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Universal mock theta functions as quantum Jacobi forms

  • Greg Carroll
  • James Corbett
  • Amanda FolsomEmail author
  • Ellie Thieu
Research
  • 64 Downloads

Abstract

Quantum Jacobi forms were defined in 2016, naturally combining Zagier’s definition of a quantum modular form with that of a Jacobi form. To date, just three examples of such functions exist in the literature. Here, we prove that the universal mock theta function \(g_2\), as well as the universal mock theta functions \(K, K_1, K_2,\) and \(\kappa \), gives rise to an infinite family of quantum Jacobi forms \(G_{a,b}(z;\tau )\) of weight 1 / 2 in dense subsets \({{\mathscr {Q}}}_{a,b} \subseteq {\mathbb {Q}} \times {\mathbb {Q}}\). We then use these quantum Jacobi transformation properties to establish polynomial expressions for \(G_{a,b}\) at pairs of rational numbers, as well as simple closed-form expressions for sums of Mordell integrals.

Mathematics Subject Classification

11F03 11F37 11F50 11F99 33D70 

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Copyright information

© SpringerNature 2018

Authors and Affiliations

  • Greg Carroll
    • 1
  • James Corbett
    • 1
  • Amanda Folsom
    • 1
    Email author
  • Ellie Thieu
    • 1
  1. 1.Department of Mathematics and StatisticsAmherst CollegeAmherstUSA

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