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

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.

12 pages, 2 figures, comments are welcome!