Solutions of linear difference equations with variable coefficients (Q1269051)

Explicit solutions of a linear difference equation of unbounded order \[ y_k = \sum_{i=1}^{k-1} b_{k,i} y_i + x_k, k \geq 1, \] and a linear difference equation of order \(N (N \geq 2)\) \[ y_{k+N} = \sum_{j=1}^{N} a_{k,j} y_{k+N-j} + x_{k+N}, k \geq 1, \] are presented. Expressions for the product of companion matrices and for the positive integer powers of a companion matrix are given.