  The transformation of the Nth- order linear difference equation into a system
of the first order difference equations is presented. The proposed
transformation gives possibility to get new forms of the N-dimensional system
of the first order equations that can be useful for analysis of the solutions
of the Nth- order difference equations. In particular, for the third-order
linear difference equation the nonlinear second-order difference equation that
plays the same role as the Riccati equation for second-order linear difference
equation is obtained. The new form of the N-dimensional system of first order
equations can be also used for finding the WKB solutions of the linear
difference equation with coefficients that vary sufficiently slowly with index.
