The periodicity and solutions of the rational difference equation with periodic coefficients

In this paper, we give necessary and sufficient conditions for generalized solution and periodicity of the difference equation xn+1=pnxn−k+xn−(k+1)qn+xn−(k+1) with (k+2)(k+2)-periodic coefficients, where k∈Nk∈N, x−k−1,x−k,⋯,x0∈Rx−k−1,x−k,⋯,x0∈R. Also, we obtain that the generalized solution is periodic with (k+1)(k+1)-period.