On some solvable systems of difference equations

摘要

We show that the following systems of difference equationsxn+1=un1+vn,yn+1=wn1+sn,n∈N0,where un, vn, wn, sn are some of the sequences xn or yn, with real initial values x0 and y0, are solvable in fourteen out of sixteen possible cases. Two cases are left unsolved. Probably the most interesting is the result in the case un = xn, vn = xn, wn = xn, sn = yn, where a fascinating formula is obtained in an elegant way by using some ad hoc ideas.