Abstract
In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expression for radial limits associated to the partition rank and crank functions previously studied by Ono, Rhoades, and Folsom.
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In honor of Ernst-Ulrich Gekeler
The research of K. Bringmann was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant agreement no. 335220—AQSER. A. Folsom is grateful for the support of NSF CAREER Grant DMS-1449679, and for the hospitality provided by the Max Planck Institute for Mathematics, Bonn, and the Institute for Advanced Study, Princeton, under NSF Grant DMS-1128155.
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Bringmann, K., Folsom, A. Quantum Jacobi forms and finite evaluations of unimodal rank generating functions. Arch. Math. 107, 367–378 (2016). https://doi.org/10.1007/s00013-016-0941-z
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DOI: https://doi.org/10.1007/s00013-016-0941-z