A note on the tetrahedral index and the Hahn-Exton q-Bessel function

TL;DR

This paper explores the relationship between the tetrahedral index in 3D quantum topology and the Hahn-Exton q-Bessel function, introducing new connections, techniques, and conjectures from physical mathematics to the q-hypergeometric community.

Contribution

It establishes a link between the tetrahedral index and the Hahn-Exton q-Bessel function, and introduces new methods and conjectures from physical mathematics to q-hypergeometric studies.

Findings

Identifies connections between the tetrahedral index and q-Bessel functions.

Provides new techniques and conjectures for q-hypergeometric functions.

Facilitates translation of relations between topology and special functions.

Abstract

The purpose of this short note is twofold: First to elucidate some connections between the ``building block'' of Dimofte--Gaiotto--Gukov's $3$D index, known as the tetrahedral index $I_\Delta (m,e)$, and Hahn--Exton's $q$-analogue of the Bessel function $J_\nu (z;q)$. The correspondence between $I_\Delta$ and $J_\nu$ will allow us to translate useful relations from one setting to the other. Second, we want to introduce to the $q$-hypergeometric community some possibly new techniques, theory and conjectures arising from applications of physical mathematics to geometric topology.