On some solvable systems of difference equations

Abstract

We show that the following systems of difference equations xn+ 1= un1+ vn,yn+ 1= wn1+ sn,n∈ N0, where u n, v n, w n, s n are some of the sequences x n or y n, with real initial values x 0 and y 0, are solvable in fourteen out of sixteen possible cases. Two cases are left unsolved. Probably the most interesting is the result in the case u n = x n, v n = x n, w n = x n, s n = y n, where a fascinating formula is obtained in an elegant way by using some ad hoc ideas.