The analytic inversion of any finite symmetric tridiagonal matrix

We use the theory of orthogonal polynomials to write down explicit expressions for the polynomials of the first and second kind associated with a given infinite symmetric tridagonal matrix H. The Green's function is the inverse of the infinite symmetric tridiagonal matrix (H - zI). By calculating the inverse of the finite symmetric tridiagonal matrix (Hpp - zIpp) we can find the analytical form of the inverse of the finite symmetric tridiagonal matrix, Hpp.