Asymptotics of hyperrgeometric 2F1 for large integer parameters

Question

I have stumbled upon $$\frac{\Gamma(n+\frac{1}{2})\Gamma(n-\frac{1}{2})}{\Gamma(2n+1)} \ {}_2F_1\left(n+\frac{1}{2}, \ n-\frac{1}{2}; \ 2n+1, \ z\right)$$

Where the integer $n \to \infty$. I need to determine the asymptotic expression of this with this condition. I can do that for the gamma function part using Stirling's approximation, but I haven't found any way to determine

$${}_2F_1\left(n+\frac{1}{2}, \ n-\frac{1}{2}; \ 2n+1, \ z\right) \ \ \text{for } \ \ n \to \infty.$$

How do I obtain this?

Any derivation or reference to the asymptotics of this specific parameters would be very helpful.

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Some edits:

Answer 1

With $z = \cosh \zeta$, $\lambda = n$, $\alpha = \frac{1}{2}$, $\beta = \frac{1}{2}$, and $\gamma = 2$, the first result in $\S9$ of G. N. Watson's paper Asymptotic expansions of hypergeometric functions gives \begin{align*} \left( {\operatorname{csch} \!\big( \tfrac{\zeta }{2} \big)} \right)^{2n + 1} & {}_2F_1\!\left( {n + \tfrac{1}{2},n - \tfrac{1}{2};2n + 1; - \operatorname{csch}^2\! \big( \tfrac{\zeta }{2} \big)} \right) \\ &\sim \frac{{2\Gamma (2n + 1)}}{{\Gamma \!\left( {n - \frac{1}{2}} \right)\Gamma \!\left( {n + \frac{3}{2}} \right)}}{\rm e}^{ - \left( {n + \frac{1}{2}} \right)\zeta } (1 - {\rm e}^{ - \zeta } )^{ - \frac{3}{2}} (1 + {\rm e}^{ - \zeta } )^{\frac{1}{2}} \sqrt {\frac{\pi }{n}} \\ &\sim 2^{2n+1}{\rm e}^{ - \left( {n + \frac{1}{2}} \right)\zeta } (1 - {\rm e}^{ - \zeta } )^{ - \frac{3}{2}} (1 + {\rm e}^{ - \zeta } )^{\frac{1}{2}} , \end{align*} as $n\to+\infty$. Here $\zeta \in \left\{\xi: \operatorname{Re}(\xi)\ge 0,\, |\operatorname{Im}(\xi)|\le \pi \right\}$. Consequently, \begin{align*} {}_2F_1\!\left( n + \tfrac{1}{2},n - \tfrac{1}{2};2n + 1; - \operatorname{csch}^2 \!\big( \tfrac{\zeta }{2}\big) \right) & \sim \Big( {2\,{\rm e}^{ - \frac{\zeta }{2}} \sinh\! \big( \tfrac{\zeta }{2} \big)} \Big)^{2n + 1} (1 - {\rm e}^{ - \zeta } )^{ - \frac{3}{2}} (1 + {\rm e}^{ - \zeta } )^{\frac{1}{2}} \\ & = \coth ^{\frac{1}{2}} \!\big( \tfrac{\zeta }{2} \big)(1 - {\rm e}^{ - \zeta } )^{2n} \end{align*} as $n\to+\infty$. If we set $w=- \operatorname{csch}^2 \!\big( \tfrac{\zeta }{2}\big)$, then the domain of validity corresponds to $\left| \arg ( - w) \right| \le \pi$: $$ {}_2F_1\!\left( n + \tfrac{1}{2},n - \tfrac{1}{2};2n + 1; w \right) \sim (1 - w)^{1/4} \left( \tfrac{1}{2} + \tfrac{1}{2}\sqrt {1 - w} \right)^{ - 2n}. $$