On the Dynamics of the Recursive Sequence xn+1 = α + xn-kp/xnq

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation x(n+1) = alpha + x(n-k)(p)/x(n)(q), n = 0, 1, . . . , where k is an element of{2, 3 . . .}, alpha, p, q, is an element of (0,infinity) and the initial conditions x(-k), . . . , x(0) are arbitrary positive numbers. We investigate the boundedness character for p is an element of (0,infinity). Also, we investigate the existence of a prime two periodic solution for k is odd. Moreover, when k is even, we prove that there are no prime two periodic solutions of the equation above.