgamma() functions giving logarithms of the Beta and Gamma functions, which should help with large parameters – Henry May 20 '17 at 10:08
  • A similar question https://math.stackexchange.com/questions/2071332/approximation-for-gamma-alpha-gamma-beta-where-alpha-and-beta/2071341#2071341 for Python. – Claude Leibovici May 21 '17 at 06:26

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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.

Mohammad W. Alomari
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