Note on the bilinear difference equation with a delay

Abstract

A representation formula for the general solution to a higher-order rational difference equation has been given recently in this journal. The formula was proved by the method of induction without giving any theoretical explanation related to it. Here we show how the corresponding representation formula for the general solution to a general higher-order rational difference equation, that is, to the bilinear difference equation with a delay, is obtained in an elegant way. We also give some theoretical explanations related to the representation, as well as some explanations related to such types of difference equations. The corresponding representation for a system of bilinear difference equations with delay is also presented.