I recently encountered the following function


$$
f(t) = \sum\_{n=0}^\infty \frac{t^{n^2}}{n^2!}.
$$


It seems familiar, though I cannot remember where I might have seen it before. I would like to know in what text, if any, it has been studied. Above all, I would like to know its asymptotic properties (how fast it grows) as $t$ tends to infinity ($t$ real and positive).


Thanks,


G.



Is this series well known?
This is a slice of the taylor series for $\exp(t)$. The terms that dominate are those near $n=t^{1/2}$. Using Stirling's approximation, as $t\to\infty$ with $q$ more or less bounded, we have
$$\frac{t^{(t^{1/2}+q)^2}}{(t^{1/2}+q)^2)!} = \frac{e^{t-2q^2}}{\sqrt{2\pi t}} (1 + O(q^3/t)).$$
So $$f(t) \sim \frac{e^t}{\sqrt{2\pi t}}c(t),$$ where
$$c(t) = \sum\_q e^{-2q^2},$$
with the sum over all $q$ for which $t^{1/2}+q$ is integer.


It is clear that $c(t)$ depends only on the fractional part of $t^{1/2}$. It is nice curve that oscillates between 1.2353 (when $t^{1/2}$ is midway between two integers) and 1.2679 (when $t^{1/2}$ is an integer).


