1. Ramanujan and coefficients of meromorphic modular forms.
- Author
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Bringmann, Kathrin and Kane, Ben
- Subjects
- *
MODULAR forms , *COEFFICIENTS (Statistics) , *MEROMORPHIC functions , *EISENSTEIN series , *MATHEMATICAL expansion - 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 Poincaré series introduced by Petersson and a new family of Poincaré 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. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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