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I have the following entire function defined $\forall x,y \in \mathbb{N}\,\,;\,\,x,y > 1$:

$$f(s) = \sum_{n=0}^{\infty} \frac{s^n}{(x \,[n]\, y)n!}$$

Where here $x\, [n] \,y$ is the n'th hyper operator, so $x\,[0]\,y = x+y$, $x\,[1]\,y = x \cdot y$ and $x\,[2]\,y = x^y$ where generally the pattern to the operators is:

$$x\, [n] \,(x\,[n+1]\,y) = x\,[n+1]\,(y+1)$$ and $$x \,[n]\,1 = x\,\,;\,\,\forall n > 0$$

This sequence of numbers tends to infinity much faster than factorial, double or triple exponential, or pretty much any elementary sequence. So the function f is entire. I'm wondering how I can find if $\lim_{s\to \infty}f(-s) = 0$ or not. Calculating or plotting this function to see evidence of this is impossible because computer memory is insufficient. I am very keen on this function and if it decays to zero I have a use for it. To generalize the question, I am wondering if perhaps there is some criteria an entire function $$g(s) =\sum_{n=0}^{\infty}a_n \frac{s^n}{n!}$$ has on $a_n$ so that $\lim_{x \to \infty} g(-x) = 0$

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