A new approach to tridiagonal matrices related to the Sylvester-Kac matrix

The Sylvester-Kac matrix, a well-known tridiagonal matrix, has been extensively studied for over a century, with various generalizations explored in the literature. This paper introduces a new type of tridiagonal matrix, where the matrix entries are defined by an integer sequence, setting it apart from the classical Sylvester-Kac matrix. The aim of this paper is to investigate several fundamental properties of this generalized matrix, including its algebraic structure, determinant, inverse, LU decomposition, characteristic polynomial, and various norms.