It is simple to see that the following series converges absolutely and uniformly on $\mathcal{H}$ for all k positive:

$F_{2k}(z) = \sum_{n \in \mathbb{Z}} q^{n^{2k}}$

And this series being a generating function for higher degree forms is like a theta series analogue. But because the Fourier transform of the said function is not so nicely behaved ( the best I could compute required some very bad hyper geometric series), we do not get a ‘Modular form’ like symmetry.

So instead of using the Fourier transform, and then the poisson summation formula to get a symmetry on the Fourier series, is it possible to use some other integral transform, and then use the corresponding eigenfunction series based on that transform to get a ‘nice’ generating function with some transformation properties?

If not, is it possible to study the obstructions that one faces when trying to find the symmetries of a generating function like that?

(p.s. this question arose from a comment by D. Zagier in one of his lectures that every generating function is in some or the other form related to some Modular form, but I suppose we’ll need something more general than just Modular forms to study such functions)