2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/225799We present an explicit expression for the $q$-difference system, which the $BC_1$-type Jackson integral ($q$-series) satisfies, as first order simultaneous $q$-difference equations with a concrete basis. As an application, we give a simple proof for the hypergeometric summation formula introduced by Gustafson and the product formula of the $q$-integral introduced by Nassrallah-Rahman and Gustafson.Classical Analysis and ODEstext