RANK GENERATING FUNCTIONS FOR ODD-BALANCED UNIMODAL SEQUENCES, QUANTUM JACOBI FORMS, AND MOCK JACOBI FORMS

Let mu(m, n) (respectively, eta(m, n)) denote the number of odd-balanced unimodal sequences of size 2n and rank m with even parts congruent to 2 mod 4 (respectively, 0 mod 4) and odd parts at most half the peak. We prove that two-variable generating functions for mu(m, n) and eta(m, n) are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single C-infinity function in R x R to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables w and q, how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size 2n with even parts congruent to 0 mod 4 and odd parts at most half the peak.