Bailey pairs, radial limits of $q$-hypergeometric false theta functions, and a conjecture of Hikami

Abstract

In the first part of this paper we prove a conjecture of Hikami on the values of the radial limits of a family of $q$-hypergeometric false theta functions. Hikami conjectured that the radial limits are obtained by evaluating a truncated version of the series. He proved a special case of his conjecture by computing the Kashaev invariant of certain torus links in two different ways. We prove the full conjecture using Bailey pairs. In the second part of the paper we show how the framework of Bailey pairs leads to further results of this type.