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

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.