I recently came across the following function which intrigues me:
\begin{equation}
f(\alpha):=\sum\_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}.
\end{equation}
For $-1\leq \alpha\leq 1$ this function is well defined. Moreover, $f(1)=e$ and $f(-1)=\cos(1)-\sin(1)$. However, I have been unable to find any literature on this function. Does someone know if this a special case of some class of functions or has it been studied in some way?



What is known about this series?
The function $$f\_\alpha(z)=\sum\_{n=0}^\infty \frac{\alpha^{n(n+1)/2}z^n}{n!}$$
was much studied, though it does not have a common name. It is indeed the Borel transform
of an incomplete theta function. This function solves the functional equation
$$f'(z)=\sqrt{\alpha}f(z\sqrt{\alpha}),\quad f(0)=1.$$
Alan Sokal posted 4 lectures on this function <http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/>
which contain a survey of what is known.


When $\alpha$ is a root of unity, this is a trigonometric polynomial, as you noticed 
for $\alpha=1,-1$. For $|\alpha|=1$ but not a root of unity, it was recently studied, for example in MR2392816.


(I clearly remember that this question has already been asked on MO, and I answered. But I cannot find it).


