Multi-Delayed Discrete Matrix Functions and Their Applications in Solving Higher-Order Difference Equations

A new class of linear non-homogeneous discrete matrix equations with multiple delays and second-order differences is considered, where the coefficient matrices satisfy pairwise permutability conditions. First, new multi-delayed discrete matrix sine- and cosine-type functions are introduced, which generalize existing delayed discrete matrix functions. Based on the introduced matrix functions and appropriate commutativity conditions, an explicit matrix representation of the solution is derived. The importance of our results is shown by comparing them with related previous works, along with suggestions about some new open problems. Finally, an example is provided to illustrate the importance of the results.