In the course of some calculations, I came across the following powers series. For fixed $C>1$ let $$ f_C(u)=\sum_{k=0}^\infty\frac{u^k}{C^{k^2}}. $$ This series converges for all $u\in\mathbb C$, hence $f_C$ is an entire function. Can it be expressed in terms of classical special functions? Does it satisfy a differential equation? Has anbybody seen this guy somewhere else?
2 Answers
$\begingroup$
$\endgroup$
2
This function has no known expression in terms of common special functions. It is called "partial theta-function", and there was some recent research on it: http://arxiv.org/pdf/1106.6262v1.pdf, http://arxiv.org/pdf/1106.1003.pdf, and literature cited there.
answered Feb 13, 2014 at 14:24
-
$\begingroup$ The corresponding $\sum_{-\infty}^\infty$ would be given in terms of the theta function, as in Carlo's answer. $\endgroup$ Commented Feb 13, 2014 at 14:41
-
$\begingroup$ And what does it tell you about $f_C(u)$? Only the functional equation $f_C(u)+f_C(-u)=1+\theta_3$ whose solution is not unique. $\endgroup$ Commented Feb 14, 2014 at 3:12
$\begingroup$
$\endgroup$
1
for $u=C$ it's an elliptic theta function,
$$f_u(u)=1+\frac{1}{2}u^{1/4}\vartheta_2(0,1/u)$$
more generally $$f_C(e^{2iz})+f_C(e^{-2iz})=1+\sum_{k=-\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$$
answered Feb 13, 2014 at 9:39
-
1$\begingroup$ This is NOT an expression of $f_C(u)$ in terms of theta-function. $\endgroup$ Commented Feb 13, 2014 at 14:25