On symmetric Tetranacci polynomials in mathematics and physics

TL;DR

This paper introduces symmetric Tetranacci polynomials as a generalization of Tetranacci numbers, with applications in physics and matrix diagonalization, providing a closed-form expression involving generalized Fibonacci polynomials.

Contribution

It presents the first comprehensive definition and closed-form solution for symmetric Tetranacci polynomials, linking them to eigenvector structures in physics and matrix analysis.

Findings

Derived a complete closed-form expression for symmetric Tetranacci polynomials.

Linked the polynomials to eigenvectors of symmetric Toeplitz matrices.

Demonstrated applications in condensed matter physics.

Abstract

In this manuscript, we introduce (symmetric) Tetranacci polynomials $\xi_j$ as a twofold generalization of ordinary Tetranacci numbers, by considering both non unity coefficients and generic initial values in their recursive definition. The issue of these polynomials arose in condensed matter physics and the diagonalization of symmetric Toeplitz matrices having in total four non-zero off diagonals. For the latter, the symmetric Tetranacci polynomials are the basic entities of the associated eigenvectors; thus, treating the recursive structure determines the eigenvalues as well. Subsequently, we present a complete closed form expression for any symmetric Tetranacci polynomial. The key feature is a decomposition in terms of generalized Fibonacci polynomials.