New method for solving non-homogeneous periodic second-order difference equation and some applications

In the present study, we are interested in solving the nonhomogeneous second-order linear difference equation with periodic coefficients of period \( p\ge 2\), by bringing two new approaches enabling us to provide both analytic and combinatorial solutions to this family of equations. First, we get around the problem by converting this kind of equations to an equivalent family of nonhomogeneous linear difference equations of order p with constant coefficients. Second, we propose new expressions of the solutions of this family of equations, using our techniques of calculating the powers of product of companion matrices and some properties of generalized Fibonacci sequences. The study of the special case \( p=2 \) is provided. And to enhance the effectiveness of our approaches, some numerical examples are discussed.