Compressed lattice sums arising from the Poisson equation

Purpose: 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). Methods: 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. Results: 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 ϕ2(x,y,d):=1π2∑m,n∈Ocos(πmx)cos(πnd√y)m2+dn2,(1) where d>0, x, y are real numbers and Odenotes the odd integers. In this paper we first survey the earlier work and then discuss the sum (1) more completely. Conclusions: 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 1/π logA, where A is an algebraic number. These evaluations suggest that a deep theory interconnects all such summations.