I'm not sure if this will help, but I know some about taylor series and can do some fractional calculus 
(Where can't we do fractional calculus)
If \( \phi \) is holo on \( \Re(z) < 1 \) and satisfies some fast growth at imaginary infinity and negative infinity. And if \( \phi(z) \neq \mathbb{N} \) in the half plane. Then fix \( 1>\tau > 0 \):
\( \frac{1}{2\pi i}\int_{\tau-i\infty}^{\tau+i\infty}\frac{\pi}{\sin(\pi z)\G(\phi(z))} w^{-z}\,dz = \sum_{n=0}^\infty \frac{w^n}{(\phi(-n))!} \)
Maybe that might help some of you? The unfortunate part is as \( w \to \infty \) we're going to get decay to zero. I'm not sure about the iterates. We can also note this is a modified fourier transform and so we can apply some of Paley Wiener's theorems on bounding fourier transforms from the original functions. I.e: We can bound the taylor series by the function in the integral. Therefore maybe if we get very fast decay to zero we can talk about asymptotics of \( 1/\exp^{0.5}(x) \).
(Where can't we do fractional calculus)If \( \phi \) is holo on \( \Re(z) < 1 \) and satisfies some fast growth at imaginary infinity and negative infinity. And if \( \phi(z) \neq \mathbb{N} \) in the half plane. Then fix \( 1>\tau > 0 \):
\( \frac{1}{2\pi i}\int_{\tau-i\infty}^{\tau+i\infty}\frac{\pi}{\sin(\pi z)\G(\phi(z))} w^{-z}\,dz = \sum_{n=0}^\infty \frac{w^n}{(\phi(-n))!} \)
Maybe that might help some of you? The unfortunate part is as \( w \to \infty \) we're going to get decay to zero. I'm not sure about the iterates. We can also note this is a modified fourier transform and so we can apply some of Paley Wiener's theorems on bounding fourier transforms from the original functions. I.e: We can bound the taylor series by the function in the integral. Therefore maybe if we get very fast decay to zero we can talk about asymptotics of \( 1/\exp^{0.5}(x) \).
