  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.
