Abstract
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.
Highlights
In [ ], we analyzed various generalized lattice sums [ ], which have been studied for many years in the mathematical physics community, for example, in [ – ]
Attention was directed to the problem of solving the Poisson equation, either in engineering scenarios or in regard to crystal structure
We have the following results which were established by factorization of lattice sums after being empirically discovered by the methods described in the few sections
Summary
In [ ], we analyzed various generalized lattice sums [ ], which have been studied for many years in the mathematical physics community, for example, in [ – ]. More recently interest was triggered by some intriguing research in image processing techniques [ ]. These developments have underscored the need to better understand the underlying theory behind both lattice sums and the associated Poisson potential functions. Attention was directed to the problem of solving the Poisson equation, either in engineering scenarios (computationally, say for image enhancement) or in regard to crystal structure (theoretically). In [ ] we studied a class of lattice sums that amount to Poisson solutions, namely the n-dimensional forms eiπ (m r +···+mnrn) φn(r , . By virtue of striking connections with Jacobi θ-function values, we were able to develop new closed forms for certain values of the coordinates rk and extend such analysis to similar lattice sums. A primary result was that for rational x, y, the natural potential φ (x, y) is π log A where
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