A multidimensional extension of the combinatorics function technique. I. Linear and homogeneous partial difference equations

Abstract

This paper is aimed at series solutions of physical phenomena that are described by linear homogeneous differential equations, like the Schrödinger differential equation. Series solutions to such equations, when they exist, lead to multidimensional, linear, and homogeneous recurrence relations among the expansion coefficients. The physical constraints imposed on the solutions of an ordinary differential equation (in the case of the Schrödinger equation that would be on the wavefunctions) then lead to a set of ’’initial values’’ on the expansion coefficients. The consistency of the initial values with the recurrence relation or partial difference equation (PDE), is one of the major problems in such cases. Until now, there was no systematic way of obtaining the solution of a PDE in terms of the initial values, and no systematic technique dealing with the consistency check. In this paper, we have been able to solve both of these problems by a natural extension of the combinatorics function technique developed by Antippa and Phares for one-dimensional linear recurrence relations.