Solvability of a general class of two-dimensional hyperbolic-cotangent-type systems of difference equations

Abstract

We show that the following class of two-dimensional hyperbolic-cotangent-type systems of difference equations xn+1=un−kvn−l+aun−k+vn−l,yn+1=wn−ksn−l+awn−k+sn−l,n∈N0, where k, l∈ N, a∈ C, u−j, w−j∈ C, j= 1 , k‾ , v−j′, s−j′, j′= 1 , l‾ , and each of the sequences un, vn, wn, sn is equal to xn or yn, is theoretically solvable. When k= 0 and l= 1 , we show that the systems are practically solvable by presenting closed-form formulas for their solutions. To do this, we employ a constructive method, which is possible to use on the complex domain, presenting in this way a new and elegant solution to the problem in this case, and giving a hint how such type of systems can be solved.