Non-Abelian elastic collisions, associated difference systems of equations and discrete analytic functions

We extend the equations of motion that describe non-relativistic elastic collision of two particles in one dimension to an arbitrary associative algebra. Relativistic elastic collision equations turn out to be a particular case of these generic equations. Furthermore, we show that these equations can be reinterpreted as difference systems defined on the ${\mathbb{Z}}^2$ graph and this reinterpretation relates (unifies) the linear and the non-linear approach of discrete analytic functions.