We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.

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Summary. We obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k−Toeplitz matrix A. The proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind. We also compute the characteristic polynomial of A which enables us to state some conditions for the existence of A −1 . Our results also extend known results for the case when the residue mod k of the order of A is equal to 0 or k − 1 (Numer. Math., 10 (1967), pp. 153-161.). Classification (2000): 15A09, 42C05, 33C45, 65Q05 Mathematics Subject

In this paper, we use the analytic theory for 2 and 3-Toeplitz matrices to obtain the explicit expressions for the eigenvalues, eigenvectors and the spectral measure associated to the corresponding infinite matrices. As an application we consider two solvable models related with the so-called chain model. Some numerical experiments are also included.