Linear $q$-Difference Equations

Abstract

We prove that a linear $q$-difference equation of order n has a fundamental set of $n$-linearly independent solutions. A $q$-type Wronskian is derived for the $n$th order case extending the results of Swarttouw–Meijer (1994) in the regular case. Fundamental systems of solutions are constructed for the $n$-th order linear $q$-difference equation with constant coefficients. A basic analog of the method of variation of parameters is established.