A Generating Operator for Solutions of Certain Partial Difference and Differential Equations

Abstract

This paper concerns an algebraic method for generating additional solutions of an n-dimensional homogeneous linear partial difference equation from a known solution. A parallel theory is developed for the continuous case via the Taylor series expansion; the pertinent partial differential equation is linear and homogeneous of order $\ell $, with constant coefficients. In both the discrete and the continuous cases, a generating operator is introduced and is shown to commute with the given difference or differential operator, respectively. Applications are presented for both cases.