Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity

Abstract

We define odd-balanced unimodal sequences and show that their generating function V ( x , q ) \mathcal {V}(x,q) has the same remarkable features as the generating function for strongly unimodal sequences U ( x , q ) U(x,q) . In particular, we discuss (mixed) mock modularity, quantum modularity, and congruences modulo 2 2 and 4 4 . We also study two related functions which share some of the properties of U ( x , q ) U(x,q) and V ( x , q ) \mathcal {V}(x,q) .