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Open Access Research

Compressed lattice sums arising from the Poisson equation

David H Bailey1 and Jonathan M Borwein2*

Author Affiliations

1 Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

2 Centre for Computer Assisted Research Mathematics and Its Applications (CARMA), University of Newcastle, Callaghan, NSW, 2308, Australia

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Boundary Value Problems 2013, 2013:75  doi:10.1186/1687-2770-2013-75

Published: 5 April 2013

Abstract

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M2">View MathML</a>, x, y are real numbers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M3">View MathML</a> denotes 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M4">View MathML</a>, 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.

Keywords:
lattice sums; Poisson equation; experimental mathematics; high-precision computation