Ramanujan and coefficients of meromorphic modular forms

TL;DR

This paper explores Fourier coefficients of meromorphic modular forms, building on Ramanujan's classical work, by introducing new Poincaré series and establishing explicit identities, including for forms with higher-order poles.

Contribution

It introduces new Poincaré series and provides the first explicit Fourier expansions for meromorphic forms with higher-order poles, expanding the understanding of their coefficients.

Findings

Constructed new Poincaré series of independent interest.

Established explicit identities for Fourier coefficients.

Provided first examples of Fourier expansions for forms with higher-order poles.

Abstract

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincar\'e series introduced by Petersson and a new family of Poincar\'e series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.