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

DOI: 10.2140/pjm.2015.274.1

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