INTRODUCTORY SETTING: Chapter 1 of the PhD Thesis by Geoffrey B Campbell submitted to The Australian National University in the School of Mathematical Sciences in October 1997, degree awarded in April 1998.

 

1.1. THE BACKGROUND: PARTITIONS, q-SERIES, AND ARITHMETICAL FUNCTIONS.

The emergence of the theory of partitions in the past twenty years as a major field of research is due in no small part to the work of Andrews. His style of applying basic hypergeometric series to partition theoretic issues has clarified and given new perspective to a beautiful, detailed and elaborate area of mathematics. Andrews’ persistence in successfully attacking combinatorial problems with an armoury of q-series has been a catalyst for others to proceed in similar researches. His enthusiasm in promoting the "Lost Notebooks" of Ramanujan and their contents, and in encouraging others to solve related problems and conjectures has led to fruitful collaborations with many and diverse interests. (see Andrews [5-10], Alladi et al [2-3], Andrews et al [17])

Others, such as Askey [21-25] have picked up applications of Ramanujan’s mathematics such as to special functions, orthogonality, q-integrals, and so on yielding fruitful connections between various theories. We note here also Andrews’ enthusiastic and productive response to Baxter’s solving of the hard hexagon model, which highlighted the (at first) unexpected connection between the Rogers-Ramanujan identities and statistical mechanics. For accounts of this story see Andrews [9, chapters 1 and 8]. Relevant papers are by Andrews [8], Andrews and Baxter [13-14], then Andrews, Baxter and Forrester [15], or Andrews and Forrester [18].

Most of the real impetus for the work underlying this thesis probably starts from the papers of Ramanujan [84]. (see also Berndt [35-36]) His work aroused interest in the development of mock theta functions, congruence properties of partitions, asymptotic formulae for partition functions, new arithmetical functions, the bilateral basic hypergeometric series 1y 1, the Rogers-Ramanujan identities and all the "spin-offs" from them including the continued fractions, modular equations and the statistical mechanics inferences. (see Andrews et al [12])

Andrews’ book on partitions [6] provides a central point from which to begin in relation to the partition-theoretic work in this thesis. His chapter 2 gives an historical setting including the identities of Euler

(1.1)

which lead us into the theory of partitions of positive integers. The first of these is the well known "pentagonal number theorem". It asserts that the number of partitions of n into even distinct parts minus the number of its partitions into odd distinct parts is given by (-1)m if and zero otherwise.

Andrews chapter 2 also displays the identities of Gauss

(1.2)

and of Jacobi

(1.3)

This result is very important in the theory of theta series, and has found broader application than simply the theory of partitions of numbers. In particular we see that the theta functions

(1.4)

are representable as infinite products. Setting z equal to respectively in (1.3) we obtain (see Whittaker and Watson [96, chapter 21])

(1.5)

The main result of Andrews’ chapter 2 is the q-binomial theorem: If then

(1.6)

and he shows how this result underpins all of the above identities and theorems.

The arithmetical function work in this thesis also starts with Ramanujan. His arithmetical function for positive integers k, and n, (see originally Ramanujan [85], then Hardy [70], Hardy and Wright [71, §16.6], or Sivaramakrishnan [89, chapter IX])

(1.7)

has an intimate link with the visible point vector identities of the author, such as for ,

(1.8)

(1.9)

However Andrews’ [6] enables us to see the combinatorial interpretation for these identities coming from considering weighted versions of the vector partition generating functions given in his chapter 12. It is shown that for , ,

, (1.10)

, (1.11)

where P(n), Q(n) enumerate vector partitions of vector <n1, n2, ... , nr > in specific "unrestricted", "distinct" senses in the first hyperquadrant of Euclidean r-space.

From chapter 5 of this thesis onward, we develop a parallel theory to the basic hypergeometric series theory. In 1935 the Cambridge tract by Bailey [27] gave a good overview of the ordinary hypergeometric series, with a small chapter on q-series or basic hypergeometric series, including the Rogers Ramanujan identities. These latter, first published by Rogers [87] in 1894, became famous after their discovery by Ramanujan. For a modern account of this which highlights Rogers’ extraordinary skills see chapters 1 and 2 of Andrews [9]. The identities are

, (1.12)

. (1.13)

The interpretation in terms of partition theory of these identities is respectively:-

"The number of partitions of n in which the differences between parts are at least 2 is equal to the number of partitions of n into parts congruent to 1 or 4 (mod 5)", and

"The number of partitions of n into parts not less than 2 in which the differences between parts are at least 2 is equal to the number of partitions of n into parts congruent to 2 or 3 (mod 5)".

The book by Bailey was followed in the 1950s to early 1960s with papers by Slater culminating in her book [90], which developed the basic hypergeometric series including many other Rogers-Ramanujan style identities. However, a more recent comprehensive account of basic hypergeometric series was published in 1990 by Gasper and Rahman [65]. It brings together work on "summation, transformation and expansion" formulas, as well as "basic contour integrals", "bilateral basic hypergeometric series" and the "Askey-Wilson q-Beta Integrals" and related similar formulas, some of the latter being q-analogues for cases of the extension of the beta function due to Selberg [88],

(1.14)

where . Although it is beyond the scope of this thesis, the author aims to find Dirichlet series analogues for some of these results after aspects of chapters 5 to 9 of this thesis are more fully developed. For a further account of the q-analogues of Selberg’s integral see Andrews [9, chapter 5], where it is stated there are essentially two approaches to finding extensions of Selberg’s integral. One is related to generalized q-Hermite polynomials, and the other uses integrals following Ramanujan.

As background to the work underpinning chapters 5 to 9 of this thesis, we state some known results on basic hypergeometric series.

First given by Heine [73] last century is

(1.15)

valid for . A proof of this using only the q-binomial theorem is given in Andrews [6, chapter 2]. (1.15) has the Gauss hypergeometric series as limiting case when q approaches unity. Easily obtainable from the beta integral, this result is (see Bailey [27])

(1.16)

where c is not zero nor a negative integer and . In a recent paper [46] the author gave the Dirichlet series analogue for (1.15) and (1.16) as: for positive integers a, b, c, with

(1.17)

.

In this thesis we find further analogues as Dirichlet series for the q-series results. Among these is a similar result to (1.17) where the sum on the left side is taken over a subset Sm of the positive integers, namely (see formula (6.2.4) in chapter 6)

(1.18)

where the right side can be expressed more concisely in terms of arithmetical functions.

The q-Dixon sum given in Gasper and Rahman [65, pages 237 and 36-37] is

 

(1.19)

This is a q-analogue of the ordinary hypergeometric series result known as Dixon’s theorem (see Bailey [27])

(1.20)

Dixon’s theorem includes as a special case, the sum of the cubes of the binomial coefficients, and reduces to Kummer’s theorem as c is permitted to increase indefinitely. In the theory of the author’s paper [46], (1.19) leads to the Dirichlet series analogue:

If and also then

We use the notation here, , and

(1.22)

where is the sum of the g th powers of the divisors of k.

Some examples of (1.21) are as follows. Let us restrict our conditions so that b = c and successively 1, 2, 3, then 4. Then we have

(1.24)

(1.25)

(1.26)

These identities are all new. Similar summations to these appear in chapter 6. Possible researches to continue for these new analogues include for example, contiguous relations leading to continued fractions, analogues to differential equations, and also analogues to the q-gamma and q-beta functions leading to analogues of Selberg type integrals.

The purpose of chapter 5 of this thesis is to give notations and transformation theorems. The objective is to provide a means to easily "at sight" write down a Dirichlet series analogue just from looking at a q-series identity. For example, (1.17) or (1.18) would follow a priori from viewing (1.15) after knowing the transform and notation. Chapters 6 and 7 give examples of this approach.

Chapter 9 is the start of an attempt to form Dirichlet series analogues for the q-integral, just as Askey [22-25] has considered q-integrals.

 

 

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