Regular Boundary Value Problems on a Path throughout Chebyshev Polynomials E. Bendito, A. Carmona, A.M. Encinas and J.M. Gesto Departament de Matem` atica Aplicada III. Universitat Polit` ecnica de Catalunya. Spain Abstract In this work we study the different types of regular boundary value problems on a path associated with the Schr¨ odinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. In any case, the Green functions are given in terms of Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Sch¨ odinger operator on a path. Key words: Discrete Schr¨ odinger operator, Path, Boundary value problems, Green function, Chebyshev polynomials 1 Introduction In this work, we analyze the linear boundary value problem in the context of second order difference equations with constant coefficients associated with the Schr¨odinger operator on a finite path. Our study runs in parallel to the known for boundary value problems associated with ordinary differential equations. In particular we concentrate on determining explicit expressions for the Green function associated with regular boundary value problems on a path. The boundary value problems here considered are of three types that corre- spond to the cases in which the boundary has two, one or no vertices. In any Email address: angeles.carmona@upc.edu (E. Bendito, A. Carmona, A.M. Encinas and J.M. Gesto). URL: http://www-ma3.upc.es/users/bencar/index.html (E. Bendito, A. Carmona, A.M. Encinas and J.M. Gesto). Preprint submitted to Elsevier 27 September 2006