2

I usually provide more details to the questions I ask on this site, but for this specific question I can't even wrap my head around how to even attempt solving it. By the way, this question has no real background or motivation besides pure curiosity on my side.

I'm trying to gain insight into whether it is possible to express the following family of infinite series in a closed form:

$$S_n(x) = \sum_{k=0}^{\infty}x^{k^n}$$

The two trivial series of this form are the cases for which $n=0$ and $n=1$, as $S_0(x)$ clearly diverges for all $x$ except $x=0$, and $S_1(x)=\frac{1}{1-x}$ is the standard geometric series, which converges for all $x$ such that $|x|<1$.

To my surprise, wolframalpha was able to compute the closed form for $n=2$ as well, and it is the following:

$$S_2(x) = \sum_{k=0}^{\infty}x^{k^2}=\frac{1+\vartheta_3(0,x)}{2}$$

where $\vartheta_a(y,q)$ is the elliptic theta function.

Is it possible to find a closed form for any other members of this family of infinite series, besides just $S_1$ and $S_2$? If so, how? What knowledge do I need to possess to arrive at such closed forms?

I can already say at the very start, that I have no clue what the theta elliptic functions are, and how to operate with them. Intuition is telling me that closed forms for other values of $n$ will be even more "non-elementary".

KKZiomek
  • 4,012
  • Possibly relevant: Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^k}$ – Dave L. Renfro May 19 '20 at 18:55
  • 2
    The identity for the theta function is a definition, hardly a closed form. That's like saying that $\zeta(s)$ is a closed form for $\sum n^{-s}$. The fact that theta functions turn out to have special symmetries (and can even be evaluated for certain special $x$) is quite deep. Still, there are many reasons why the case $n=2$ is special relative to $n > 2$, related to the fact that $e^{-x^2}$ is (almost) its own Fourier transform. As for $S_n$ for $n > 2$, well, there isn't any particular reason to expect them to have any nice properties at all. – user760870 May 19 '20 at 19:06
  • As to: "What knowledge do I need to possess to arrive at such closed forms?" there are two answers. One possibility is that these functions are pretty much random and one can't say anything intrinsic about them (rather than trivialities like $\int_{0}^{1} (S_3(x) -1) dx/x = \zeta(3)$, for example). The other is that they have amazing properties that nobody has discovered, so the only knowledge you require is an insight that has eluded mathematicians in the two millennia or so since they considered infinite sums in one form or another. – user760870 May 19 '20 at 19:09

0 Answers0