On a class of solvable difference equations generalizing an iteration process for calculating reciprocals

Abstract

The well-known first-order nonlinear difference equation yn+1=2yn−xyn2,n∈N0, naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.