The general solution of the r-generalized Fibonacci matrix recurrence equation with noncommutative coefficients

Abstract:
The construction of the general solution of the r-generalized Fibonacci matrix recurrence equation with noncommutative coefficients is reported. The reduction of the general result to the case of commutative coefficients is presented. Our resolutive formula allows the introduction of new permutationally invariant functions of two solvents of a quadratic matrix equation with matrix coefficients, which play the role of the two elementary symmetric functions of the two roots of a quadratic scalar equation.