A high order $q$-difference equation for $q$-Hahn multiple orthogonal polynomials

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to 1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case.