On the Rational Recursive Sequence x_{n+1}=Ax_{n}+Bx_{n-k}+frac{beta x_{n}+gamma x_{n-k}}{Cx_{n}+Dx_{n-k}}

In this article, we study the global and asymptotic properties of the solutions of the difference equation $$x_{n+1}=Ax_{n}+Bx_{n-k}+(beta x_{n}+gamma x_{n-k})/(Cx_{n}+Dx_{n-k}),quad n=0,1,2,ldots,$$ where the initial conditions x _?k ,…,x _?1,x ₀ are arbitrary positive real numbers and the coefficients A,B,C,D,β and γ are positive constants, while k is a positive integer number. Some numerical examples will be given to illustrate our results.