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I have an infinite summation of the kind: $$ \Gamma(\alpha-a)\Gamma(\alpha-b)\sum_{n,m\ge 0}\frac{\Gamma(\alpha-a-b+m+n-1)}{\Gamma(\alpha-2a+n)\Gamma(\alpha-2b+m)}f(n,m)$$ for some function $f(n,m)$.

I want to understand the asymptotics of this when $\alpha\to\infty$. I suspect the term $n=m=0$ will then be leading and I can discard the rest of the sum, but I am not sure.

I know that $\Gamma(1+z)\approx z^ze^{-z}\sqrt{2\pi z}$ when $z$ is large, but I don't know how to use this fact here.

For instance, when I find something like $(\alpha-a)^{\alpha-a}$, how do I approximate this further? As $(\alpha-a)^\alpha$, as $\alpha^{\alpha-a}$, as $\alpha^\alpha$, or what?

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Regarding your last question: $(\alpha-a)^{\alpha-a}=\alpha^{\alpha-a}(1-\frac{a}{\alpha})^{\alpha-a}$ and this goes to $\alpha^{\alpha-a}e^{-a}$

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