On a system of difference equations of third order solved in closed form

Abstract

In this work, we show that the system of difference equations xn+1 = ayn􀀀2xn􀀀1yn + bxn􀀀1yn􀀀2 + cyn􀀀2 + d yn􀀀2xn􀀀1yn , yn+1 = axn􀀀2yn􀀀1xn + byn􀀀1xn􀀀2 + cxn􀀀2 + d xn􀀀2yn􀀀1xn , where n 2 N0, x􀀀2, x􀀀1, x0, y􀀀2, y􀀀1 and y0 are arbitrary nonzero real numbers and a, b, c and d are arbitrary real numbers with d , 0, can be solved in a closed form. We will see that when a = b = c = d = 1 the solutions are expressed using the famous Tetranacci numbers. In particular, the results obtained here extend those in our recent work.