Powers of a matrix and combinatorial identities

TL;DR

This paper derives a general polynomial identity in multiple variables to provide a closed-form expression for matrix powers and uses these results to establish new combinatorial identities.

Contribution

It introduces a novel polynomial identity applicable to any size matrix, enabling explicit computation of matrix powers and derivation of combinatorial identities.

Findings

Closed-form expressions for matrix powers

New combinatorial identities derived from the polynomial identity

General polynomial identity valid for any positive integer k

Abstract

In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix. Finally, we use these results to derive various combinatorial identities.