No-neighbours recurrence schemes for space-time Green’s functions on a 3D simple cubic lattice

Abstract

Application of multivariate creative telescoping to a finite triple sum representation of the discrete space-time Green’s function for an arbitrary numeric (non-symbolic) lattice point on a 3D simple cubic lattice produces a fast, no-neighbours, seventh-order, eighteenth-degree, discrete-time recurrence scheme.For arbitrary numeric lattice points outside the diagonal symmetry planes, the seven numeric eighteenth-degree polynomial coefficients of the recurrence scheme are products of polynomials with integer coefficients that are linear in the recurrence index n, and two polynomials of degree four, and five polynomials of degree twelve that are irreducible over the field of integers. Owing to the symmetry of the scalar Green’s function upon interchanging any of the lattice point coordinates, the twelfth degree polynomials with integer coefficients may each be expanded in terms of 102 elementary symmetric polynomials in symbolic lattice point coordinates. The recurrence schemes determined by the telescoper for 102 distinct numeric lattice points can be used to form linear systems of equations. These are solved for the coefficients of the elementary symmetric polynomials required to construct the symbolic polynomial coefficients of the generic 3D recurrence scheme.Given its compact and straightforward 2D counterpart, this 3D recurrence scheme is far more intricate than expected, and is most efficiently presented through tables of coefficients. However, the scheme and the resulting lattice Green’s function sequences also exhibit more features. The complexity reduces for lattice points on diagonal symmetry planes, yielding a fast no-neighbours, fifth-order, twelfth-degree, discrete-time recurrence scheme. An illustrative example reveals unexpected phenomena, e.g. a late-time, high-frequency interplay of resonances that appears anomalous but can be fully explained, and the possible occurrence of removable recurrence scheme singularities. These effects are studied in detail in separate papers.