On compact representations for the solutions of linear difference equations with variable coefficients

A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both nth and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested sums of their variable coefficients. It makes explicit all the sum of products involved in the previous representations of such solutions. Some basic applications are also illustrated.