Quantum Jacobi forms and balanced unimodal sequences

The notion of a quantum Jacobi form was defined in 2016 by Bringmann and the second author in [1], marrying Zagier's notion of a quantum modular form [12] with that of a Jacobi form. Only one example of such a function has been given to-date (see [1]). Here, we prove that two combinatorial rank generating functions for certain balanced unimodal sequences, studied previously by Kim, Lim and Lovejoy [8], are also natural examples of quantum Jacobi forms. These two combinatorial functions are also duals to partial theta functions studied by Ramanujan. Additionally, we prove that these two functions have the stronger property that they exhibit mock Jacobi transformations in C×H as well as quantum Jacobi transformations in Q×Q. As corollaries to these results, we use quantum Jacobi properties to establish new, simpler expressions for these functions as simple Laurent polynomials when evaluated at pairs of rational numbers.