On the matrix $pth$ root functions and generalized Fibonacci sequences

This study is devoted to the polynomial representation of the matrix $p$th root functions. The Fibonacci-H"orner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the Binet formula, serves as a triggering factor to provide explicit formulas for the matrix $p$th roots. Special cases and illustrative numerical examples are given.