Sequences Defined by Non-Linear Algebraic Difference Equations

Abstract

when the values of y(x) are assigned at the m points a, a + 1, * * *, a + m -1. Obviously every equation (1) defines an infinite number of sequences: one for each set of values y(a), y(a + 1), * * *, y(a + m 1). We propose to obtain some of the properties of such sequences by studying the difference equations which define them. In the study of infinite sequences one is mainly interested in their ultimate behavior. Hence we are here interested in the behavior of the solutions of the difference equations, for integral values of x, in the neighborhood of infinity. Although, at present, it is not possible to write out explicitly the solutions of most non-linear difference equations, it seems that it might be feasible to determine whether a given equation defines sequences that approach zero as x becomes infinite by considering the solutions of the linear difference equation formed by omitting the non-linear terms. (We assume that at least one linear term appears.) For when y(x) approaches zero the linear terms are infinitesimals of the first order, while the non-linear terms are infinitesimals of higher order. Therefore, it would be expected that the behavior of such sequences is largely determined by the linear terms of the difference equation. With this idea in mind, we attempt to gain information about the sequences defined by a non-linear difference equation by considering the solution of the difference equation formed from its linear terms.3 It is clear that once we have criteria for determining when a difference equation defines sequences which approach zero we can easily determine whether one defines sequences that approach a constant limit a. For, if after the application of the transformation y(x) = g(x) + a, the transformed difference equa-