I have stuck on an asymptotic expansion question for the Gaussian Hypergeometric function for large $a$ and bounded $z$ as the following form with $\lambda$ a constant.
\begin{equation} {}_2F_{1}(a,a+\lambda;a+2\lambda;-z) \tag{1} \end{equation}
I have found this reference that may be related to my question,
but they consider a question like this
\begin{equation} {}_2F_{1}(a+e_{1}\lambda,a+e_{2}\lambda;a+e_{3}\lambda;-z) \end{equation}
where
\begin{equation} e_{j}\in {0,1,-1},j=1,2,3. \end{equation}
So if we could investigate the asymptotic behavior of the above expression $(1)$ for sufficienty large $a$ and bounded $z$?
Thanks a lot for any helpful suggestions!
Best regareds, Liu !
Just one more thing, the worlframealpha tells me that for sufficiently large $a$, the following expression tends to zero for $a$ large but bounded $z$. \begin{equation} \frac{\Gamma(a+\lambda )2^{a-1}}{2az^{-a}}{}_2F_{1}(a,a+\lambda,a+2\lambda,-z)\tag{2} \end{equation} where $\Gamma$ denotes the gamma function.
And here is a WorlframAlpha example of the above integral,
I set $a=32768/2$ and $z=4*6.5^2$ and $\lambda=1/2$ and wa show me that the integral equals $0$.
I will assume that $|\arg(1+z)|<\pi$ and that $\lambda\in \mathbb{C}$ is fixed. By the known linear transformation, $$ {}_2F_1(a,a + \lambda ;a + 2\lambda ; - z) = (1 + z)^{\lambda - a} {}_2F_1 (2\lambda ,\lambda ;a + 2\lambda ; - z) $$ Hence, by $(15.12.3)$, $$ {}_2F_1 (a,a + \lambda ;a + 2\lambda ; - z) \sim (1 + z)^{\lambda - a} \frac{{\Gamma (a + 2\lambda )}}{{a^\lambda \Gamma (a + \lambda )}}\sum\limits_{n = 0}^\infty {\frac{{q_n ( - z)(\lambda )_n }}{{a^n }}} \tag{1} $$ as $a\to \infty$ in $|\arg a|\le \frac{\pi}{2}-\delta<\frac{\pi}{2}$, where $(\lambda)_n$ is the Pochhammer symbol and the $q_n ( - z)$ are given by the generating function $$ \left( {\frac{{{\rm e}^t - 1}}{t}} \right)^{\lambda - 1} {\rm e}^{t(1 - 2\lambda )} (1 + z - z{\rm e}^{ - t} )^{ - 2\lambda } = \sum\limits_{n = 0}^\infty q_n ( - z)t^n . $$ You may simplify the expansion $(1)$ further by employing the known asymptotic expansion for the ratio of two gamma functions. If $\operatorname{Re}(z)\ge-\frac{1}{2}$, the expansion $(1)$ applies in the larger sector $|\arg a|\le \pi-\delta<\pi$.