Ramanujan-like formulas for Fourier coefficients of all meromorphic cusp forms

TL;DR

This paper proves that all negative-weight meromorphic modular forms, including quasi-meromorphic ones, have Fourier expansions similar to classical Ramanujan formulas, extending previous special case results.

Contribution

It generalizes Ramanujan-like Fourier expansion formulas to all negative-weight meromorphic modular forms within the space of polar harmonic Maass forms.

Findings

All negative-weight meromorphic modular forms have Ramanujan-like Fourier expansions.

The results include quasi-meromorphic modular forms as well.

Fourier expansions are valid for forms bounded towards i∞.

Abstract

In this paper, we investigate Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight $6$ Eisenstein series which was computed by Hardy and Ramanujan. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maass forms, we prove in this paper that all negative-weight meromorphic modular forms (and furthermore all quasi-meromorpic modular forms) have Fourier expansions of this type, granted that they are bounded towards $i\infty$.