On some systems of difference equations

Abstract

In this note we show that the following systems of difference equations un+1 = vn/1+ vn, vn+1 = u n/1 + un, un+1 = vn/1+ u n, vn+1 = un/1 + vn, un+1 = un/1+ vn, vn+1 = vn/1 + u n with n∈ℕ0 and complex initial values u oand vo, are solvable explicitly by means of Riccati equations, and based on the formulae for the general solutions we present the behaviour of their solutions as n→∞. While the result is natural for the first system, it is a bit surprising for the other two systems.