“Strange” combinatorial quantum modular forms

Abstract

Motivated by the problem of finding explicit q-hypergeometric series which give rise to quantum modular forms, we define a natural generalization of Kontsevich's “strange” function. We prove that our generalized strange function can be used to produce infinite families of quantum modular forms. We do not use the theory of mock modular forms to do so. Moreover, we show how our generalized strange function relates to the generating function for ranks of strongly unimodal sequences both polynomially, and when specialized on certain open sets in C. As corollaries, we reinterpret a theorem due to Folsom–Ono–Rhoades on Ramanujan's radial limits of mock theta functions in terms of our generalized strange function, and establish a related Hecke-type identity.