Vol. 274, No. 1, 2015

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Unimodal sequences and “strange” functions: a family of quantum modular forms

Kathrin Bringmann, Amanda Folsom and Robert C. Rhoades

Vol. 274 (2015), No. 1, 1–25
Abstract

We construct an infinite family of quantum modular forms from combinatorial rank “moment” generating functions for strongly unimodal sequences. The first member of this family is Kontsevich’s “strange” function studied by Zagier. These results rely upon the theory of mock Jacobi forms. As a corollary, we exploit the quantum and mock modular properties of these combinatorial functions in order to obtain asymptotic expansions.

Keywords
quantum modular forms, mock modular forms, Jacobi forms, unimodal sequences, partitions, asymptotics, moment generating functions
Mathematical Subject Classification 2010
Primary: 11F99
Secondary: 11F37, 33D15
Milestones
Received: 13 November 2013
Revised: 25 April 2014
Accepted: 29 April 2014
Published: 2 March 2015
Authors
Kathrin Bringmann
Mathematical Institute
University of Cologne
Weyertal, 86-90
D-50931 Köln
Germany
Amanda Folsom
Mathematics
Yale University
P.O. Box 208283
New Haven, CT 06520-8283
United States
Robert C. Rhoades
Center for Communications Research
805 Bunn Drive
Princeton, NJ 08540
United States