A generalized solution expression for linear homogeneous constant-coefficient difference equations

We present here what is, to our knowledge, a completely new and general solution expression for the complementary solution of an arbitrary Nth order linear homogeneous constant-coefficient difference equation which, unlike the solution expressions usually presented in textbooks, does not a priori assert the specific structural form of the solution. This method easily handles the case of repeated zero roots, a case of practical importance for which the classical solution expression fails, as recently shown by Johnson. Furthermore, we show that both the classical solution expression, and Johnsonʹs “singular solution” expression for the case of repeated zero roots, are special cases of our more general expression. Finally, we present an example illustrating the interrelationships amongst the different solution expressions as well as the solution obtained via the generating-function method.