  We study linear difference equations with variable coefficients in a ring
using a new nonlinear method. In a ring with identity, if the homogeneous part
of the linear equation has a solution in the unit group of the ring (i.e., a
unitary solution) then we show that the equation decomposes into two linear
equations of lower orders. This decomposition, known as a semiconjugate
factorization in the nonlinear theory, generalizes the classical operator
factorization in the linear context. Sequences of ratios of consecutive terms
of a unitary solution are used to obtain the semiconjugate factorization. Such
sequences, known as eigensequences are well-suited to variable coefficients;
for instance, they provide a natural context for the expression of the
classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear
difference equations with periodic coefficients and also derive formulas for
the general solutions of linear functional recurrences satisfied by the
classical special functions such as the modified Bessel and Chebyshev.
