Compressed lattice sums arising from the Poisson equation
David H. Bailey, Jonathan Michael Borwein · Boundary Value Problems · 2013
In recent years attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). In (Bailey et al. in J. Phys. A, Math. Theor. 46:115201, 2013, doi:10.1088/1751-8113/46/11/115201) we studied a class of lattice sums that amount to solutions of Poisson’s equation, utilizing some striking connections between these sums and Jacobi ϑ-function values, together with high-precision numerical computations and the PSLQ algorithm to find certain polynomials associated with these sums. We take a similar approach in this study. We were able to develop new closed forms for certain solutions and to extend such analysis to related lattice sums. We also alluded to results for the compressed sum 1 where , x, y are real numbers and denotes the odd integers. In this paper we first survey the earlier work and then discuss the sum (1) more completely. As in the previous study, we find some surprisingly simple closed-form evaluations of these sums. In particular, we find that in some cases these sums are given by , where A is an algebraic number. These evaluations suggest that a deep theory interconnects all such summations. PACS Codes:02.30.Lt, 02.30.Mv, 02.30.Nw, 41.20.Cv. MSC:06B99, 35J05, 11Y40.