A Connection Formula of the Hahn-Exton $q$-Bessel Function

TL;DR

This paper derives a connection formula for the Hahn-Exton $q$-Bessel function relating its behavior near zero and infinity, using $q$-transformations, and examines its limit as $p$ approaches 1.

Contribution

It introduces a new connection formula for the Hahn-Exton $q$-Bessel function using $q$-Borel and $q$-Laplace transformations, extending previous methods.

Findings

Derived a connection formula for the Hahn-Exton $q$-Bessel function.

Analyzed the limit of the connection formula as $p o 1^-$.

Applied $q$-transformations to obtain the connection formula.

Abstract

We show a connection formula of the Hahn-Exton $q$-Bessel function around the origin and the infinity. We introduce the $q$-Borel transformation and the $q$-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit $p\to 1^-$ of the connection formula.