I recently came across the following function which intrigues me: $$f(\alpha):=\sum_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}.$$ 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?

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).

I don't know if this helps, but $g(\alpha,t) = \sum_{i=0}^\infty \alpha^{i(i+1)/2} t^{i}$ is related to Jacobi Theta functions, and $f(\alpha)$ would be the Borel transform of this with respect to $t$, evaluated at $t=1$.