On a class of solvable higher-order difference equations

Abstract

Closed form formulas for well-defined solutions of the next difference equation (Formula prsented), where k ϵ ℕ, (an)nϵℕ0 and initial values x-i, i = (Formula prsented) are real numbers, are given. Long-term behavior of well-defined solutions of the equation when (an)nϵℕ0 and (bn)nϵℕ0are constant sequences is described in detail by using the formulas. We also describe the domain of undefinable solutions of the equation. Our results explain and considerably improve some recent results in the literature.