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

TL;DR

This paper proves Hikami’s conjecture on the radial limits of certain $q$-hypergeometric false theta functions by leveraging Bailey pairs. Using a specific Bailey pair relative to $(q,q)$ and the Bailey lemma, it shows that the radial limits at roots of unity are obtained by evaluating truncated series, formalized as $\lim_{q\to\zeta_N^M}\widetilde{\Phi}_m^{(a)}(q) = {\zeta_N}^{\frac{M(m-a-1)^2}{4m}} Y_{m,N}^{(a)}(\zeta_N^M)$, and extends this framework to additional families of false theta functions. The paper also provides three explicit examples with corresponding truncations, establishing analogous radial-limit formulas and enriching the toolkit for quantum modular-type identities derived from Bailey pairs. Overall, the work connects radial limits of $q$-series to finite truncations and suggests deep links with quantum invariants such as the Kashaev invariants of torus links. This Bailey-pair based method offers a unified approach to generate and verify similar radial-limit identities for a broad class of false theta functions.

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.

Theorems & Definitions (28)

• Conjecture 1.1: Hikami Hik
• Theorem 1.2
• Lemma 2.1: Hikami
• Lemma 2.2
• proof : Proof of Theorem \ref{['mainthm']}
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Proposition 3.4
• proof