Explicit solutions of regular linear discrete-time descriptor systems with constant coefficients

Explicit solution formulas are presented for systems of the form Exk+1 = Axk + fk with k ∈ K, where K ⊂ Z is a discrete interval and the pencil λE − A is regular. Different results are obtained when one starts with an initial condition at the point k = 0 and calculates into the future (i.e., Exk+1 = Axk + fk with k ∈ N) and when one wants to get a complete solution (i.e., Exk+1 = Axk + fk with k ∈ Z).