Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices

Abstract

We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many contexts in pure and applied mathematics. Therefore, the study of the inverse of these matrices becomes of specific interest. However, explicit formulas for inverses are known only in a few cases, in particular when the coefficients of the diagonal entries are subjected to some restrictions. We will show that the inverse of generalized Jacobi matrices can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. In fact, recent advances in the study of linear difference equations, allow us to compute the solution of this kind of boundary value problems. So, the conditions that ensure the uniqueness of the solution of the boundary value problem leads to the invertibility conditions for the matrix, whereas that solutions for suitable problems provide explicitly the entries of the inverse matrix. Keywords Second order difference equation • Boundary value problem • Generalized Jacobi matrix 1 Introduction and Preliminaries Throughout the paper, R denotes the field of real numbers. Given a positive integer n, we denote by R n the real coordinate space of n dimensions. Moreover, the components of the vector v ∈ R n are denoted by v j , j = 1,. .. , n, i.e., v = (v 1 ,. .. , vn) .

Staircase patterns in words: Subsequences, subwords, and separation number

“…The intimate relation between Chebyshev polynomials and Fibonacci, Lucas, and more generally Horadam's numbers (i. e., solutions to a general second-order linear recursion with constant coefficients) has been intensively explored, among other areas, in enumerative combinatorics for computing generating functions (see, for instance, [20,27,30]), number theory for representation of integer sequences (see, for instance, an early work [6] and recent [33,35]), and applied linear algebra for computing determinants and inverse matrices (see, for instance, [13,14,17,40]). The formula (3) also provides a natural link between the theory of Chebyshev polynomials and a discrete Poisson equation and its Sturm-Liouville theory [7,[13][14][15]22], and subsequently to the harmonic functions and first passage time of random walks (see, for instance, the classical [24,29] and more recent [10,12,18]). Altogether, the link fundamentally bridges between analytic and probabilistic methods exploited in this paper.…”