Explicit inverse of a tridiagonal k-Toeplitz matrix

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 enable us to state some conditions for the existence of A−1. Our results extends some other ones in the literature known for the case when the residue mod k of the order of A equals 0 or k − 1.