Ramanujan and coefficients of meromorphic modular forms

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• 作者：Kathrin Bringmann^a ; ¹ ; ^{kbringma@math.uni-koeln.de} ; Ben Kane^b ; ² ; ^{bkane@maths.hku.hk}
• 关键词：11F30 ; 11F37 ; 11F11
• 刊名：Journal de Mathématiques Pures et Appliquées
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：107
• 期：1
• 页码：100-122
• 全文大小：483 K
• 卷排序：107

文摘

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.