Asymptotic behaviour of generalized hypergeometric function

I have been attempting to study the following hypergeometric function when $\lambda\gg1$ for a while: $$ _{1}F_{2}(-\lambda;3,1,-x\lambda^2) $$ with $x\in\mathbb{R}$. All the asymptotic studies available for the generalized hypergeometric functions don't focus on the polynomial case, i.e. only consider the case where the factors are positive ($a_p>0$).

Solution 1:

It's not going to be pretty. Let $x > 0$, then $${_1 \hspace {-1px} F_2}(-\lambda; 1, 3; -\lambda^2 x) = \sum_{k \geq 0} \frac {(1 - k + \lambda)_k} {(1)_k \, (3)_k} \frac {(\lambda^2 x)^k} {k!}.$$ If $k = \alpha \lambda$, then the summand is asymptotic to $f(\alpha) e^{\lambda \phi(\alpha)}$ with $$ f(\alpha) = (2 \pi^3 \alpha^7 (1 - \alpha) \lambda^7)^{-1/2}, \\ \phi(\alpha) = (\alpha - 1) \ln(1 - \alpha) + \alpha \ln (e^2 \alpha^{-3} x)$$ for large $\lambda$ and $0 < \alpha < 1$. The maximum of $\phi$ is located at $\alpha_0 \in (0, 1)$ s.t. $\alpha_0^3 + x \alpha_0 - x = 0$. The sum over $k \geq \lambda$ is negligible. Then, same as here or here, $${_1 \hspace {-1px} F_2}(-\lambda; 1, 3; -\lambda^2 x) \sim \lambda f(\alpha_0) e^{\lambda \phi(\alpha_0)} \int_{\mathbb R} e^{\lambda \phi''(\alpha_0) \alpha^2/2} d\alpha = \\ \frac {e^{2 \alpha_0 \lambda} (\alpha_0^2 + \alpha_0 + x + 1)^\lambda} {\pi x \lambda^3 \sqrt {7 \alpha_0^2 + 2 (x - 4) \alpha_0 - 2 x + 3}}, \quad \lambda \to \infty$$ ($\lambda$ does not have to be an integer). The case $x < 0$ seems more difficult because the hypergeometric series is alternating near the maximum.