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The classical Euler beta function can be defined by $$ B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t $$ for $\Re(p),\Re(q)>0$. The beta function and the classical Euler gamma function $\Gamma(z)$ have the relation $$ B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}. $$ We know that the reciprocal $\frac{1}{\Gamma(z)}$, which is an entire function on $\mathbb{C}$, and the ratio $\frac{\Gamma(z+a)}{\Gamma(z+b)}$ have asymptotic expansions.

My question is:

Is there an asymptotic expansion for the reciprocal $$\frac{1}{B(p,q)}=\frac{\Gamma(p+q)}{\Gamma(p)\Gamma(q)}$$ when $p$ and $q$ tend to $\infty$?

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    $\begingroup$ You have the Stirling asymptotics for $\Gamma(p)$ and $\Gamma(q)$, except when their arguments are close to $\pm\pi$, and you have the Stirling asymptotic for $\Gamma(p+q)$ iff $p+q\to\infty$ with $\arg(p+q)$ away from $\pm\pi$. So, it is unclear what your question is about. $\endgroup$
    – Iosif Pinelis
    Commented Oct 21 at 12:35

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