Zeitschrift f¨ ur Analysis und ihre Anwendungen c  European Mathematical Society Journal for Analysis and its Applications Volume 26 (2007), 481–494 Linear q -Difference Equations M. H. Abu Risha, M. H. Annaby, M. E. H. Ismail and Z. S. Mansour 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. Keywords. q-Difference equations, q-Wronskian, q-type Liouville’s formula Mathematics Subject Classification (2000). Primary 39A13, secondary 15A03, 34A30 1. Introduction and basic definitions In the following, q is a positive number, 0 <q< 1, and I is an open interval containing zero. Now we state the basic definitions used in this article, cf. [4, 9]. Then we introduce a brief account about the q-calculus established in [3]. Let n ∈ N. The q-shifted factorial (a; q) n of a ∈ C is defined by (a; q) 0 := 1 and, for n> 0, (a; q) n := n Y k=1 ( 1 - aq k-1 ) . The multiple q-shifted factorial for complex numbers a 1 ,...,a k is defined by (a 1 ,a 2 ,...,a k ; q) n := k Y j =1 (a j ; q) n . M. H. Abu Risha, M. H. Annaby, Z. S. Mansour: Department of Mathematics, Cairo University, Faculty of Science, Giza, Egypt; moemenha@yahoo.com, mhannaby@yahoo.com, zeinabs98@hotmail.com M. E. H. Ismail: Department of Mathematics, University of Central Florida, Orlando, Florida 33620-5700; ismail@math.ucf.edu