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Taylor coefficients of mock-Jacobi forms and moments of partition statistics

Published online by Cambridge University Press:  09 July 2014

KATHRIN BRINGMANN ,
KARL MAHLBURG  and
ROBERT C. RHOADES
KATHRIN BRINGMANN
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany e-mail kbringma@math.uni-koeln.de
KARL MAHLBURG
Affiliation:
Department of Mathematics, Princeton University, NJ, U.S.A. e-mail mahlburg@math.princeton.edu
ROBERT C. RHOADES
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail rhoades@math.stanford.edu
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Abstract

We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.

As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 2 , September 2014 , pp. 231 - 251
Copyright
Copyright © Cambridge Philosophical Society 2014 

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