Behaviour of the series $\exp_p(x)=\sum_{k=0}^{\infty}\frac{x^k}{(k!)^p}$ depending on $p\approx 2$?
Solution 1:
For p = 2, $exp_2(x) = J_0(2 \sqrt{x})$, where $J_0$ is a Bessel function. The asymptotics of this are known: as $x \to -\infty$, $exp_2(x) = \frac {\sin \left( 2\,\sqrt {x}+1/4\,\pi \right) }{\sqrt {\pi }\, x^{1/4}} + O(x^{-3/4})$.