Integer powers of complex anti-tridiagonal matrices and some complex factorizations

TL;DR

This paper derives a general formula for the entries of integer powers of complex anti-tridiagonal matrices and explores complex factorizations of Fibonacci-related sequences, providing new algebraic insights.

Contribution

It introduces a novel general expression for powers of complex anti-tridiagonal matrices and presents new factorizations of Fibonacci polynomials, Fibonacci, and Pell numbers.

Findings

Explicit formulas for matrix powers based on parity of n and r

New complex factorizations of Fibonacci polynomials

Enhanced understanding of matrix-based sequence factorizations

Abstract

In this paper, we obtain a general expression for the entries of the rth power of a certain n-square complex anti-tridiagonal matrix where if n is odd, r is integer or if n is even, r is natural number. In addition, we get the complex factorizations of Fibonacci polynomials, Fibonacci and Pell numbers.