On symmetric Tetranacci polynomials in mathematics and physics

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.