Has there been any exploration on Cubic Power Series?

Question

I was interested in finding some identities/special values involving the function $$\gamma(z) = \sum_{i=0}^{\infty} z^{i^3} = 1 + z + z^8 + z^{27} + ... $$

which can be thought of as a "cubic generalization of the famous $$\theta(z) = \sum_{i=0}^{\infty} z^{i^2} = 1 + z + z^4 + z^9 + ... $$

Which can be cooked up using jacobi theta functions.

Unfortunately the term "cubic theta function" doesn't lead to any insight on this series since the "cubic" is reserved for a type of identity as opposed to the form of the series.

Surely these have been looked at before does anyone have any links/intel about them?

See some posts in https://approach0.xyz/search/?q=%24%5Csum_%7Bi%3D0%7D%5E%7B%5Cinfty%7D%20z%5E%7Bi%5E3%7D%24&p=1 — Sil, Nov 27 '20 at 21:38
The Mellin transform of $\gamma(e^{-x})$ is $\zeta(3s)\Gamma(s)$, the equivalent of $\theta(e^{-\pi x})=x^{-1/2}\theta(e^{-\pi/x})$ involves a convolution. — reuns, Nov 27 '20 at 23:28

score 1 · Answer 1 · answered Aug 09 '22 at 01:25

1

Results found here: https://mathoverflow.net/questions/427918/is-anything-known-about-the-series-sum-n-0-infty-x-sqrtn are general enough to be applied to the cubic power series as well

answered Aug 09 '22 at 01:25

Sidharth Ghoshal

18,359

Has there been any exploration on Cubic Power Series?

1 Answers1