The Partition Method for a Power Series Expansion: Theory and Applications

By Victor Kowalenko

The Partition Method for a Power Series Expansion: Theory and Applications explores how the method known as 'the partition method for a power series expansion', which was developed by the author, can be applied to a host of previously intractable problems in mathematics and physics.

In particular, this book describes how the method can be used to determine the Bernoulli, cosecant, and reciprocal logarithm numbers, which appear as the coefficients of the resulting power series expansions, then also extending the method to more complicated situations where the coefficients become polynomials or mathematical functions. From these examples, a general theory for the method is presented, which enables a programming methodology to be established.

Finally, the programming techniques of previous chapters are used to derive power series expansions for complex generating functions arising in the theory of partitions and in lattice models of statistical mechanics.

• Explains the partition method by presenting elementary applications involving the Bernoulli, cosecant, and reciprocal logarithm numbers
• Compares generating partitions via the BRCP algorithm with the standard lexicographic approaches
• Describes how to program the partition method for a power series expansion and the BRCP algorithm

• Language: English
• Publisher: Academic Press
• Release date: Jan 19, 2017
• ISBN: 9780128045114

Victor Kowalenko

Dr Victor Kowalenko is a Senior Research Fellow in the Department of Mathematics and Statistics, University of Melbourne, Australia. Since 2009, he has been associated with the ARC Centre of Excellence in Mathematics and Statistics of Complex Systems. He began his research career by joining the DSTO’s railgun project in Maribyrnong in the early 1980’s before transferring to the DSTO facility at Fishermen’s Bend to work on aeronautical systems. He then returned to the Department of Physics, University of Melbourne as one of the inaugural Australian Research Fellows to work on particle-anti-particle plasmas and general relativistic magnetohydrodynamics. It was here that he introduced the partition method for a power expansion. Between 2001 and 2003, when he was a Senior Research Fellow in the School of Computer Science and Software Engineering, Monash University, he was able to develop the method further and to extend it to intractable problems in mathematics and physics.

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Preface

Victor Kowalenko     Melbourne, Australia

, all taken to arbitrary powers and all three elliptic integrals. All these applications together with the interesting mathematics discovered in the process have appeared in separate publications. Thus, the first aim of this book has been to combine these separate works and extend them in one reference.

Another aim of this book is to provide the general theoretical framework behind the method since the previous applications only sketched out the method as it applied to the specific problem under investigation. Consequently, this book remedies the present situation by presenting a very general theory that is capable of going beyond the previously mentioned applications. In particular, the later chapters introduce further applications and developments resulting from the method that have not been previously published.

In the partition method for a power series expansion the coefficient of each power in the resulting series is determined by summing all the contributions due to all the integer partitions summing to the power under consideration. In calculating the contribution from each partition each part or element is assigned a specific value based on a partial expansion of the original function. In addition, one needs to know the number of occurrences or the multiplicity of each part in the partition. Hence the method requires the composition of each partition in order to calculate the coefficients in the resulting power series.

Whilst it is possible to write down or calculate by hand the first few orders of the resulting power series, it becomes impractical to do so for higher orders typically greater than the tenth order, where there are already 42 partitions required to determine the coefficient. At this stage it is better to develop a computational approach for determining the coefficients. Therefore, this book also presents the programming methodology behind the method. The methodology begins with the problem of generating partitions in the appropriate form so that a code based on the general theory can be created. By appropriate form it is meant that the partitions are expressed in the multiplicity representation rather than the standard lexicographic form, which most existing codes adopt. Consequently, a code based on representing the partitions in the form of a special non-binary tree known as a partition tree is presented. The bivariate recursive central partition or BRCP algorithm has the major advantage that it can be adapted with minor modification to handle a host of problems in the theory of partitions, which often require completely different programs.

With the aid of the BRCP algorithm, the general theory is developed into a programming methodology whereby a C/C++ code expresses the coefficients of the resulting power series in a general symbolic form. Then the output from the code can be transported to different computing platforms where mathematical packages such as Mathematica can be used to calculate the coefficients for specific problems either in rational or algebraic form, neither of which can be handled effectively in standard C/C++ with its floating point arithmetic. By importing the general symbolic forms for each power into Mathematica, one can exploit the package's integer arithmetic routines to express the coefficients in integer form, thereby avoiding (1) round-off errors due to scientific notation, and (2) under(over)flow due to under(over)sized numbers. On the other hand, one can use the package's symbolic routines to express the coefficients in algebraic form, e.g., as polynomials, for problems involving other variables and parameters unrelated to the variable in the power series.

Although the method has been applied in a number of interesting and challenging problems in this book, a sure sign of its power, it is still in its infancy with many more exciting developments waiting to be uncovered. For example, the material in the final chapters represents a cursory introduction into modifying the method to infinite products, which abound in applied mathematics and theoretical physics, particularly statistical mechanics. Therefore, it is hoped that this book will not only interest, but will also encourage the reader to apply the method in new and unforeseen applications.

I am indebted to Dr Glyn Jones and his colleagues at Elsevier for encouraging and supporting the production of this book, especially as it presents a totally novel approach from enumerative combinatorics to the subject of power series expansions. I also wish to thank Professor A.J. Guttmann for extending the facilities of the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, where this work was carried out. Finally, I thank Dr. D.X. Balaic, Australian X-Ray Capillary Optics Pty Ltd, for producing the first programs incorporating the BRCP algorithm. Although not used here, they were helpful in developing the programming methodology for the method and enabling the codes to be developed to tackle various problems in the theory of partitions mentioned in this book. Many of these programs, which represent a by-product of the BRCP algorithm, are listed in Chapter 7 and the second appendix for the reader, who may not necessarily be interested in power series expansions, but may have an interest in the properties and classes of partitions, a discipline in its own right.

August, 2016

Chapter 1

Introduction

Abstract

. The latter polynomials are also found to possess numerous and interesting properties.

Keywords

Bernoulli numbers/polynomials; Cosecant numbers/polynomials; Euler numbers/polynomials; Multiplicity; Partition tree; Power series expansion; Reciprocal logarithm numbers; Secant numbers/polynomials

The partition method for a power series expansion was first introduced in the derivation of an asymptotic expansion for the particular Kummer or confluent hypergeometric function that arises in the response theory of the charged Bose gas , which was unusual because it was the parameters and not the variable that affected the final asymptotic result. Consequently, it was possible to determine for the first time ever the physical properties of this fundamental quantum system of condensed matter physics in small or weak magnetic fields [2]. Although the charged Bose gas was considered initially, it was soon realized that the new expansion could be applied to the magnetized degenerate electron gas because the Bose Kummer function is a variant of the incomplete gamma function, upon which the response theory of this alternative, and far more important, system in condensed matter physics is based. Moreover, in an interesting twist the weak magnetic field expansion for the Bose Kummer function was later used in Ref. [1] to derive a large or strong magnetic field expansion, thereby allowing the strong magnetic field behaviour of the system to be presented in Ref. [2].

Because the partition method for a power series expansion had been applied effectively to an important special function of mathematical physics, a natural question was whether it could be applied to other special functions, particularly those where the standard techniques, e.g. Taylor/Maclaurin series, for obtaining power series expansions broke down. In response to this question, over the next decade or so, the partition method for a power series expansion was developed further so that it could be applied to a host of mathematical functions culminating in a series of papers, Refs. , and generalizations of them. Moreover, as a result of this activity another application was found where the method was used in the evaluation of trigonometric sums with inverse powers of cosine or sine [6,7]. These, however, will not be discussed here, even though they have produced some fascinating number-theoretical results.

Before proceeding any further, a good question to ask at this stage is: What do we mean by a power series expansion? In this book we shall use the following definition from Ref. [8], which states that

A power series in a variable z is an infinite sum of the form , where are integers, real numbers, complex numbers, or any other quantities of a given type.

This broad definition becomes confusing when we ask ourselves what a generating function is. According to Ref. . So as a rule, we shall regard generating functions as power series expansions whose coefficients involve special functions. It should also be mentioned here that in the literature the partition-number function is often called the partition function, but we shall refrain from doing so here because it means something quite different to those working in statistical mechanics.

As stated in the introduction to Ref. [5], the partition method for a power series expansion is not only capable of yielding power series where standard techniques, e.g., Taylor/Maclaurin series or the Hardy-Littlewood circle method [10], break down, but also in those cases where a standard technique can be applied. In these cases the power series expansions are ultimately identical to one another, but give rise to a totally different perspective. As we shall see, the cross-fertilization of both approaches is frequently responsible for the derivation of new mathematical results and properties. A particularly fascinating property of the partition method for a power series expansion is that the discrete mathematics of integer partitions results in power series expansions for continuous functions. Thus, discrete mathematics turns out to be the base of continuous mathematics.

Broadly speaking, the partition method for a power series expansion is composed of two major steps. For first step one requires the composition of each integer partition summing to the order k for each part i in a partition. Because the first step involves the generation of (integer) partitions, it can also be of importance to the general theory of partitions, especially when it results in alternative algorithms to the standard methods of generating partitions as discussed in Refs. [11–13]. Despite this, however, the more important step is the second one since it is responsible for producing the power series expansions for mathematical functions. In this step each partition summing to k provides a specific contribution to the coefficient of the k. Occasionally, there is a third step, but this will be addressed at the appropriate place.

For the lowest order terms in the resulting power series expansion it is possible to calculate the coefficients by hand, but after these have been determined, it becomes increasingly onerous to calculate the coefficients. This is due to the exponential increase in the number of partitions summing to higher powers, which is known as combinatorial explosion. For example, the number of partitions that sum to 10 is 42. Therefore, one needs to sum 42 distinct contributions in order to calculate the tenth order term in the power series expansion. At this stage it is far more expedient to calculate the coefficients by computer programs, which in turn means developing a programming methodology for the method. Such a methodology requires not only being able to generate the partitions at each order, but also calculating their contributions in symbolic form stemming from a general theory for the method. However, before a general theory can be created, first we need to examine some simple examples in this chapter to understand how the method can be applied to problems. Then in the following chapter we shall study more advanced examples, which will enable the theory to become as general as possible.

1.1 Cosecant Expansion

The first application of the partition method for a power series expansion after the derivation of the asymptotic power series expansion for the Bose Kummer function was the cosecant power series expansion. According to No. 1.411(11) of Gradshteyn and Ryzhik [14], the power series expansion for cosecant is given by

(1.1)

. In this result, which has been taken originally from Ref. represent the Bernoulli numbers. These famous numbers are defined by the following generating function:

(1.2)

The condition below (1.1) implies that x must be real, but as we shall see, x . For those values of t, one can replace the equivalence symbol by an equals sign. Since the process of regularization will be applied throughout this book, it is suggested that the reader, who is unaware of this concept, consult Appendix A for a more detailed exposition.

via the partition method for a power series expansion with the following theorem.

Theorem 1.1

There exists a power series expansion for cosecant, which can be written as

(1.3)

, etc. More generally, they are given by

(1.4)

represents the number of occurrences or multiplicity of the part or element i in each partition that sums to k represents the length or the total number of parts in a partition summing to k.

Remark

As in the case of the generating function for the Bernoulli numbers, viz. , not less than π as stated in the remark to Theorem 1 of Ref. [5].

Proof

By introducing the convergent power series expansion for sine, e.g., see Ref. as

(1.5)

As explained in Appendix , which means in turn that the series does not require regularization for these values of zas in the domain of convergence. Therefore, for all values of z, the regularized value of the geometric series can be expressed as

(1.6)

Whenever we are only concerned with the values of z for which the geometric series is convergent, which will arise in various problems or theorems in this book, we shall replace the equivalence symbol by an equals sign. Now if we introduce (1.6) into (1.5), then we obtain

(1.7)

Note that the introduction of an equivalence statement into (1.5) has resulted in another equivalence (statement). Since the above equivalence is a function of even powers, it can be expressed as

(1.8)

The series on the right-hand side (rhs) of . From the inequality, No. 3.7.29 in Ref. [23], this means that

(1.9)

Therefore, from the left-hand and right-hand sides or members of as stated in the theorem. For these values of xdoes not possess singularities at these values of x.

in .

So far, everything has been relatively simple except perhaps for the concept of regularization. Nevertheless, we can see that the evaluation of the higher orders of the cosecant numbers is becoming more complex. Consequently, two algorithms for evaluating them will be presented. Although the second of these methods is better from an optimization point of view, the first needs to be described as the second evolves from it.

For the first algorithm we still require the partition method for a power series expansion, which as stated earlier, was first introduced by the author in Refs. . One begins by drawing branch lines to all pairs of numbers that can be summed to the order, 2k, where the first number in the tuple is an even integer less than or equal to k . Its omission will be explained shortly. This recursive approach is applied repeatedly until all paths emanating from the seed number terminate with a tuple containing a zero. Hence the number of tuples with a zero in them represents the total number of paths, which in turn, as we shall see, gives the total number of partitions summing to 10 with only even