Analytical Study of a Class of Rational Difference Equations

We obtain the solution of the fourth order difference equation $$x_{n+1}=\frac{ \alpha x_{n-3}}{A+B x_{n-1}x_{n-3}}$$ with the initial conditions; $x_{-3}=d,$ $x_{-2}=c,$ $x_{-1}=b,$ and $x_{0}=a$ are arbitrary nonzero real numbers, $\alpha$, $A$ and $B$ are arbitrary constants. The result is used to study the convergence of solutions, the existence of unbounded solutions and the convergence to periodic solutions. We illustrate the results by several numerical examples.