# Linear $q$-Difference Equations

### M.H. Abu Risha

Cairo University, Egypt### M.H. Annaby

Cairo University, Egypt### Z.S. Mansour

Cairo University, Egypt### Mourad E. H. Ismail

University of Central Florida, Orlando, United States

## 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.

## Cite this article

M.H. Abu Risha, M.H. Annaby, Z.S. Mansour, Mourad E. H. Ismail, Linear $q$-Difference Equations. Z. Anal. Anwend. 26 (2007), no. 4, pp. 481–494

DOI 10.4171/ZAA/1338