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 $I_{\mathrm{\Delta }}(m,e)$, and Hahn–Exton’s $q$-analogue of the Bessel function $J_{\nu }(z;q)$. The correspondence between $I_{\mathrm{\Delta }}$ 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.