I'm looking at hypergeometric functions at the moment in relation to the $n$'th term of the Taylor series of $\sqrt{1-a x^2}$. From this consideration a $_{2}F_{1}$ arises that I'd like to approximate using for example techniques outlined in this answer (or similar).
The specific case of interest is $_{2}F_{1}\left(\frac12 - \frac n2, -\frac n2 ,\frac32 - n,z\right)$ where $-\frac1z\rightarrow0$ and $n\in\mathbb{Z}^+$.
Finding a useful transformation to something of an integral representation does not seem straightforward: reduction and integral formulas always seem to divide by $\Gamma(b)$, $\Gamma(c)$, or spawn Pochhammer symbols that I can't justify due to negative integer values appearing in the Gamma functions through which I will divide, leading to divergences.
Something close to a solution that I could see would be the Gegenbauer polynomial double $\nu$ relations inspired by this answer: \begin{align} C_{2\nu}^{(\lambda)}(x) &= \binom{2(\nu+\lambda)-1}{2\nu} {}_2F_{1}\left(-\nu,\nu+\lambda,\lambda+\frac12,1-x^2\right) \end{align} with the obvious caveat that $c=1+a+b$ such that the exact parameters of the ${}_2F_1$ are not satisfied. Parameter $\lambda$ may become negative with the extension mentioned in this paper so this should not be a problem.
Something else that came close was the limit representation from corollary 6.1 in this paper: \begin{align} \lim_{q\rightarrow-\gamma} \frac{{}_2F_1\left(\alpha,\beta,q,x\right)}{\Gamma(q)} &= x^{\gamma+1} \frac{(\alpha)_{\gamma+1}(\beta)_{\gamma+1}}{(\gamma+1)!} {}_2F_1\left(\alpha+\gamma+1,\beta+\gamma+1,\gamma+2,x\right) \end{align}
So just to be clear: the goal is to find an asymptotic approximation to the aforementioned ${}_2F_1$ with the lowest error possible. I myself thought the integral representation would lend itself to that the best, but I'm very much open to other suggestions. (Expanding the hypergeometric function to first order in $1/z$ is too imprecise to me and I feel like I'm leaving something on the table that way.)
Partial answer but too long for comment.
Using the following relations from Prudnikov the hypergeometric equation can be rewritten into Gamma functions and associated Legendre polynomials. Some of the identities I used include the following: \begin{align} \begin{split} {}_2F_{1}\left(a,b,c,z\right) &= \frac{\Gamma\left(c\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right)\Gamma\left(c-a\right)}(-z)^{-a} {}_2F_{1}\left(a,1+a-c,1+a-b,\frac1z\right) +\cdots\\ &\phantom=\cdots+\frac{\Gamma\left(c\right)\Gamma\left(a-b\right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}(-z)^{-b} {}_2F_{1}\left(b,1+b-c,1+b-a,\frac1z\right) \end{split}\\ \begin{split} {}_2F_{1}\left(a,b,\frac12,z\right) &= \frac{2^{a-b-1}}{\sqrt{\pi}}\Gamma\left(a+\frac12\right) \Gamma\left(1-b\right) (1-z)^{-(a+b)/2}\times\cdots\\ &\phantom=\cdots\times\left[P^{b-a}_{a+b-1}\left(-\sqrt{\frac{z}{z-1}}\right) + P^{b-a}_{a+b-1}\left(\sqrt{\frac{z}{z-1}}\right)\right] \end{split}\\ \begin{split} {}_2F_{1}\left(a,b,\frac32,z\right) &= \frac{2^{a-b-5/2}}{\sqrt{\pi z}}\Gamma\left(a-\frac12\right) \Gamma\left(b-\frac12\right) (1-z)^{3/4-(a+b)/2}\times\cdots\\ &\phantom=\cdots\times\left[P^{3/2-(a+b)}_{a-b-1/2}\left(-\sqrt{z}\right) - P^{3/2-(a+b)}_{a-b-1/2}\left(\sqrt{z}\right)\right] \end{split} \end{align} which can be found in Prudnikov et al. Integrals and Series Vol.3, page 454 equation 6 and page 458 equations 74 through 78. Note that the above relations are subject to conditions mentioned there.
I may update this answer, going in a different direction without the associated Legendre polynomials if the asymptotics turn out a little nicer.