Unimodal sequences and quantum and mock modular forms

• Type of Material: Article
• Authors: Jennifer Bryson, Texas A&M University; Ken Ono, Emory University; Sarah Pitman, Emory University; Robert C. Rhoades, Stanford University
• Language: English
• Date: 2012-10-02
• Publisher: National Academy of Sciences
• Publication Version: Final Published Version
• Copyright Statement: Copyright Bryson et al.
• Final Published Version (URL): http://dx.doi.org/10.1073/pnas.1211964109
• Title of Journal or Parent Work: Proceedings of the National Academy of Sciences
• ISSN: 0027-8424
• Volume: 109
• Issue: 40
• Start Page: 16063
• End Page: 16067
• Grant/Funding Information: The authors thank the National Science Foundation and the Asa Griggs Candler Fund for their generous support.
• Research Categories: Mathematics; Physics, General

Abstract

We show that the rank generating function U(t; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U(-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F(q). As a result, we obtain a new representation for a certain generating function for L-values. The series U(i; q) = U(-i; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.