This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

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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Citation and License

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


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/75


Received:4 December 2012
Accepted:15 March 2013
Published:5 April 2013

© 2013 Bailey and Borwein; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1 Background

In [1], we analyzed various generalized lattice sums [2], which have been studied for many years in the mathematical physics community, for example, in [2-4]. More recently interest was triggered by some intriguing research in image processing techniques [5]. These developments have underscored the need to better understand the underlying theory behind both lattice sums and the associated Poisson potential functions.

In recent times, 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 [1] we thus studied a class of lattice sums that amount to Poisson solutions, namely the n-dimensional forms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M5">View MathML</a>

By virtue of striking connections with Jacobi ϑ-function values, we were able to develop new closed forms for certain values of the coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M6">View MathML</a> and extend such analysis to similar lattice sums. A primary result was that for rationalx, y, the natural potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M7">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M8">View MathML</a>whereAis an algebraic number. Various extensions and explicit evaluations were given. We also touched on results for the compressed sum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M9">View MathML</a>

(2)

In this paper we discuss (2) in more detail. More detailed motivation can be found in [1,5].

Such work is made possible by number-theoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 25,000-digit arithmetic.

In Section 1.1, we describe, consistent with [5], the underlying equations along with ‘natural’ Madelung constants and relate them to the classical Madelung constants. In Section 1.2, we produce the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10">View MathML</a> which, especially with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M11">View MathML</a>, provide the central objects of our study. In Section 2.1, we describe rapid methods of evaluating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M12">View MathML</a> and add computational details in Section 2.2. In Section 3.1 we discuss closed forms for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M7">View MathML</a>. In Section 3.2 we discuss closed forms for related sums in two dimensions. This leads to the introduction of theta function formalism in the next two subsections and allows us to resolve much else. In Sections 3.5 and 3.6 we are then able to evaluate compressed potential sums. Finally, we further discuss computational matters in Section 3.7.

1.1 Madelung-type sums

In a recent treatment of ‘natural’ Madelung constants [5], it is pointed out that the Poisson equation for an n-dimensional point-charge source,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M14">View MathML</a>

(3)

gives rise to an electrostatic potential - we call it the bare-charge potential - of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M15">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M16">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M17">View MathML</a>. Since this Poisson solution generally behaves as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M18">View MathML</a>, the previous work [5] defines a ‘natural’ Madelung constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M19">View MathML</a> as (here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M20">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M21">View MathML</a>

(6)

where in cases such as this log sum one must infer an analytic continuation [5], as the literal sum is quite non-convergent. This <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M19">View MathML</a> coincides with the classical Madelung constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M23">View MathML</a>

(7)

only for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M24">View MathML</a> dimensions, in which case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M25">View MathML</a>. But in all other dimensions there is no obvious ℳ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M26">View MathML</a> relation.

A method for gleaning information about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M19">View MathML</a> is to contemplate the Poisson equation with a crystal charge source, modeled after NaCl (salt) in the sense of alternating lattice charges:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M28">View MathML</a>

(8)

Accordingly - based on the Poisson equation (3) - solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10">View MathML</a> can be written in terms of the respective bare-charge functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M30">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M31">View MathML</a>

(9)

1.2 The crystal solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10">View MathML</a>

In [5] it is argued that a solution to (8) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M33">View MathML</a>

(10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M34">View MathML</a> denotes the odd integers (including negative odds). These <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10">View MathML</a> do give the potential within the appropriate n-dimensional crystal, in that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10">View MathML</a> vanishes on the surface of the cube <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M37">View MathML</a>, as is required via symmetry within an NaCl-type crystal of any dimension - thus we have solved the Poisson equation subject to Dirichlet boundary conditions.

To render this representation more explicit and efficient, we could write equivalently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M38">View MathML</a>

It is also useful that - due to the symmetry inherent in having odd summation indices - we can cavalierly replace the cosine product in (10) with a simple exponential:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M39">View MathML</a>

(11)

This follows from the simple observation that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M40">View MathML</a>, so when the latter product is expanded out, the appearance of even a single sin term is annihilating due to the bipolarity of every index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M41">View MathML</a>.

We observe that the convergence of these conditionally convergent sums is by no means obvious, but that results such as [[2], Thm. 8.3 and Thm. 8.5] ensure that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M42">View MathML</a>

(12)

is convergent and analytic with abscissa <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M43">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M44">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M45">View MathML</a>. For the central case herein, summing over increasing spheres is analytic in two dimensions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M46">View MathML</a> and in three dimensions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M47">View MathML</a>, but in general the best estimate we have is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M48">View MathML</a>, so for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M49">View MathML</a>, to avoid ambiguity, we work with the analytic continuation of (12) from the region of absolute convergence with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M50">View MathML</a>. Indeed, all our transform methods are effectively doing just that.

2 Methods

2.1 Fast series for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M10">View MathML</a>

From previous work [5] we know a computational series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M52">View MathML</a>

(13)

suitable for any nonzero vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M53">View MathML</a>. The previous work also gives an improvement in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M54">View MathML</a> dimensions, namely the following form for which the logarithmic singularity at the origin has been siphoned off:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M55">View MathML</a>

(14)

These series, (13) and (14) are valid, respectively, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M56">View MathML</a>.

For clarification, we give here the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M57">View MathML</a>-dimensional case of the fast series:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M58">View MathML</a>

(15)

Though it may not be manifest in this asymmetrical-looking series, it turns out that for any dimension n the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M59">View MathML</a> is invariant under permutations and sign-flips. For example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M60">View MathML</a> and so on. It thus behoves the implementor to consider x - which appears only in the first sum of (15) - to be the largest in magnitude of the three coordinates for optimal convergence. A good numerical test case is the exact evaluation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M61">View MathML</a>, which we have confirmed to 500 digits.

2.2 Computational techniques

In this study, we employ an experimental scheme similar to that used in [1], as well as in numerous other studies that have been performed. In particular, we compute the key expressions in this study to very high precision, then use the PSLQ algorithm in an attempt to recognize the computed numerical values in terms of relatively simple, closed-form expressions. Given an n-long input vector (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M62">View MathML</a>) of real numbers, the PSLQ algorithm finds a nontrivial set of integer coefficients (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M63">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M64">View MathML</a> if it exists (to within the tolerance of the numeric system being used).

Numerous experimental evaluations of minimal polynomials satisfied by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M65">View MathML</a> (the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66">View MathML</a>), and a detailed description of the underlying computational methodology employed to find these evaluations, are presented in [1]. A similar computational scheme was employed in this study to obtain minimal polynomials satisfied by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M67">View MathML</a>. For instance, we recover, among a myriad of other evaluations,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M68">View MathML</a>

This computational scheme is as follows:

1. Given x, y and d, select a conjectured polynomial degree m and a precision level P. We typically set the numeric precision level P somewhat greater than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M69">View MathML</a> digits.

2. Compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M70">View MathML</a> to P-digit precision using formula (47). Evaluate the four theta functions indicated using the very rapidly convergent formulas given in [[6], p.52] or [7].

3. Generate the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M71">View MathML</a>-long vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M72">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M73">View MathML</a>. Note: we have found that without the eight here, the degree of the resulting polynomial would be eight times as high (but the larger polynomials were in fourth or eighth powers). Given the very rapidly escalating computational cost of higher degrees, many of the results in [1] and herein would not be feasible without this factor.

4. Apply the PSLQ algorithm (we employed the two-level multipair variant of PSLQ for d up to 19, and the three-level multipair variant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M74">View MathML</a>[8]) to find a nontrivial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M71">View MathML</a>-long integer vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M76">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M77">View MathML</a>, if such a vector exists. PSLQ (or one of its variants) either finds a vector A, which then is the vector of coefficients of an integer polynomial satisfied by α (certified to the ‘epsilon’ of the numerical precision used), or else exhausts precision without finding a relation, in which case the algorithm nonetheless provides a lower bound on the Euclidean norm of the coefficients of any possible degree-m integer polynomial A satisfied by α.

5. If no numerically significant relation is found, try again with a larger value of m and correspondingly higher precision. If a relation is found, try with somewhat lower m, until the minimal m is found that produces a numerically significant relation vector A. Here ‘numerically significant’ means that the relation holds to at least 100 digits beyond the level of precision required to discover it. To obtain greater assurance that the polynomial produced by this process is in fact the minimal polynomial for α, use the polynomial factorization facilities in Mathematica or Maple to attempt to factor the resulting polynomial.

3 Results and discussion

3.1 Closed forms for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M78">View MathML</a> potential

Provably we have the following results which were established by factorization of lattice sums after being empirically discovered by the methods described in the next few sections.

Theorem 1 ([1])

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M79">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M80">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M81">View MathML</a>

(18)

Using the integer relation method PSLQ [9] to hunt for results of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M82">View MathML</a>

(19)

for α algebraic, we may obtain and further simplify many results as follows.

Conjecture 2 ([1])

We have discovered and subsequently proven

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M83">View MathML</a>

where the notation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M84">View MathML</a>indicates that we originally only had experimental (extreme-precision numerical) evidence of an equality.

Such hunts are made entirely practicable by (14). Note that for general x and y we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M85">View MathML</a>, so we can restrict searches to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M86">View MathML</a>.

Remark 3 (Algebraicity)

In light of our current evidence, we conjectured that for x, y rational,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M87">View MathML</a>

(20)

for α algebraic. Theorem 5 proved this conjecture. We note that Theorem 5 proves that all values should be algebraic but does not, a priori, establish the precise values we have found. This will be addressed in Section 3.4.

3.2 Madelung and ‘jellium’ crystals, and Jacobi ϑ-functions

We have studied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M88">View MathML</a>

(21)

as a ‘natural’ potential for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M11">View MathML</a> dimensions in the Madelung problem. There is another interesting series, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M90">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M91">View MathML</a> denotes the even integers.

Now it is explained and illustrated in [5] that this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M92">View MathML</a> function is the ‘natural’ potential for a classical jellium crystal and relates to Wigner sums [2]. This involves a positive charge at every integer lattice point, in a bath - a jelly - of uniform negative charge density. As such, the ψ functions satisfy a Poisson equation but with different source term [5]. Note, importantly, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M92">View MathML</a> satisfies Neumann boundary conditions on the faces of the Delord cube - in contrast with the Dirichlet boundary conditions satisfied in the Madelung case.

Briefly, a fast series for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M94">View MathML</a> has been worked out [5] as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M95">View MathML</a>

(23)

valid on the Delord n-cube, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M96">View MathML</a>.

In the following, we shall also use the general form of the general Jacobi theta functions, defined as in [10] and [[6], Sec. 2.6]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M97">View MathML</a>

(24)

One useful relation is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M98">View MathML</a>

(25)

3.3 Closed forms for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M78">View MathML</a>

Using the series (23) for high-precision numerics, it was discovered (see [[5], Appendix]) that previous lattice-sum literature had concealed a longtime typographical error for certain two-dimensional sums, and that a valid closed form for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M101">View MathML</a> is actually

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M102">View MathML</a>

(26)

In the Madelung case, it then became possible to cast <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M78">View MathML</a> likewise in closed form, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M104">View MathML</a>

(27)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M106">View MathML</a>. (See [5] and [[1], Appendix II] for details.)

Theorem 4 ([1])

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M107">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M108">View MathML</a>

(28)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M109">View MathML</a>

(29)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M110">View MathML</a>

(30)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M111">View MathML</a>

(31)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M105">View MathML</a>.

We recall the general ϑ-transform giving for all z with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M113">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M114">View MathML</a>

(32)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M115">View MathML</a> (while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M116">View MathML</a>). In particular, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M117">View MathML</a> we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M118">View MathML</a>

(33)

which directly relates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M120">View MathML</a> in (31). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M121">View MathML</a>

(34)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M122">View MathML</a>

(35)

or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M123">View MathML</a>, and (32) then shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M124">View MathML</a>

(36)

Likewise (28) is unchanged on replacing λ by κ. Hence, it is equivalent to (20) to prove that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M125">View MathML</a> with x, y rational <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126">View MathML</a> in (30) is algebraic.

Theorem 5 (Algebraic values of λ and μ, [1])

For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M125">View MathML</a>withx, yrational, the values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M129">View MathML</a>in (30) are algebraic. It follows that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M130">View MathML</a>withαalgebraic. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M131">View MathML</a>forβalgebraic.

This type of analysis also works for any singular value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M133">View MathML</a> and so applies to sums with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M134">View MathML</a> in the denominator, as we see in the next section.

3.4 Explicit equations for degree 2, 3 and 5

We illustrate the complexity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M135">View MathML</a>, the algebraic polynomial linking the input and output as we move from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M136">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M137">View MathML</a>, by first considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M139">View MathML</a>.

Example 6 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M141">View MathML</a>)

As described in [1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M142">View MathML</a>

(37)

and so letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M143">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M144">View MathML</a>

(38)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M145">View MathML</a>

(39)

Iteration yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M146">View MathML</a>. From this one may recursively compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M147">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M148">View MathML</a> and watch the tower of radicals grow. Correspondingly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M149">View MathML</a>

(40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M140">View MathML</a> are given by (38) and (39). After simplification, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M153">View MathML</a>

(41)

In addition, since we have an algebraic relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M154">View MathML</a>, this allows us computationally to prove the evaluation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M156">View MathML</a> once <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M7">View MathML</a> is determined as it is for the cases of Theorem 1.

We may perform the same work inductively for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M158">View MathML</a> using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M159">View MathML</a> in the known addition formulas for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M160">View MathML</a>[1] to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M161">View MathML</a> and so on. The inductive step is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M162">View MathML</a>

(42)

Example 7 (Empirical computation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M163">View MathML</a>)

Given z and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M165">View MathML</a>, we compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M166">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M168">View MathML</a> up to degree J, K and look for a relation to precision D. This is a potential candidate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M163">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M170">View MathML</a> at a precision significantly greater than D and for various choices of w, we can reliably determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M163">View MathML</a> in this way [1].

Example 8

We conclude this subsection by proving the following empirical evaluation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M172">View MathML</a>

(43)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M173">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M174">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M175">View MathML</a>. We may - with help from a computer algebra system - solve for the stronger requirement that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M176">View MathML</a> using Example 6. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M177">View MathML</a>

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M179">View MathML</a>, as required. To convert this into a proof, we can make an a priori estimate of the degree and length of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M180">View MathML</a> - using (39), (41) and (42) while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M181">View MathML</a> - and then perform a high precision computation to show no other algebraic number could approximate the answer well enough. The underlying result we appeal to [[6], Exercise 8, p.356] is given next in Theorem 9.

In this particular case, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M182">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M184">View MathML</a> and need to confirm that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M185">View MathML</a>. A very generous estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M186">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M187">View MathML</a> shows it is enough to check <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M188">View MathML</a>. This is very easy to confirm. Relaxing to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M190">View MathML</a> requires verifying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M191">View MathML</a>. This takes only a little longer.

Recall that the length of a polynomial is the sum of the absolute value of the coefficients.

Theorem 9 (Determining a zero)

SupposePis an integral polynomial of degreeDand lengthL. Suppose thatαis algebraic of degreedand length. Then either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M192">View MathML</a>or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M193">View MathML</a>

Example 10 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M194">View MathML</a>)

Making explicit the recipe for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M194">View MathML</a>, we eventually arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M196">View MathML</a>

(44)

This shows how generous our estimates were in Example 8.

In similar fashion we can now computationally confirm all of the exact evaluations in Conjecture 2 and the like. For example, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M197">View MathML</a>. This solves to produce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M198">View MathML</a> and establishes (16) of Theorem 1. Now Example 6 can be used to produce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M199">View MathML</a> is as given in Conjecture 2, and so on.

We now turn to our main focus.

3.5 ‘Compressed’ potentials

Yet another solution to the Poisson equation with crystal charge source, for <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>, is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M201">View MathML</a>

(45)

This is the potential inside a crystal compressed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M202">View MathML</a> on the y-axis, in the sense that the Delord-cube now becomes the cuboid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M203">View MathML</a>. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M204">View MathML</a> vanishes on the faces of this 2-cuboid (rectangle) for <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> and integer. This is technically only a compression when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M206">View MathML</a>.

Along the same lines as the analysis of (13), we can posit a fast series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M207">View MathML</a>

(46)

valid on the Delord cuboid, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M209">View MathML</a>. Moreover, the log-accumulation technique of [[5], Appendix] applied as with Theorem 4 yields the following.

Theorem 11 (ϑ-representation for compressed potential)

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M210">View MathML</a>

(47)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M211">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M212">View MathML</a>.

This theorem is consistent with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M213">View MathML</a> in the sense <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M214">View MathML</a>. In addition, for integers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M215">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M216">View MathML</a>

(48)

We note that there are various trivial evaluations such as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M217">View MathML</a> for all x. Moreover, for each positive rational d, there is an analogue of Theorem 5 in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M218">View MathML</a> is replaced by the dth singular value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M219">View MathML</a>[2,6]. Thence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M221">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M222">View MathML</a>. Precisely, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M223">View MathML</a>

(49)

and have (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M224">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M225">View MathML</a>

(50)

(see [[6], Prop. 2.1]). We observe that (47) can be written for all x, y as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M226">View MathML</a>

(51)

on appealing to (49) and (50). Hence we may now study <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M227">View MathML</a> exclusively.

Fix an integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M228">View MathML</a>. The addition formulas for the ϑ’s as given in [[6], Section 2.6]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M229">View MathML</a>

(52)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M230">View MathML</a>

(53)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M231">View MathML</a>

(54)

when replacing z by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M232">View MathML</a> and w by z, and appealing to (50), allow one recursively to write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M233">View MathML</a> algebraically in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M234">View MathML</a>. We first give the equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M235">View MathML</a>.

Example 12 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M235">View MathML</a>)

As for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M140">View MathML</a>, which is the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66">View MathML</a>, we have analytically

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M239">View MathML</a>

(55)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M240">View MathML</a>

(56)

This becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M241">View MathML</a>

(57)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M242">View MathML</a>.

We turn to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M243">View MathML</a>.

Example 13 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M243">View MathML</a>)

As for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M141">View MathML</a>, which is the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M247">View MathML</a>

(58)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M248">View MathML</a>

(59)

In tandem with (57) this becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M249">View MathML</a>

(60)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M242">View MathML</a>.

This can also be determined by using the methods of Example 7 to obtain the algebraic equation linking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M252">View MathML</a> and then carefully factoring the result - guided by results such as Example 10. It could also have been rigorously proven by the method used for corresponding formula with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M66">View MathML</a> in [1].

We next exhibit several sample compressed-sum evaluations, of which (62) is the most striking.

Theorem 14 (Some explicit compressed sums)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M254">View MathML</a>

(61)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M255">View MathML</a>

(62)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M256">View MathML</a>

(63)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M257">View MathML</a>

(64)

Proof Let us first sketch a proof of (62). This requires showing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M258">View MathML</a>. Direct computer algebra using (51) shows that it suffices to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M259">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M260">View MathML</a>. Now (60) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M261">View MathML</a>

(65)

However, 3z is in the compressed lattice generated by 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M262">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M263">View MathML</a>

From this, we may deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M264">View MathML</a> is a zero of the denominator of (65), whose only real root is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M265">View MathML</a>.

To prove (61), we similarly consider (51) and (60) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M266">View MathML</a> so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M220">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M268">View MathML</a> is in the compressed lattice. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M269">View MathML</a>, and we now deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M270">View MathML</a>

and the result again follows.

To prove (63) requires an application of this method with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M273">View MathML</a>. We likewise may prove (64) (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M274">View MathML</a>). In this final case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M275">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M276">View MathML</a>. In these cases, we need to appeal to both (59) and (60) at least once. □

In general, the polynomial for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M277">View MathML</a> with x, y rational can be calculated as we describe in the next section.

3.6 Further results for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M278">View MathML</a>

As noted, we anticipated similarly algebraic evaluations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M279">View MathML</a> for nonnegative integer values of a, b, c, d. And, as we will show in more examples in the next section, this is indeed true. All such results can, in principle, be rigorously established by the techniques of Example 7 and Example 8 or indeed Theorem 14, but we will not do this. We do however give the first few polynomials, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M280">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M281">View MathML</a>, that is, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M283">View MathML</a>.

Example 15 (Some explicit compressed polynomials for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M284">View MathML</a>)

We present here the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M280">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M286">View MathML</a>.

Note that reversing the order of the variables leads to a more intractable computation since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M324">View MathML</a> is not of the form covered by Theorem 11.

3.7 Computational results

Results that we have obtained for the special set of cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M325">View MathML</a>, for integers d up to 40, are shown in Table 1. Our computations required up to 25,000-digit precision, and, for large degrees and correspondingly high precision levels, were rather expensive (over 20 processor-hours in some cases). We employed the ARPREC arbitrary precision software [11].

Table 1. PSLQ runs to recover minimal polynomials satisfied by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M326">View MathML</a>

The degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M329">View MathML</a> of the minimal polynomial for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M330">View MathML</a>, the number of zeroes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M331">View MathML</a> among the minimal polynomial coefficients, the numeric precision level P, the run time in seconds T, and the approximate base-10 logarithm M of the absolute value of the central coefficient are also shown in the table, together with the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M332">View MathML</a>.

We were not able to obtain relations for all cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M333">View MathML</a>. Evidently, the precision levels we employed in these cases (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M335">View MathML</a>) were still insufficient to recover the underlying relations; a method to obtain the result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M336">View MathML</a> is described in Remark 16 - which works well because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M337">View MathML</a>. It is interesting that in this work, as opposed to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M338">View MathML</a> constants we studied in [1], odd values and prime values of d generally yielded simpler relations than the even instances.

In the earlier study [1], we included Jason Kimberley’s observation that the degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M329">View MathML</a> of the minimal polynomial for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M340">View MathML</a> appeared to be given for odd primes by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M341">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M342">View MathML</a>. If one sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M343">View MathML</a>, for notational convenience, then it appears that for any prime factorization of an integer greater than 2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M344">View MathML</a>

(66)

Unfortunately, we have not yet been able to find a similar formula for the minimal polynomial degree in the compressed cases under study in Table 1. We are actively investigating this at the present time.

With regards to the run times listed in Table 1 (given to 0.01 second accuracy), it should be recognized that like all computer run times, particularly in a multicore or multiprocessor environment, they are only repeatable to two or three significant digits. They are listed here only to emphasize the extremely rapid increase in computational cost as the degree m and corresponding precision level P increase.

Remark 16 (A refined approach)

A referee has suggested making more more explicit the relations with elliptic functions. We had opted to limit the theory needed so as to better understand what experimental computation could produce for a wider audience.

(a) The basic quantity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126">View MathML</a>, which appears in Theorem 4, can be written in terms of the Jacobian elliptic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M346">View MathML</a>[10]. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M347">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M348">View MathML</a> as above, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M349">View MathML</a>

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M350">View MathML</a> in this case. All multiplication formulas for λ are then essentially special cases of addition formulas for Jacobian elliptic functions. At least in outline this makes it clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M126">View MathML</a> is algebraic, since elliptic functions evaluated at rational multiples of the periods are always algebraic with respect to k. In [12] some closely related lattice sums (essentially linear combinations of some of the lattice sums being considered herein) are evaluated in part by such reasoning.

(b) Similarly, for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M352">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M353">View MathML</a>

Then, in terms of the elliptic function cn[6,10], we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M354">View MathML</a>

(67)

and the corresponding duplication formula for f is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M355">View MathML</a>. Now the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M356">View MathML</a> and the Landen transform [6] yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M357">View MathML</a>

[[2], Appendix] which makes the minimal polynomial for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M358">View MathML</a> easily expressible

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M359">View MathML</a>

(68)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M360">View MathML</a>. Observe, much as in earlier examples, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M361">View MathML</a>. We may now obtain an algebraic equation relating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M363">View MathML</a> by computing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M364">View MathML</a> while reducing repeatedly using (68); finally, resolvant computations produce the equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M336">View MathML</a> that we failed to find in Table 1.

4 Conclusions

In this study, we have obtained some explicit closed-form evaluations of certain summations that arise as solutions of the Poisson equation, which in turn arises in numerous arenas ranging from engineering to studies of crystal structures. Many of these evaluations were first found experimentally by means of a process we have employed in other settings: computing the relevant expressions to high precision, then applying the PSLQ algorithm to recognize the resulting numerical values in closed form. In this case, we computed the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/75/mathml/M366">View MathML</a>, for various integers d, to a numeric precision of up to 25,000 digits, then found the integer coefficients of a polynomial that is satisfied by the constant up to a tolerance corresponding to the precision level. We then demonstrated a theoretical framework whereby such experimentally discovered relations can, in many cases, be formally proven.

For further details of the computation of lattice sums and related theta functions the reader is referred to [13-22].

Competing interests

Neither author has any personal, financial or other competing interests relevant to the topics studied in this paper.

Authors’ contributions

Both authors participated in the research work. DHB performed many of the numerical computations; JMB did much of the theoretical work.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors thank many colleagues and the referees for fruitful discussions about lattice sums and theta functions. The first author supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. The second author is supported in part by the Australian Research Council.

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