The Pennsylvania State University

The Graduate School

Department of Mathematics

THE BAILEY TRANSFORM AND CONJUGATE BAILEY

PAIRS

A Thesis in

Mathematics

by

Michael J. Rowell

c© 2007 Michael J. Rowell

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

August 2007

The thesis of Michael J. Rowell was reviewed and approved* by the following:

George AndrewsEvan Pugh Professor of MathematicsThesis Co-AdviserChair of Committee

Ae Ja YeeAssistant Professor of MathematicsThesis Co-Adviser

James SellersAssociate Professor of Mathematics

Donald RichardsProfessor of Statistics

John RoeProfessor of MathematicsHead of Department of Mathematics

*Signatures are on file in the Graduate School.

iii

Abstract

This thesis introduces a new generalized conjuagate Bailey pair and infinite fam-

ilies of conjugate Bailey pairs. We discuss the applications of each in conjuction with

the Bailey transform. Results range over many different applications: generalized Lam-

bert series, infinite products, Ramanujan-like identities, partitions, indefinite quadratics

forms and sums of triangular numbers.

We close with some partition-related remarks on two of the identities which appear

in previous chapters, and use this interpretation to prove generalizations and finite forms

of each of the identities.

iv

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Different Sets of Partitions . . . . . . . . . . . . . . . . . . . 6

2.2 Hypergeometric q-series . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The Bailey Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Umbral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 3. Conjugate Bailey Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 A General Conjugate Bailey Pair . . . . . . . . . . . . . . . . . . . . 17

3.2 Specific Conjugate Bailey Pairs . . . . . . . . . . . . . . . . . . . . . 20

Chapter 4. A Comprehensive Look into a Conjugate Bailey Pair . . . . . . . . . 25

4.1 Known Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Generalized Lambert Series and Related Identities . . . . . . . . . . 32

4.3 Infinite Products and Ramanujan-like Identities . . . . . . . . . . . . 37

v

4.4 Weighted sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 5. A General Discussion of Various Conjugate Bailey Pairs . . . . . . . 48

5.1 Bailey Pairs and the Symmetric Bilateral Bailey Transform . . . . . 48

5.2 Lambert Series, Infinite Products and Ramanujan-like Identities . . . 50

5.3 Indefinite Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5 Sums of Triangular Numbers . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 6. Infinite Families of Conjugate Bailey Pairs . . . . . . . . . . . . . . . 69

6.1 A generalization of Watson’s 8φ7 transformation formula . . . . . . . 69

6.2 Our Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Infinite families of conjugate Bailey pairs and Identities . . . . . . . 74

Chapter 7. Combinatorial and Partition-Related Remarks . . . . . . . . . . . . . 78

7.1 Some Generalizations of Fine’s Identity . . . . . . . . . . . . . . . . 79

7.1.1 A General Case of a Simple Bijection . . . . . . . . . . . . . . 79

7.1.2 Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.1.3 The Eulerian Number Triangle and the Polylogarithm Function 87

7.1.4 Combining Eulerian Polynomials and our Generalization . . . 92

7.1.5 Another choice for An(q) . . . . . . . . . . . . . . . . . . . . 95

7.2 Combinatorial Interpretations of One of Ramanujan’s Entries . . . . 100

vi

7.2.1 Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2.2 Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Chapter 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

vii

List of Tables

3.1 Conjugate Bailey Pairs for when a 7→ 1. . . . . . . . . . . . . . . . . . . 22

3.2 Conjugate Bailey Pairs for when a 7→ −1. . . . . . . . . . . . . . . . . . 23

3.3 Conjugate Bailey Pairs for when a 7→ 0. . . . . . . . . . . . . . . . . . . 24

3.4 Conjugate Bailey Pairs for when a, b 7→ ∞. . . . . . . . . . . . . . . . . 24

viii

List of Figures

2.1 A Young Diagram of the partition (4, 4, 2, 1). . . . . . . . . . . . . . . . 5

7.1 Our map, φ, used in 7.2.1 illustrated above. . . . . . . . . . . . . . . . . 104

ix

Acknowledgments

I would first and foremost like to thank my parents, Jim and Cindy Rowell. It

was and continues to be their constant support that enables me to take the steps that

I have taken in my life. They have been my most instrumental teachers throughout my

life and without them I would be lost.

This work would have never begun had it not been for the time and effort put

forth by both Dr. George Andrews and Dr. Ae Ja Yee. I cannot begin to thank them

enough for their patience and guidance.

And lastly I would like to thank Lisa Johansen. It has been her unwavering love

and support that has not only made me a better mathematician, but a better person as

well.

1

Chapter 1

Introduction

In 1949 W.N. Bailey introduced the Bailey transform [12], and using this trans-

form was able to give a simple proof of the Rogers-Ramanujan identities; for |q| < 1,

1 +∞∑n=0

qn2

(1− q)(1− q2) · · · (1− qn)=

1(1− q)(1− q6) · · · (1− q4)(1− q9) · · ·

(1.1)

and

1 +∞∑n=0

qn2+n

(1− q)(1− q2) · · · (1− qn)=

1(1− q2)(1− q7) · · · (1− q3)(1− q8) · · ·

(1.2)

as well as many other Ramanujan-like identities. The ingredients for the Bailey transform

are two pairs, a Bailey pair and a conjugate Bailey pair. In 1951 Slater published a

long list of known and new Bailey pairs [23] which Slater soon followed in 1952 by

publishing a list of 130 Ramanujan-like identities, many of which were new. Since the

introduction of the Bailey transform, there have many adavances in pairs, both Bailey

and conjugate Bailey, and a long list of identities. One work in particular which served

as the main motivation for this paper is a joint work by Andrews and Warnaar [9] in

which a number of new conjugate Bailey pairs are introduced. It is the purpose of this

manuscript to investigate the conjugate Bailey pairs used in their paper, generalize them,

2

apply the Bailey transform to them, and to interpret the results both analytically and

combinatorially.

The new pairs introduced by Andrews and Warnaar involved an indefinite sum

which appeared to make things more complicated than previous pairs, but when used

with appropriate Bailey pairs, produced interesting and less convoluted results. After an

in-depth dissection of the proofs used by Andrews and Warnaar, I was able to consolidate

the method used and generalize the pairs extensively. It turned out that not only were

all of the conjugate Bailey pairs used in Andrews and Warnaar encompassed in this

new pair, but also all of the conjugate Bailey pairs used in Bailey’s and Slater’s work

were included in this new generalization. Chapter 3 of this manuscript discusses the

previously mentioned steps.

Once the generalized conjugate Bailey pair was found, it was left to show that

it had interesting applications with the use of the Bailey transform. With many steps

similar to those taken by Bailey, Slater, Andrews and Warnaar, we are able to present

in Chapters 4 and 5 some of our results. While many of the new identities were easily

simplified using classic identities such as the Jacobi triple product, some were unable to

be simplified. It is with the use of Umbral methods that these identities were further

simplified and were able to be interpreted as an elegant partition identity. Details of

these Umbral methods can be found in Chapter 2.

In Chapter 6, we again generalize our conjugate Bailey pair so that we are able to

discuss infinite families of such pairs. In order to do so we make use of a generalization

of Watson’s transformation formula which can be proved with the use of Bailey Chains.

3

Such an investigation leads to infinite families of some of the identities found in previous

chapters.

Also explored in this thesis are some alternative methods of proof to some of the

identities found using Andrews’ and Warnaar’s conjugate Bailey pairs. In Chapter 7,

interpreting the identities as partition identities we are able to present new finite forms

of identities as well as many new interesting generalizations.

4

Chapter 2

Preliminaries

This section is intended to introduce the reader to the basic definitions and no-

tations that will be used later in this manuscript. I have chosen to introduce the topic

as it was introduced to me, starting with partitions. It was after I was roped in by the

elegance and simplicity of partition identities that I was shown the complicated world of

q-series and the horrific notation that comes with it.

2.1 Partitions

We define a partition as a finite nonincreasing sequence of positive integers, λ =

(λ1, . . . , λk). We refer to each λi as the parts of our partition. We say that λ is a

partition of n, |λ| = n, if the sum of the parts is equal to n. For example, there are 7

partitions of 5,

(5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1). (2.1)

We also define the following statistics for a partition, λ,

• λ1 the largest part,

• µ(λ) the number of parts,

• µk(λ) the number of parts equal to k,

5

• #d(λ) the number of different parts.

Another statistic made famous by Dyson is the rank of a partition, r(λ), which is

equal to the largest part minus the number of parts, λ1 − µ(λ). We define P to be the

set of all partitions.

Example 2.1.1. If we consider the partition λ = (10, 4, 4, 3, 2, 2, 2, 1), we have the

following statistics:

|λ| = 28, λ1 = 10, µ(λ) = 8, µ4(λ) = 2, #d(λ) = 5, r(λ) = 2.

To each partition we can associate a graphical representation [3, p. 6], in which

case each row corresponds to a part of the partition (See Figure 2.1). In the example

we have shown, nodes are expressed using boxes. It is also common to see dots used, in

which case our representation is referred to as a Ferrers graph.

Fig. 2.1. A Young Diagram of the partition (4, 4, 2, 1).

6

2.1.1 Different Sets of Partitions

In later sections, we will use partitions to help us prove identities by showing that

the coefficient of qn on either side of the identity counts the same set of partitions. Since

it will not always be the case that the set of partitions that we need is going to be P,

we define some other useful subsets of P. For example, we might restrict ourselves to

partitions with each part less than a given bound or only allow parts which are odd. We

start by defining D, the set of partitions in which all of our parts are distinct. If we

continue our previous example we see that there are only 3 partitions of 5 into distinct

parts,

(5), (4, 1), (3, 2). (2.2)

We can continue to further complicate our restrictions, but we need more specific

statistics for our partitions. We define µik(λ) as the number of parts of λ which are

congruent to i modulo k. We define the sets of partitions:

1. Dk to be all partitions into distinct parts such that each part is congruent to k

modulo k.

2. Dik

to be all partitions into distinct parts such that each part is congruent to i, k

or k − i modulo k.

3. Dk,i to be all partitions in Dik

such that µik(λ) > µk−i

k(λ).

4. D=k,i

to be all partitions in Dik

such that µik(λ) ≥ µk−i

k(λ).

We also define the set of partitions without gaps. A partition without gaps has

the property that if k occurs as a part, then all positive integers less than k must occur

7

as parts. For example, there are 3 partitions of 5 without gaps,

(2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1). (2.3)

We note that it is not a coincidence that the number of partitions of 5 without

gaps is equal to the number of partitions of 5 into distinct parts. There is a simple

bijection between the two sets using conjugation. The conjugate of a partition can be

found by considering the Young Diagram of a partition and referring to the columns as

parts (rather than the rows). More can be read on the conjugate of a partition in [3,

p. 7]. We can also consider partitions into odd parts in which there are no gaps. For

example, the partition λ = (7, 5, 5, 3, 1, 1, 1).

Our last set of partitions we will define is the set of overpartitions. An overparti-

tion of n is a partition of n in which the first occurence of a number may be overlined.

There are many more overpartitions of a number n than there are partitions of n. For

example, there are 24 overpartitions of 5, compared to the 7 partitions. Below we show

the 8 overpartitions of 3,

(3), (3), (2, 1), (2, 1), (2, 1), (2, 1), (1, 1, 1), (1, 1, 1). (2.4)

2.2 Hypergeometric q-series

Often we are concerned with the number of partitions of a number. For example,

we can define p(n) to be the number of partitions of n. As with many sequences, it is

8

often helpful to express it as a generating function which we will denote as P(q) (|q| < 1),

P(q) =∞∑n=0

p(n)qn. (2.5)

We first note that this can easily be expressed as

∞∑n=0

p(n)qn =∑λ∈P

q|λ| =∞∏n=1

11− qn

. (2.6)

We can see this by noting that each term on the right hand side, 1/(1− qn) = 1 + qn +

q2n+ · · · , contributes the number of parts of size n. If we want to express the generating

function for all partitions with distinct parts, we need to ensure that no part is chosen

more than once. Thus, ∑λ∈D

q|λ| =∞∏n=1

(1 + qn). (2.7)

It is apparent at this point that we need to introduce some notation if we would like to

cut down on the number of infinite product symbols we use. The following is standard

q-series notation [19, p. xvi]:

(a)k = (a; q)k = (1− a)(1− aq) · · · (1− aqk−1) =k−1∏i=0

(1− aqi) (2.8)

and (2.9)

(a)∞ = (a; q)∞ = limk→∞

(a; q)k =∞∏i=0

(1− aqi). (2.10)

9

We can also combine our infinite products in the following way:

(a1; q)k(a2; q)k · · · (an; q)k = (a1, a2, · · · , an; q)k. (2.11)

We also define an n+1φn basic hypergeometric series as [19, p. 4]

n+1φn

[a1, · · · an+1b1, · · · , bn

; q, z]

=∞∑k=0

(a1, · · · an+1; q)k(q, b1, · · · , bn; q)k

zk. (2.12)

With our new notation we can now easily express two of our generating functions,

∑λ∈P

qn =1

(q)∞(2.13)

and ∑λ∈D

qn = (−q)∞. (2.14)

It is with this notation that we can introduce some of the more classic identities

of q-series.

Theorem 2.2.1 (Euler). [3, p. 19]

∞∑n=0

zn

(q)n=

1(z)∞

. (2.15)

Theorem 2.2.2 (q-binomial Theorem). [3, p. 17]

∞∑n=0

(a)n(q)n

zn =(az)∞(z)∞

. (2.16)

10

Theorem 2.2.3 (Gauss). [3, p. 23]

∑n=0

qn(n+1)/2 =(q2; q2)∞(q; q2)∞

. (2.17)

Theorem 2.2.4 (Jacobi Triple Product). [3, p. 21]

∞∑n=−∞

(−1)nanqn(n−1)/2 = (a, q/a, q; q)∞. (2.18)

Theorem 2.2.5 (Euler’s Pentagonal Number). [3, p. 11]

(q)∞ =∞∑

n=−∞(−1)kqk(3k+1)/2. (2.19)

It should be noted that all of these identities are very closely related to partitions

and their generating functions. For example, we can view the right hand side of (2.19)

as

(q)∞ =∑λ∈D

(−1)µ(λ)q|λ|. (2.20)

So we can interpret this as the generating function for counting strict partitions

in which their sign will be assigned based on the number of parts. From (2.19) we can

see that when n is not a pentagonal number (i.e. of the form qn(3n+1)/2, n ∈ Z), there

are the same number of strict partitions into an even number of parts as there are into

an odd number of parts. For example, let us consider n = 8. Below are the 6 strict

partitions of 8. Note that exactly half of them have an odd number of parts.

11

(8), (7, 1), (6, 2), (5, 3), (5, 2, 1), (4, 3, 1).

2.3 The Bailey Transform

In 1949 [12], W.N. Bailey introduced what is now known as the Bailey Transform:

If

βn =n∑r=0

αrun−rvn+r, (2.21)

and

γn =∞∑r=n

δrur−nvr+n, (2.22)

then∞∑n=0

αnγn =∞∑n=0

βnδn, (2.23)

subject to conditions on the four sequences αn, βn, γn and δn which make all the infinite

series absolutely convergent. For the purpose of this thesis, we will need a slight variation

on the Bailey transform. The following is referred to as the symmetric bilateral Bailey

transform: If

βn =n∑

r=−nαrun−rvn+r, (2.24)

and

γn =∑r≥|n|

δrur−nvr+n, (2.25)

then∞∑

n=−∞αnγn =

∞∑n=0

βnδn, (2.26)

12

with the same convergent conditions on αn, βn, γn and δn. In either form, we refer to

the series αn and βn as a Bailey pair and the series δn and γn as a conjugate Bailey pair.

As the symmetric bilateral Bailey transform is the only version of the Bailey transform

we will use for this thesis, we will refer to it as just the Bailey transform.

As mentioned previously, Bailey used the Bailey transform to introduce a new

method of proof for the Rogers-Ramanujan Identities and many other Ramanujan-like

identities [12].

2.4 Umbral Methods

In later chapters, we implement the use of Jacobi’s triple product to simplify our

results to a Ramanujan-like identity. There are however, a list of results which appear

to be closely related to these Ramanujan-like identities in which our triple product does

not apply. It is for these identities that we can use an Umbral mapping in our triple

product and simplify our results.

For the purpose of this manuscript, an Umbral mapping allows replacing the

powers of a given paramter with a sequence, (an → an). As long as our series are

already expanded in terms of our parameter and the series will continue to converge

after our substitution, we can use this Umbral mapping. For further discussion and

examples please see [6].

13

In this section we will investigate Jacobi’s triple product identity with the use of

Umbral methods. Let us first recall Jacobi’s triple product identity,

∞∑n=−∞

znqn2

= (−zq,−z−1q, q2; q2)∞. (2.27)

Using the following simple application of (2.16) (with q → q2, z → −q/a followed by

a→∞) ,

∞∑n=0

znqn2

(q2; q2)n= (−zq; q2)∞ (2.28)

we can rewrite the triple product identity as

∞∑n=−∞

znqn2

= (q2; q2)∞∑i,j≥0

zi−jqi2+j2

(q2; q2)i(q2; q2)j

. (2.29)

We have now written our triple product identity in a way such that the variable z is

conducive for an Umbral mapping.

Let us consider the function,

s(n) =

0 for n ≥ 0

1 for n < 0.

With this function we consider our Umbral mapping,

14

Theorem 2.4.1.

∞∑n=−∞

(−1)s(n)znqn2

= (q2; q2)∞∑

i,j≥0i≥j

zi−jqi2+j2

(q2; q2)i(q2; q2)j

−(q2; q2)∞∑

i,j≥0i<j

zi−jqi2+j2

(q2; q2)i(q2; q2)j

. (2.30)

Proof. We invoke the Umbral mapping, zn 7→ (−1)s(n)zn,

∞∑n=−∞

znqn2

= (q2; q2)∞∑i,j≥0

(−1)s(i−j)zi−jqi2+j2

(q2; q2)i(q2; q2)j

(2.31)

= (q2; q2)∞∑

i,j≥0i≥j

zi−jqi2+j2

(q2; q2)i(q2; q2)j

−(q2; q2)∞∑

i,j≥0i<j

zi−jqi2+j2

(q2; q2)i(q2; q2)j

. (2.32)

While the generating functions for such identities may not be aesthetically pleas-

ing to the eye, the combinatorial interpretations turn out to be much more elegant.

With the notation we defined earlier and the use of Umbral methods, we are now able

to translate these results into partition identities.

15

Theorem 2.4.2. Let αn be a double-sided sequence which ensures convergence and

k ∈ Z with k > 2. Then

∞∑n=−∞

αn(−1)nqn(kn+k−2)/2 =∑λ∈D1

k

αµk−1

k (λ)−µ1k(λ)

(−1)µ(λ)q|λ|. (2.33)

Proof. Let z 7→ −zq(k−2)/2 and q 7→ qk/2 in the Jacobi triple product. Then we have

∞∑n=−∞

zn(−1)nqn(kn+k−2)/2 = (z−1q, zqk−1, qk; qk)∞ (2.34)

=∑λ∈D1

k

zµk−1

k(λ)−µ1

k(λ)(−1)µ(λ)q|λ|. (2.35)

We now invoke the umbral mapping zn 7→ αn.

One of the more simple but elegant partition identities that follows from this

Umbral map is the following:

Corollary 2.4.3.

∞∑n=0

(−1)nqn(3n+1)/2(1 + q2n+1) =∞∑

n=−∞(−1)n+s(n)qn(3n+1)/2 (2.36)

=∑λ∈D

µ23(λ)−µ1

3(λ)≥0

(−1)µ(λ)q|λ|

−∑λ∈D

µ23(λ)−µ1

3(λ)<0

(−1)µ(λ)q|λ|. (2.37)

And so we are given a variation of the well-known Pentagonal Number Theorem.

16

Chapter 3

Conjugate Bailey Pairs

It appears in the literature that the main focus when using the Bailey transform

is to find new Bailey pairs and use them in conjunction with one of the well-known

conjugate Bailey pairs [12], [23], [24]. For whatever reason, there has been little interest

expressed in finding new conjugate Bailey pairs. One obvious reason for this is that the

relationship which must hold in these pairs is far more complicated.

In a recent paper [9], Andrews and Warnaar made a monumental step in the

direction of this thesis by introducing many new conjugate Bailey pairs. Two of the

pairs introduced were very similar in appearance, but unfortunately the resemblance

was all but lost at the proof level. They were the following,

Lemma 3.0.4 (Andrews, Warnaar). The identity,

γn =∑j≥n

δjuj−nvj+n (3.1)

holds for un = vn = 1/(q2; q2)n with

δn =(q2; q2)2n

(−q; q)2n+1qn, γn = q−n

2 ∑j≥n

qj2+j (3.2)

17

and

δn = (q)2nqn, γn = q−2n2 ∑

j≥2nqj(j+1)/2. (3.3)

One of the more striking observations of these new pairs is the existence of a

restricted sum in γn, a characteristic not seen in previous conjugate Bailey pairs.

Andrews, who had already moved on to other research, encouraged me to look

into these pairs feeling that he had only touched a small amount of what appeared to

be a larger picture. My goal, upon seeing the pairs, was to unify the two pairs above, as

well as the others in the paper, into one generalization. After a great deal of studying

the pairs and their proofs, which differed in style, I was able to generalize the pairs a

great deal. Seldom as this occurs, I was able to achieve my goal; bringing together the

pairs as well as their proofs. In this chapter, I present this generalization and its proof.

As Andrews had hinted, the pairs were in fact just the tip of an iceberg; the

generalization found contained an infinite number of new conjugate Bailey pairs. To

give an indication of the vastness of such pairs, tables of some of the simple pairs are

presented later in this chapter.

3.1 A General Conjugate Bailey Pair

The following theorem is our main result regarding conjugate Bailey pairs. We

present a very general conjugate Bailey pair and it’s proof. It should be mentioned that

the proof, while proving a much more general identity, was able to simplify greatly those

steps taken by Andrews and Warnaar in their paper [9].

18

Theorem 3.1.1.

γn =∑j≥n

δj

(q; q)j−n(fq; q)j+n(3.4)

=(efq/a, a; q)n(fq, fq/a; q)n

(−1a

)nq−n(n−1)/2

×∑j≥n

(efqn+1/a, fq/a, b, c; q)j(eq/a; q)j−n(efq2/ab, efq2/ac, fqn+1, eq; q)j(q; q)j−n

×(1− efq2j+1/a)(−efbc

)jqj(j+3)/2 (3.5)

where

δn =(efq2/abc, efq/a; q)∞(efq2/ab, efq2/ac; q)∞

· (a, b, c; q)n(eq; q)n

(efq2

abc

)n. (3.6)

19

Proof. Our proof is an application of Watsons 8φ7 transformation:

γn =∑j≥n

δj

(q; q)j−n(fq; q)j+n(3.7)

=(efq2/abc, efq/a; q)∞(efq2/ab, efq2/ac; q)∞

∑j≥n

(a, b, c; q)j(eq; q)j(q; q)j−n(fq; q)j+n

(efq2

abc

)j(3.8)

=(efq2/abc, efq/a; q)∞(efq2/ab, efq2/ac; q)∞

· (a, b, c; q)n(eq; q)n(fq; q)2n

(efq2

abc

)n3φ2

(aqn, bqn, cqn

eqn+1, fq2n+1 ; q,efq2

abc

)

(3.9)

=(efq2/abc, efq/a; q)∞(efq2/ab, efq2/ac; q)∞

· (a, b, c; q)n(eq; q)n(fq; q)2n

(efq2

abc

)n· (efqn+2/ab, efqn+2/ac; q)∞

(efq2n+2/a, efq2/abc; q)∞

× limd 7→∞ 8φ7

efq2n+1

a ,

√efq2n+3

a ,−√efq2n+3

a , fqn+1

a , eqa , bqn, cqn, d√

efq2n+1

a ,−√efq2n+1

a , eqn+1, fq2n+1, efqn+2

ab , efqn+2

ac , 0; q,

efq2+n

bcd

(3.10)

by eq.(III.17) [19, p. 360] with a = efq2n+1/a, b = fqn+1/a, c = eq/a, d 7→ ∞, e = bqn

and f = cqn. After some simplification we see that this

=(efq/a, a; q)n(fq, fq/a; q)n

(−1a

)nq−n(n−1)/2

·∑j≥n

(efqn+1/a, fq/a, b, c; q)j(eq/a; q)j−n(efq2/ab, efq2/ac, fqn+1, eq; q)j(q; q)j−n

(1− efq2j+1/a)(−efbc

)jqj(j+3)/2.

(3.11)

It should be noted that all conjugate Bailey pairs introduced in Andrews’ and

Warnaar’s work [9] are encompassed in this theorem. It should also be noted that the

20

conjugate Bailey pair used by Bailey [12] and Slater [24] in their work,

δn =(y)n(z)nx

n

ynzn, γn =

(x/y)∞(x/z)∞(x)∞(x/yz)∞

· (y)n(z)nxn

(x/y)n(x/z)nynzn

(3.12)

is also a special case of our theorem. We can see this by allowing a = eq followed by

some simple change of variables.

3.2 Specific Conjugate Bailey Pairs

Throughout the paper corollaries and theorems will use special cases of Theorem

3.1.1. However, in most cases there will still be at least one open parameter. In order to

show the number of pairs that our theorem can create, we give some tables of some of the

more simple pairs. As we saw in the previous section, allowing a = eq allowed for a large

simplification. Our theorem will also simplify greatly if we allow e = a. The following

conjugate Bailey pair has the mapping q 7→ q2 followed by e = a, f = 1, a 7→ aq, b 7→ bq

and c 7→ cq in Theorem 3.1.1.

Corollary 3.2.1.

γn =∑j≥n

δj

(q2; q2)j−n(q2; q2)j+n(3.13)

=(aq; q2)n(q/a; q2)n

(−1a

)nq−n

2 ∑j≥n

(q/a, bq, cq; q2)j(q3/b, q3/c, aq3; q2)j

(1− q4j+2)(− a

bc

)jqj(j+2)

(3.14)

21

where

δn =(q2/bc, q2; q2)∞(q3/b, q3/c; q2)∞

· (1− aq)(bq, cq; q2)n(1− aq2n+1)

(q2

bc

)n. (3.15)

All pairs written in the following tables are special cases of Corollary 3.2.1. The

values for a, b and c accompany each pair. We note that the two lemmas of Andrews

and Warnaar previously mentioned appear on our list: a = −1, b = 1, c = q and a =

0, b = 1, c = q.

22

a b c γn δn

1 1 1 (−1)nq−n

2 ∑j≥n

(1+q2j+1

)

(1−q2j+1)2(−1)

jq

j2+2j

(−q)2

∞(q2; q

2)2

∞(q;q

2)2n

(1−q2n+1)q2n

1 1 −1 (−1)nq−n

2 ∑j≥n

qj2+2j

(1−q2j+1)(−q

2; q

2)∞(q

4; q

4)∞

(q2;q

4)n

(1−q2n+1)(−1)

nq2n

1 1 q (−1)nq−n

2 ∑j≥n

(1+q2j+1

)

(1−q2j+1)(−1)

jq

j2+j (q)2n

(1−q2n+1)q

n

1 1 −q (−1)nq−n

2 ∑j≥n

(1+q2j+1

)

(1−q2j+1)q

j2+j

(−q; q2)2

∞(q2; q

2)∞

(q;−q)2n

(1−q2n+1)(−1)

n(q)

n

1 −1 −1 (−1)nq−n

2 ∑j≥n

(−1)jq

j2+2j

(1+q2j+1)(q)

2

∞(−q2; q

2)2

∞(−q;q

2)2n

(1−q2n+1)q2n

1 −1 q (−1)nq−n

2 ∑j≥n

qj2+j (−q;−q)2n

(1−q2n+1)· (−1)

nq

n

1 −1 −q (−1)nq−n

2 ∑j≥n

(−1)jq

j2+j

(q)∞(q; q2)∞

(−q)2n

(1−q2n+1)q

n

1 ∞ 1 (−1)nq−n

2 ∑j≥n

(1+q2j+1

)

(1−q2j+1)q2j

2+2j

(−q)∞(q2; q

2)∞

(q;q2)n

(1−q2n+1)(−1)

nq

n2+2n

1 ∞ −1 (−1)nq−n

2 ∑j≥n

(−1)jq2j

2+2j

(q)∞(−q2; q

2)∞

(−q;q2)n

(1−q2n+1)q

n2+2n

1 ∞ q (−1)nq−n

2 ∑j≥2n

qj(j+1)/2 (q

2;q

2)n

(1−q2n+1)(−1)

nq

n2+n

1 ∞ −q (−1)nq−n

2 ∑j≥2n

(−1)bj/2c

qj(j+1)/2

(q)∞(−q; q2)∞

(−q2;q

2)n

(1−q2n+1)q

n2+n

1 ∞ −q2

(−1)nq−n

2 ∑j≥n

(1 + q2j+1

)2(−1)

jq2j

2

(q)∞(−q2; q

2)∞

(−q;q2)n+1

(1−q2n+1)q

n2

1 ∞ ∞ (−1)nq−n

2 ∑j≥3n

j≡0,1 mod 3(−1)

b j3 cq

j(j+2)/3(q

2; q

2)∞

q2n2+2n

(1−q2n+1)

Table 3.1. Conjugate Bailey Pairs for when a 7→ 1.

23

a b c γn δn

−1 1 1 q−n

2 ∑j≥n

qj2+2j

(1−q2j+1)(−q)

2

∞(q2; q

2)2

∞(q;q

2)2n

(1+q2n+1)q2n

−1 1 −1 q−n

2 ∑j≥n

(−1)jq

j2+2j

(1+q2j+1)(−q

2; q

2)∞(q

4; q

4)∞

(q2;q

4)n

(1+q2n+1)(−1)

nq2n

−1 1 q q−n

2 ∑j≥n

qj2+j (q)2n

(1+q2n+1)(q)

n

−1 1 −q q−n

2 ∑j≥n

(−1)jq

j2+j (−q,q

2;q

2)∞

(q;−q)∞· (q;−q)2n

(1+q2n+1)(−1)

n(q)

n

−1 −1 −1 q−n

2 ∑j≥n

(1−q2j+1

)

(1+q2j+1)2q

j2+2j

(q)2

∞(−q2; q

2)2

∞(−q;q

2)2n

(1+q2n+1)q2n

−1 −1 q q−n

2 ∑j≥n

(1−q2j+1

)

(1+q2j+1)(−1)

jq

j2+j (−q;−q)2n

(1+q2n+1)· (−1)

nq

n

−1 −1 −q q−n

2 ∑j≥n

(1−q2j+1

)

(1+q2j+1)q

j2+j (q)∞

(−q)∞· (−q)2n

(1+q2n+1)q

n

−1 ∞ 1 q−n

2 ∑j≥n

(−1)jq2j

2+2j (q

2;q

2)∞

(q;q2)∞· (q;q

2)n

(1+q2n+1)(−1)

nq

n2+2n

−1 ∞ −1 q−n

2 ∑j≥n

(1−q2j+1

)

(1+q2j+1)q2j

2+2j (q

2;q

2)∞

(−q;q2)∞· (−q;q

2)n

(1+q2n+1)q

n2+2n

−1 ∞ q q−n

2 ∑j≥n

(1− q2j+1

)(−1)jq2j

2+j (q

2;q

2)n

(1+q2n+1)(−1)

nq

n2+n

−1 ∞ −q q−n

2 ∑j≥n

(1− q2j+1

)q2j

2+j (q

2;q

2)∞

(−q2;q2)∞· (−q

2;q

2)n

(1+q2n+1)q

n2+n

−1 ∞ q2

q−n

2 ∑j≥n

(1− q2j+1

)2(−1)

jq2j

2 (q2;q

2)∞

(q;q2)∞· (q;q

2)n+1

(1+q2n+1)(−1)

nq

n2

−1 ∞ ∞ q−n

2 ∑j≥3n

j≡0,1 mod 3(−1)

j−3b j3 cq

j(j+2)/3(q

2; q

2)∞

q2n2+2n

(1+q2n+1)

Table 3.2. Conjugate Bailey Pairs for when a 7→ −1.

24

a b c γn δn

0 1 1 q−2n

2 ∑j≥n

(1+q2j+1

)

(1−q2j+1)q2j(j+1)

(−q)2

∞(q2; q

2)2

∞(q; q2)2

nq2n

0 1 −1 q−2n

2 ∑j≥n

(−1)jq2j(j+1)

(q2; q

2)∞(−q

2; q

2)2

∞(q2; q

4)n(−1)

nq2n

0 1 q q−2n

2 ∑j≥2n

qj(j+1)/2

(q)2nqn

0 1 −q q−2n

2 ∑j≥2n

(−1)bj/2c

qj(j+1)/2

(q2; q

2)∞(−q; q

2)2

∞(q;−q)2n(−1)nq

n

0 ∞ 1 q−2n

2 ∑j≥3n

j≡0,1 mod 3(−1)

b j3 cq

j(j+2)/3 (q2;q

2)∞(q;q

2)n

(q;q2)∞(−1)

nq

n2+2n

0 ∞ q q−2n

2 ∑j≥3n

j≡0,1 mod 3(−1)

b j+13 c

qj(j+1)/3

(q2; q

2)n(−1)

nq

n2+n

0 ∞ −q q−2n

2 ∑j≥3n

j≡0,1 mod 3(−1)

j+b j+13 c

qj(j+1)/3 (q

2;q

2)∞(−q

2;q

2)n

(−q2;q2)∞q

n2+n

0 ∞ q2

q−2n

2 ∑j≥n

(1− q2j+1

)(1− q4j+2

)(−1)jq3j

2

(−q)∞(q2; q

2)∞(q; q

2)n+1(−1)

nq

n2

0 ∞ ∞ q−2n

2 ∑j≥2n

(−1)jq

j(j+1)(q

2; q

2)∞q

2n2+2n

Table 3.3. Conjugate Bailey Pairs for when a 7→ 0.

a b c γn δn

∞ ∞ 1 (−1)nq

n2 (q

2;q

2)∞

(q;q2)∞(q; q

2)n(−1)

nq

n2

∞ ∞ −1 qn2 (q

2;q

2)∞

(−q;q2)∞(−q; q

2)nq

n2

∞ ∞ q (1− q2n

)(−1)nq

n2−n

(q2; q

2)n(−1)

nq

n2−n

∞ ∞ −q (1 + q2n

)qn2−n (q

2;q

2)∞

(−q2;q2)∞(−q

2; q

2)nq

n2−n

∞ ∞ q2

(1− q2n−1

)(1− q2n+1

)(−1)nq

n2−2n (q

2;q

2)∞

(q;q2)∞(q; q

2)n+1(−1)

nq

n2−2n

∞ ∞ −q2

(1 + q2n−1

)(1 + q2n+1

)qn2−2n (q

2;q

2)∞

(−q;q2)∞(−q; q

2)n+1q

n2−2n

∞ ∞ ∞ q2n

2

(q2; q

2)∞q

2n2

Table 3.4. Conjugate Bailey Pairs for when a, b 7→ ∞.

25

Chapter 4

A Comprehensive Look into a Conjugate Bailey Pair

As mentioned in Chapter 3, the specific pairs that can be obtained from our

general result are endless. In order to give some understanding as to what Theorem

3.1.1 is capable of producing in conjunction with the Bailey transform, we make some

simple assumptions for our parameters.

We first consider the mapping of a 7→ 1 into Corollary 3.2.1. In such a case our

theorem reduces to

γn = (−1)nq−n2 ∑j≥n

(bq, cq; q2)j(q3/b, q3/c; q2)j

(1 + q2j+1)(− 1bc

)jqj(j+2) (4.1)

where

δn =(q2/bc, q2; q2)∞(bq, cq; q2)n(q3/b, q3/c; q2)∞(1− q2n+1)

(q2

bc

)n. (4.2)

We would like to consider this conjugate Bailey pair with three Bailey pairs and

combine them to produce results using the Bailey transform. We first consider the

following Bailey pair:

αn = (−1)ndnqn2

βn =(dq; q2)n(q/d; q2)n

(q2; q2)2n(4.3)

found in [3, p. 49, ex. 1]. Combining our two pairs we get the following:

26

Theorem 4.0.2.

(q2/bc, q2; q2)∞(q3/b, q3/c; q2)∞

∞∑n=0

(bq, cq, dq, q/d; q2)n(q2; q2)2n(1− q2n+1)

(q2

bc

)n

=∞∑j=0

(bq, cq; q2)j(1− d2j+1)

(q3/b, q3/c; q2)j(1− d)(1 + q2j+1)

(− 1bcd

)jqj(j+2). (4.4)

Proof. Using the Bailey tranform we get the following:

∞∑j=0

(bq, cq; q2)j(1− d2j+1)

(q3/b, q3/c; q2)j(1− d)(1 + q2j+1)

(− 1bcd

)jqj(j+2)

=∞∑

n=−∞dn

∑j≥|n|

(bq, cq; q2)j)

(q3/b, q3/c; q2)j(1 + q2j+1)

(− 1bc

)jqj(j+2) (4.5)

=∞∑

n=−∞αnγn (4.6)

=∞∑n=0

βnδn (4.7)

=(q2/bc, q2; q2)∞(q3/b, q3/c; q2)∞

∞∑n=0

(bq, cq, dq, q/d; q2)n(q2; q2)2n(1− q2n+1)

(q2

bc

)n. (4.8)

Notice the Bailey pair above was chosen so that when combined with our pair,

the term∑αnγn which had two sums was able to collapse into one sum. There are

other Bailey pairs which offer this simplifiaction into one sum and we consider some of

them below. Before we do we present the following simple result which can be directly

proven with induction.

27

Lemma 4.0.3.n∑

j=−n(−1)jqj(j+1)/2 = (−1)nqn(n+1)/2 (4.9)

We can now present the next two identities.

Theorem 4.0.4. We have the following,

(q2/bc, q2; q2)∞(q3/b, q3/c; q2)∞

∞∑n=0

(bq, cq; q2)n(q)2n+1

(q2

bc

)n

=∞∑j=0

(bq, cq; q2)j(q3/b, q3/c; q2)j

(1 + q2j+1)(

1bc

)jqj(2j+3), (4.10)

and

(q2/bc, q2; q2)∞(q3/b, q3/c; q2)∞

∞∑n=0

(bq, cq; q2)n(q,−q, q2; q2)n(1− q2n+1)

(q2

bc

)n

=∞∑j=0

(bq, cq; q2)j(q3/b, q3/c; q2)j

(1 + q2j+1)(− 1bc

)j(−1)b

j2cq

j(j+2)+2b j2c(b j2c+1

).

(4.11)

Proof. To prove (4.10) we use our Conjugate Bailey pair defined with a = 1 in Corollary

3.2.1 in the Bailey transform with the Bailey pair

αn = q2n2+n βn =

1(q; q)2n

(4.12)

28

found in [23, H(3)]. Thus,

∞∑j=0

(bq, cq; q2)j(q3/b, q3/c; q2)j

(1 + q2j+1)(

1bc

)jqj(2j+3)

=∞∑j=0

(bq, cq; q2)j(q3/b, q3/c; q2)j

(1 + q2j+1)(− 1bc

)jqj(j+2)

j∑n=−j

(−1)nqn(n+1) (4.13)

=∞∑

n=−∞(−1)nqn(n+1) ∑

j≥|n|

(bq, cq; q2)j(q3/b, q3/c; q2)j

(1 + q2j+1)(− 1bc

)jqj(j+2) (4.14)

=∞∑

n=−∞αnγn. (4.15)

Using our Bailey transform,

=∞∑n=0

βnδn (4.16)

=(q2/bc, q2; q2)∞(q3/b, q3/c; q2)∞

∞∑n=0

(bq, cq; q2)n(q)2n+1

(q2

bc

)n. (4.17)

To prove our second identity we use our Conjugate Bailey pair defined with a = 1

in the Bilateral Symmetric Bailey Transform with the Bailey pair [23, C(1)]

α2n = (−1)nq6n2+2n, α2n+1 = 0 βn =

1(q2; q4)n(q2; q2)n

. (4.18)

Allowing for different values of the open parameters in our theorems yields a

number of results, some new and some known. Of the known identities, it is interesting

to see them arise in the manner in which they occur. For those identities which appear

29

to be new, they seem to be in a number of different forms; generalized Lambert series,

weighted identities, infinite products and partition identities, and will be categorized

accordingly.

4.1 Known Identities

While working through some of the special cases, as one might expect, not all

identities that were found were new. In this section we mention four of the many classic

identities that turned up in the research. Our first is a weighted series made famous

by Jacobi, which can be deduced from the triple product identity [3, p.21, Thm 2.8,

z → −q, q2 → q]:

Corollary 4.1.1 (Jacobi).

∞∑j=0

(2j + 1)(−1)jqj(j+1)/2 = (q)3∞. (4.19)

30

Proof. We consider b = −1, c = −q and d = 1 in Theorem 4.0.2. Thus,

(1 + q)∞∑j=0

(2j + 1)(−1)jqj(j+1) (4.20)

=(q, q2; q2)∞

(−q3,−q2; q2)∞

∞∑n=0

(−q,−q2, q, q; q2)n(q2; q2)2n(1− q2n+1)

qn (4.21)

=(q)∞

(−q2)∞(1− q) 2φ1

(q, q

q3; q2; q

)(4.22)

=(q)∞

(−q2)∞(1− q)· (q2, q2; q2)∞

(q, q3; q2)∞(4.23)

Using III.2 [19, p. 359]

= (1 + q)(q2; q2)3∞. (4.24)

Our final result is obtained from dividing both sides by (1+q) and mapping q 7→ q1/2.

Our next two identities fall on Slater’s list of identities in [24]. While she proves

them using the Bailey transform as well, it is curious to see that they appear with the

use of different pairs.

Corollary 4.1.2 ( Slater (27)).

(−q,−q5, q6; q6)∞ = (q2; q2)∞∞∑n=0

(−q; q2)2nq2n(n+1)

(q2; q2)2n(1− q2n+1). (4.25)

Proof. We first note that with the triple product identity,

(−q,−q5, q6; q6)∞ =∞∑

n=−∞qj(3j+2) (4.26)

=∞∑n=0

(1 + q2n+1)qn(3n+2). (4.27)

31

We now consider b 7→ ∞, c 7→ ∞ and d = −1 in Theorem 4.0.2. Thus,

∞∑n=0

(1 + q2n+1)qn(3n+2) = (q2; q2)∞∞∑n=0

(−q; q2)2nq2n(n+1)

(q2; q2)2n(1− q2n+1). (4.28)

We note that the following corollary can be found on Slater’s list [24], but can be

proven in alternative way using Euler’s formula (2.15) with the following observation,

∞∑n=0

qn

(q)2n+1=

∞∑n=0

(1− (−1)n)2

· q(n−1)/2

(q)n. (4.29)

Corollary 4.1.3 (Slater (38)).

(−q,−q7, q8; q8)∞ = (q)∞∞∑n=0

qn

(q)2n+1. (4.30)

Proof. As with our previous corollary, we again use the triple product identity to see

that

(−q,−q7, q8; q8)∞ =∞∑

n=−∞qn(4n+3) (4.31)

=∞∑n=0

(1 + q2n+1)qn(4n+3). (4.32)

32

We then consider b, c 7→ ∞ in Theorem 4.11,

∞∑n=0

(1 + q2n+1)qn(4n+3)

= (q2; q2)∞∞∑n=0

q2n(n+1)

(q)2n+1(4.33)

=(q2; q2)∞(1− q)

lima7→∞ 2φ1

(aq, aq

q3; q2;

q2

a2

)(4.34)

=(q2; q2)∞(1− q)

lima7→∞

(q; q2)∞(q2/a2; q2)∞

2φ1

(q2/a, q2/a

q3; q2; q

)(4.35)

using III.3 [19, p. 359],

= (q)∞∞∑n=0

qn

(q)2n+1. (4.36)

4.2 Generalized Lambert Series and Related Identities

We define the following as a generalized Lambert series,

∞∑n=−∞

anqn(n+1)/2

1− bzn. (4.37)

It has been shown that Lambert series can be useful in furthering our understand-

ing of sums of even squares of integers, sums of an even number of triangular numbers

[15], Dyson’s rank of a partition [11] and many other applications. In [1] Andrews also

shows that such series are readily transformable and remarks on their close relationship

33

with theta functions. We present a number of generalized Lambert series which follow

from Theorems 4.0.2 and 4.0.4.

Corollary 4.2.1.

∞∑j=−∞

(−1)jq2j(j+1)

(1− q2j+1)= (q)∞(−q; q2)∞

∞∑n=0

(−q;−q)2n(q; q)2n+1

qn. (4.38)

Proof. We allow b = 1 and c = −q in (4.10).

One of the positive aspects of the Bailey pair used to prove Theorem 4.0.2 is its

ability to create weighted identities when allowing d → 1. The following identity is an

example of one such identity.

Corollary 4.2.2.

∞∑j=−∞

(2j + 1)qj(j+1)

(1 + q2j+1)=

(q)∞(−q)∞

∞∑n=0

(−q; q2)2n

(q)2n(1 + q2n+1)qn. (4.39)

Proof. We first note that

∞∑j=−∞

(2j + 1)qj(j+1)

(1− q2j+1)=

∞∑j=0

(1 + q2j+1)(1− q2j+1)

(2j + 1)qj(j+1). (4.40)

34

We then consider b = 1, c = −q and d = 1 in Theorem 4.0.2.

∞∑n=0

(1− q)(1 + q2n+1)(1− q2n+1)

(2n+ 1)qn(n+1)

=(−q, q2; q2)∞(q3,−q2; q2)∞

∞∑n=0

(q,−q2, q, q; q2)n(q2; q2)2n(1− q2n+1)

(−q)n (4.41)

=(−q;−q)∞(−q2;−q)∞

∞∑n=0

(q; q2)2n

(−q;−q)2n(1− q2n+1)(−q)n. (4.42)

We divide each side by (1− q) followed by allowing q 7→ −q to obtain our result.

Corollary 4.2.3.

∞∑j=−∞

(2j + 1)q2j(j+1)

(1− q2j+1)=

(q2; q2)∞(q; q2)∞

∞∑n=0

(q; q2)3n

(q2; q2)2n(1− q2n+1)(−1)nqn

2+2n. (4.43)

Proof. We first note that

∞∑j=−∞

(2j + 1)q2j(j+1)

(1− q2j+1)=

∞∑j=0

(1 + q2j+1)(1− q2j+1)

(2j + 1)q2j(j+1). (4.44)

We then consider b 7→ ∞ and c = d = 1 in Theorem 4.0.2.

We also consider series in which our sum is only one-sided.

Corollary 4.2.4.

∞∑j=0

qj(j+2)

1 + q2j+1 =(q4; q4)∞

(−q2; q4)∞

∞∑n=0

(q; q2)3n(−q; q2)n

(q2; q2)2n(1 + q2n+1)(−1)nq2n (4.45)

35

and

∞∑j=0

(2j + 1)(−1)jqj(j+2)

1 + q2j+1

= (−q2; q2)∞(q4; q4)∞∞∑n=0

(−q; q2)3n(q; q2)n

(q2; q2)2n(1 + q2n+1)q2n (4.46)

= (q)2∞(−q2; q2)2∞

∞∑n=0

(q2; q4)2n

(q2; q2)2n(1− q2n+1)q2n. (4.47)

Proof. To prove (4.45), we consider b = −1, c = 1 and d = −1 in Theorem 4.0.2. To

prove (4.46) and (4.47) we use b = 1, c = −1 and d = 1 and b = c = −1 and d = 1 in

Theorem 4.0.2, respectively. As with previous proofs, a minimal amount of simplification

yields our results.

Corollary 4.2.5.

∞∑j=0

(1 + q2j+1)(1− q2j+1)

qj(j+1) =∞∑n=0

(q2; q4)n(q;−q)2n+1

qn (4.48)

and

∞∑j=0

(1 + q2j+1)(1− q2j+1)

(2j + 1)(−1)jqj(j+1) =∞∑n=0

(q; q2)2n

(−q; q)2n(1− q2n+1)qn. (4.49)

Proof. To prove (4.48) and (4.49) we consider b = 1, c = q and d = −1 and b = d = 1

and c = q in Theorem 4.0.2, respectively.

36

Corollary 4.2.6.

∞∑j=0

qj(2j+3)

1 + q2j+1 =(q)∞

(−q; q2)2∞

∞∑n=0

(−q2; q2)2n

(q; q)2n+1qn (4.50)

and

∞∑j=−∞

(−1)jq3j(j+1)

1− q2j+1 = (q2; q2)∞∞∑n=0

qn

(q; q2)n+1. (4.51)

Proof. Both corollaries are consequences of (4.10). We allow b = −1 and c = −1 and

b→∞ and c = 1, respectively.

We will refer to the following identities as Order 2 generalized Lambert series.

Corollary 4.2.7.

∞∑j=−∞

qj(2j+3)

(1− q2j+1)2= (−q)∞(q2; q2)∞

∞∑n=0

(q2; q2)2nqn

(q; q)2n+1, (4.52)

∞∑j=−∞

qj(j+2)

(1− q2j+1)2=

(q2; q2)2∞(−q; q2)2∞

∞∑n=0

(q2; q4)nq2n

(q4; q4)n(1− q2n+1)(4.53)

and

∞∑j=−∞

(2j + 1)(−1)jqj(j+2)

(1− q2j+1)2=

(q2; q2)2∞(−q; q2)2∞

∞∑n=0

(q; q2)4nq2n

(q2; q2)2n(1− q2n+1). (4.54)

37

Proof. Our first identity is a consequence of (4.10). We allow b = 1 and c = 1. For the

last two, we note that

∞∑j=−∞

qj(j+2)

(1− q2j+1)2=

∞∑n=0

(1 + q2n+1)(1− q2n+1)2

qn(n+2). (4.55)

and

∞∑j=−∞

(2j + 1)(−1)jqj(j+2)

(1− q2j+1)2=

∞∑n=0

(1 + q2n+1)(1− q2n+1)2

(2n+ 1)(−1)nqn(n+2). (4.56)

We then apply Theorem 4.0.2 with b = c = 1 and d = −1 and b = c = d = 1.

4.3 Infinite Products and Ramanujan-like Identities

In many cases, our∑αnγn can be reduced to an infinite product using Jacobi’s

triple product. We are not the first to realize this application of the Bailey Trans-

form. Slater’s list [24] is made up entirely of such infinite products and are referred

to as Ramanujan-like identities because of their similarity to the well known Rogers-

Ramanujan Identities, (1.1) and (1.2). In this section we present some infinite product

identities which appear to not be on Slater’s list.

Corollary 4.3.1.

(q4, q8,−q8; q8)∞ = (q)∞(−q2; q2)∞∞∑n=0

(−q; q2)3nqn

2+2n

(q2; q2)2n(1− q2n+1). (4.57)

Proof. We then consider c = d = −1 and b 7→ ∞ in Theorem 4.0.2.

38

Corollary 4.3.2.

(−q2,−q2, q4; q4)∞ + 2q(−q4,−q4, q4; q4)∞

= (q)∞(−q2; q2)∞∞∑n=0

(1 + q2n+1)(−q; q2)3n

(1− q2n+1)(q2; q2)2nqn

2. (4.58)

Proof. We consider c = −q2, d = −1 and b 7→ ∞ in Theorem 4.0.2.

(q2; q2)∞(1 + q)(−q; q2)∞

∞∑n=0

(1 + q2n+1)(−q; q2)3n

(1− q2n+1)(q2; q2)2nqn

2(4.59)

=1

1 + q

∞∑n=0

(1 + q2n+1)2q2n2

(4.60)

=1

1 + q

∞∑n=−∞

(1 + q2n+1)q2n2

(4.61)

=1

1 + q

[(−q2,−q2, q4; q4)∞ + 2q(−q4,−q4, q4; q4)∞

](4.62)

where the last step taken was an application of the Jacobi triple product.

Corollary 4.3.3.

(−q4, q8,−q8; q8)∞ =(q)∞

(−q)∞

∞∑n=0

(−q; q)2n(q; q)2n+1

qn. (4.63)

Proof. We consider b = −1 and c = −q in (4.10).

Corollary 4.3.4.

(−q6,−q12, q12; q12)∞ =(q)∞

(−q; q2)∞

∞∑n=0

(−q2; q2)n(q; q)2n+1

qn. (4.64)

39

Proof. We consider b 7→ ∞ and c = −1 in (4.10).

Corollary 4.3.5.

(−q2,−q4, q6; q6)∞ + 2q(−q6,−q6, q6; q6)∞

=(q)∞

(−q3; q2)∞

∞∑n=0

(−1; q2)n(q; q)2n+1

qn. (4.65)

Proof. We consider b 7→ ∞ and c = −q2 in (4.10).

Corollary 4.3.6.

(q4, q16, q20; q20)∞ =(q)∞

(q2; q4)∞

∞∑n=0

qn(n+2)

(q)2n+1. (4.66)

Proof. We consider b = ∞, c = −1 in Theorem 4.11.

Corollary 4.3.7.

(q4; q4)3∞ = (q)∞(−q2; q2)∞∞∑n=0

(q; q2)2n(−q; q2)n

(q2; q2)2n(1− q2n+1)qn

2+2n. (4.67)

Proof. We consider c = −1, d = 1 and b 7→ ∞ in Theorem 4.0.2.

We also present identities which are not infinite products, but are similar to

Ramanujan-like identities due to Umbral methods. We note that using Jacobi’s triple

product, for 0 < i < k we have

(−qk+i,−qk−i, q2k; q2k)∞ =∞∑n=0

(1 + q(k+i)(2n+1))qkn2−in. (4.68)

40

This can be interpreted as the generating function for strict partitions of n with

parts only congruent to k ± i and 2k modulo 2k in which each partition is counted as

(−1)d where d is the number of parts divisible by 2k. In terms of our notation defined

in Chapter 2, we have

(−qk+i,−qk−i, q2k; q2k)∞ =∑

λ∈D2kk+i

(−1)µ2k2k

(λ)q|λ|. (4.69)

In applying the Bailey transform, it is often the case that our result cannot be

used in conjunction with Jacobi’s triple product because of a nearly harmless negative

sign. It is with these identities that we implement the use of Umbral methods. It is with

these methods that we have a combinatorial interpretation of

∞∑n=0

(1− q(k+i)(2n+1))qkn2−in. (4.70)

The following identities are examples in which we have used this technique to

represent one side of our identity as a generating function for partitions.

Corollary 4.3.8.

∑λ∈D=

12,10

(−1)µ(λ)q|λ| −∑

λ∈D12,2

(−1)µ(λ)q|λ| =∞∑n=0

(−1)nqn

(q; q2)n+1. (4.71)

Proof. We first allow b = −1 and c = q into (4.11). Our final result can be obtained by

applying Theorem 2.4.2.

41

Corollary 4.3.9.

∑λ∈D=

20,14

(−1)µ(λ)q|λ| −∑

λ∈D20,6

(−1)µ(λ)q|λ| (4.72)

+q∑

λ∈D=20,18

(−1)µ(λ)q|λ| − q∑

λ∈D20,2

(−1)µ(λ)q|λ| (4.73)

=∞∑n=0

(−1)nqn(n+1)

(q2; q4)n(1− q2n+1). (4.74)

Proof. We first allow b 7→ ∞ and c = q into (4.11) to get

∞∑j=0

(1 + q4j+1 + q8j+3 + q12j+6)(−1)jq2j(5j+2) =∞∑n=0

(−1)nqn(n+1)

(q2; q4)n(1− q2n+1). (4.75)

Our final result can be obtained by applying Theorem 2.4.2.

4.4 Weighted sums

As previously stated, Theorem 4.0.2 is capable of producing weighted q-series

identities. The following section presents more identities of this type.

Corollary 4.4.1.

∞∑j=0

(2j + 1)(1 + q2j+1)qj(2j+1) =∞∑n=0

(q; q2)n(−q; q)2n(1− q2n+1)

(−1)nqn2+n. (4.76)

Proof. We consider Theorem 4.0.2 with c = q, d = 1 and b 7→ ∞.

42

Corollary 4.4.2.

∞∑j=−∞

(2j + 1)(−1)jqj(2j+1)

= (q)∞(−q; q2)∞∞∑n=0

(q; q2)nqn2+n

(−q;−q)2n(1− q2n+1). (4.77)

Proof. We note that

∞∑j=−∞

(2j + 1)(−1)jqj(2j+1) =∞∑j=0

(2j + 1)(−1)j(1 + q2j+1)qj(2j+1). (4.78)

We then consider Theorem 4.0.2 with c = −q, d = 1 and b 7→ ∞.

Corollary 4.4.3.

∞∑j=−∞

(2j + 1)(−1)jqj(3j+2) = (q2; q2)∞∞∑n=0

(q; q2)2nq2n(n+1)

(q2; q2)2n(1− q2n+1). (4.79)

Proof. We note that

∞∑j=−∞

(2j + 1)(−1)jqj(3j+2) =∞∑j=0

(2j + 1)(−1)j(1 + q2j+1)qj(3j+2). (4.80)

We then consider Theorem 4.0.2 with d = 1 and b, c 7→ ∞.

43

Corollary 4.4.4.

∞∑j=−∞

(1 + q2j+1)(2j + 1)(−1)jq2j2

(4.81)

= (q)∞(−q2; q2)∞∞∑n=0

(1 + q2n+1)(q; q2)n(1− q2n+1)(q4; q4)n

qn2. (4.82)

Proof. We note that

∞∑j=−∞

(1 + q2j+1)(2j + 1)(−1)jq2j2

=∞∑

j=−∞(1 + q2j+1)2(2j + 1)(−1)jq2j

2. (4.83)

We then consider Theorem 4.0.2 with c = −q2, d = 1 and b 7→ ∞.

4.5 Partitions

As was shown in Chapter 2, q-series can play a key role in partition identities.

In this section we take q-series identities and interpret them combinatorially to equate

different classes of partitions.

Corollary 4.5.1. Let a(n) denote the number of ways of choosing a not overlined part,

λi, in any overpartition of n such that no overlined part exceeds 2λi and no other part

exceeds 2λi+ 1. Let b(n) be the number of tripartitions of n in which the first partition

has distinct parts, the second partition has no parts divisible by 8 and the last partition

has distinct parts with all parts divisible by 4. Then b(n) = a(n).

44

Proof. Recall Corollary 4.3.3. We then see that

∞∑n=0

(−q; q)2n(q; q)2n+1

qn = (−q)∞ · (q8; q8)∞(q)∞

· (−q4; q4)∞. (4.84)

It is left to observe that a(n) and b(n) are the coefficients of qn in (4.84).

Corollary 4.5.2. Let a(n) denote the number of ways of choosing a not overlined part,

λi, in any overpartition of n with all overlined parts even such that no overlined part

exceeds 2λi and no other part exceeds 2λi+1. Let b(n) be the number of overpartitions

of n in which the overlined parts are not congruent to ±2,±4 and 12 modulo 12 and all

other parts are not divisible by 24. Then b(n) = a(n).

Proof. Recall Corollary 4.3.4. We then see that

∞∑n=0

(−q2; q2)n(q; q)2n+1

qn =(−q)∞

(−q2,−q4,−q8,−q10,−q12; q12)∞· (q24; q24)∞

(q)∞. (4.85)

It is left to observe that a(n) and b(n) are the coefficients of qn in (4.85).

Corollary 4.5.3. Let a(n) denote the number of ways of choosing a part, λi, not

overlined in any overpartition of n with overlined parts even (zero allowed) such that

no overlined part exceeds 2λi − 2 and no other part exceeds 2λi + 1. Let b(n) be the

number of overpartitions of n in which all parts are not divisible by 6 and the overlined

parts are ≥ 2. Let c(n) be the number of overpartitions in which the overlined parts are

not congruent to ±2 modulo 6, parts not overlined are ≥ 2 and the parts not overlined

are not divisible by 12. Then a(n) = b(n) + 2c(n− 1).

45

Proof. Recall Corollary 4.3.5. We then see that

(q6; q6)∞(q)∞

· (−q2)∞(−q6; q6)∞

+ 2q(q12; q12)∞

(q)∞· (−q2)∞(−q2,−q4; q6)∞

=∞∑n=0

(−1; q2)n(q; q)2n+1

qn. (4.86)

It is left to observe that a(n), b(n) + 2c(n− 1) are the coefficients of qn.

Corollary 4.5.4. Let A(q) denote the generating function for partitions of n with parts

either even or equal to one in which the largest part does not exceed twice the number of

parts and each partition is counted as (−1)k where k is the number of even parts. Then

A(q) =∑

λ∈D=6,5

(−1)µ(λ)q|λ| −∑

λ∈D6,1

(−1)µ(λ)q|λ|. (4.87)

Proof. We first allow b 7→ ∞ and c = q into (4.10) to get

∞∑j=0

(1 + q2j+1)(−1)jqj(3j+2) (4.88)

= (q; q2)∞∞∑n=0

(q; q2)n(q; q)2n+1

qn (4.89)

= (q; q2)∞∞∑n=0

qn

(q2; q2)n(1 + q2n+1 + q2(2n+1) + · · · ) (4.90)

= (q; q2)∞

(1

(q; q2)∞+

q

(q3; q2)∞+

q2

(q5; q2)∞+ · · ·

)(4.91)

=∞∑n=0

(q; q2)nqn. (4.92)

46

Here our last step is merely the application of (2.15). To complete our proof we observe

that our last line is the generating function A(q) and that by applying Theorem 2.4.2

we see that

∞∑j=0

(1 + q2j+1)(−1)jqj(3j+2) =∑

λ∈D=6,5

(−1)µ(λ)q|λ| −∑

λ∈D6,1

(−1)µ(λ)q|λ|. (4.93)

Remark 4.5.5. We note that the above corollary is a new combinatorial interpretation

of a known identity found in Ramanujan’s Lost Notebook [7, Entry 9.5.2, p. 239],

∞∑n=0

(q; q2)nqn =

∞∑n=0

(−1)nq3n2+2n(1 + q2n+1). (4.94)

More will be said about other combinatorial interpretations of the identity in Chapter

7.

Corollary 4.5.6. Let a(n) denote the number of overpartitions of n with all parts odd,

all overlined parts ≥ 3, and the size of each part not overlined not exceeding the total

number of overlined parts. Let b(n) denote the number of partitions of n with parts

either odd or congruent to ±8 modulo 20. Then, b(n) = a(n).

Proof. Recall Corollary 4.3.6. We can then see that,

1(q; q2)∞(q8, q12; q20)∞

=∞∑n=0

qn(n+2)

(q)2n+1. (4.95)

It is left to observe that a(n) and b(n) are the coefficients of qn.

47

4.6 Closing Remarks

It should be noted that there are more Bailey pairs which when combined with our

conjugate Bailey pair (with a = 1) have similar results and proofs to (4.10) and (4.11).

For example, we could have considered theorems with the use of the the following Bailey

pairs,

αn = (−1)nq2n2+n, βn =

1(−q;−q)2n

(4.96)

and

αn = qn, βn =q−n

(q)2n(4.97)

found in [23, H(2) and F(3)].

48

Chapter 5

A General Discussion of Various Conjugate Bailey Pairs

In the previous chapter we discussed the possible applications of one of our special

cases with multiple Bailey pairs. This chapter provides a broad set of results which can

be obtained from many different conjugate pairs, not just those limited to the case a = 1.

In order to give an alternate presentation and order to things, we present our

results in a different manner to the previous chapter. We first use the Bailey transform

with some well-known Bailey pairs to prove some general theorems regarding conjugate

Bailey pairs. We then use these theorems in conjunction with our new conjugate Bailey

pairs to prove a wide assortment of results.

5.1 Bailey Pairs and the Symmetric Bilateral Bailey Transform

In this section we present eight known Bailey pairs and their corresponding the-

orems when used in the Symmetric Bailey Transform.

Theorem 5.1.1. If

γn =∑j≥|n|

δn(q2; q2)j−n(q2; q2)j+n

(5.1)

49

then we have

1.∞∑n=0

(q; q2)n(q2; q2)2n

δn =∞∑

n=−∞(−1)nqn(3n+1)/2γn (5.2)

2.∞∑n=0

(q; q2)n(q2; q2)2n

(−1)nqn2δn =

∞∑n=−∞

(−1)nqn(n+1)/2γn (5.3)

3.∞∑n=0

δn(−q;−q)2n

=∞∑

n=−∞(−1)nqn(2n+1)γn (5.4)

4.∞∑n=0

δn(q2; q2)n(q2; q4)n

=∞∑

n=−∞(−1)nq2n(3n+1)γ2n (5.5)

5.∞∑n=0

qn(n−1)δn(q2; q2)n(q2; q4)n

=∞∑

n=−∞(−1)nq2n(n+1)γ2n (5.6)

6.∞∑n=0

(−1)nq−n(n+1)δn(q2; q2)n

=∞∑

n=−∞(−1)nq−n(n+1)γn (5.7)

7.∞∑n=0

(−1)nδn(q4; q4)n

=∞∑

n=−∞(−1)nγn (5.8)

8.∞∑n=0

q−nδn(q; q)2n

=∞∑

n=−∞qnγn. (5.9)

Proof. Equation (5.2) follows from specializing [12, p. 5, Sec. 6, (ii)] with a = 1, b→∞

and x replaced by q. Equation (5.3) follows from the same source with a = 1, b→ 0 and

x replaced by q.

All other Bailey pairs can be found in Slater [23]. Equation (5.4) follows from

F(1) with q → q2 followed by q → −q. Equations (5.5) and (5.6) follow from C(1) and

C(5) with q → q2. Equations (5.7) and (5.8) follow from the fourth and seventh row of

the second table on p. 468, respectively, with q → q2. Equation (5.9) follows from F(3)

with q → q2.

50

The following four sections include lists of selected identities that can be obtained

from Theorem 5.1.1. Each proof will merely give the values for a, b and c which should

be substituted into Corollary 3.2.1 in order to obtain the proper conjugate Bailey pair,

(δn, γn). To see the specific conjugate Bailey pair, refer to the tables in the end of

Chapter 3. As in Chapter 4, proofs of the following identities might require the use of

the triple product identity, Gauss’ formula, Euler’s formula, and/or the Umbral methods

described in Chapter 2. While most of the details are omitted, they are similar to those

proofs presented in Chapter 4.

5.2 Lambert Series, Infinite Products and Ramanujan-like Identities

As with the identities in Chapter 4, we again explore applications of the Bailey

transform in which∞∑

n=−∞αnγn (5.10)

collapses into either a generalized Lambert series, an infinite product or a Ramanujan-like

identity.

Corollary 5.2.1.

∞∑j=0

(1− q2j+1)(1 + q2j+1)

qj(j+1)/2 =∞∑n=0

qn(n+1)

(−q2; q2)n(1 + q2n+1). (5.11)

Proof. We consider a = −1, b = −1 and c = q in Corollary 3.2.1 with equation (5.3).

51

Corollary 5.2.2.

∞∑j=−∞

(−1)jq3j(j+1)/2

(1− q2j+1)= (−q)2∞(q2; q2)2∞

∞∑n=0

(q; q2)3n

(q2; q2)2nq2n. (5.12)

Proof. We consider a = 0, b = 1 and c = 1 in Corollary 3.2.1 with equation (5.2).

Corollary 5.2.3.

∞∑j=−∞

(−1)jq3j(j+1)/2

(1 + q2j+1)= (q)∞

∞∑n=0

(q; q2)n(−1)nqn (5.13)

= (q)∞(−q2; q2)∞∞∑n=0

(q2; q4)n(−1)nq2n(n+1)

(q2; q2)2n(1 + q2n+1).(5.14)

Proof. We consider a = 0, b = 1, c = 1 followed by q → −q in Corollary 3.2.1 with

equation (5.2) and a = −1, b = ∞, c = −1 in Corollary 3.2.1 with equation (5.3).

Corollary 5.2.4.

∞∑j=0

(1− q2j+1)(1 + q2j+1)

q3j(j+1)/2 =∞∑n=0

(−q; q2)n(−q)2n+1

(−1)nqn. (5.15)

Proof. We consider a = −1, b = −1 and c = q in Corollary 3.2.1 with equation (5.2).

Corollary 5.2.5.

∞∑j=−∞

(1 + q2j+1)(1− q2j+1)

(−1)jqj(2j+1)

= (1 + q)(q2; q2)∞(−q; q2)∞

∞∑n=0

(−1, q2; q2)n(q; q)2n+1

qn. (5.16)

52

Proof. We consider a = 0, b = 1 and c = 1 in Corollary 3.2.1 with equation (5.4).

Corollary 5.2.6.

∞∑n=0

qn2

(−q; q2)n(1− q2n+1)=

∞∑j=−∞

1 + q|4j+1|

1− q|4j+1| (−1)jq2j2. (5.17)

Proof. Let a = b = −1, c = q in Corollary 3.2.1 with (5.6).

Corollary 5.2.7.

(q2; q2)∞(−q; q2)∞

∞∑n=0

q2n2+n

(q)2n(1 + q2n+1)=

∞∑j=−∞

1− q4j+1

1 + q4j+1 (−1)jqj(6j+2). (5.18)

Proof. Let b = ∞, a = c = −1 in Corollary 3.2.1 with (5.6).

Corollary 5.2.8.

(q2; q2)∞(−q; q2)∞

∞∑n=0

qn2+2n

(q)2n(1 + q2n+1)=

∞∑j=−∞

1− q4j+1

1 + q4j+1 (−1)jq2j(5j+3). (5.19)

Proof. Let b = ∞, a = c = −1 in Corollary 3.2.1 with (5.5).

Corollary 5.2.9.

(q3, q6,−q6; q6)∞ =(q)∞

(−q)∞

∞∑n=0

(−q; q2)n(q)2n+1

qn. (5.20)

Proof. We consider a = −1, b = 1 and c = q in Corollary 3.2.1 with equation (5.2).

53

Corollary 5.2.10.

(−q,−q2, q3; q3)∞ = (q)∞∞∑n=0

(−1)2n(q2; q2)n

qn. (5.21)

Proof. We consider a = 0, b = 1 and c = −q in Corollary 3.2.1 with equation (5.2).

Corollary 5.2.11.

(−q,−q2, q3; q3)∞ − 2q(−q3,−q3, q3; q3)∞ (5.22)

=(q2; q2)∞(q; q2)∞

∞∑n=0

(q; q2)n+1q2n2

(−q)2n+1(q2; q2)n. (5.23)

Proof. We consider a = −1, b = ∞ and c = q2 in Corollary 3.2.1 with equation (5.3).

Corollary 5.2.12.

(−q2,−q2, q4; q4)∞ + 2q(−q4,−q4, q4; q4)∞

= (1 + q)(q)∞

(−q)∞

∞∑n=0

(−1; q)2n(q; q)2n+1

qn. (5.24)

Proof. We consider a = 0, b = 1 and c = −q in Corollary 3.2.1 with equation (5.4).

Corollary 5.2.13.

(q6, q6, q12; q12)∞ − q(q2, q10, q12; q12)∞ =(q)∞

(−q)∞

∞∑n=0

(−q2; q2)n(q)2n

qn. (5.25)

Proof. Let a = 0, b = 1 and c = −q followed by q → −q in Corollary 3.2.1 with (5.5).

54

Corollary 5.2.14.

(−q2,−q3, q5; q5)∞ − 2q(−q5,−q5, q5; q5)∞ (5.26)

=(q2; q2)∞(q; q2)∞

∞∑n=0

(q; q2)n+1(q2; q2)n(−q)2n+1

(−1)nqn2. (5.27)

Proof. We consider a = −1, b = ∞, c = q2 in Corollary 3.2.1 with (5.2).

Corollary 5.2.15.

(q5, q10,−q10; q10)∞ =(q2; q2)∞(−q; q2)∞

∞∑n=0

(−q; q2)2n

(q2; q2)2n(1− q2n+1)qn

2+2n. (5.28)

Proof. We consider a = −1, b = ∞, c = 1 in Corollary 3.2.1 with (5.2).

Corollary 5.2.16.

∑λ∈D=

3,2

(−1)µ(λ)q|λ| −∑

λ∈D3,1

(−1)µ(λ)q|λ| =∞∑n=0

(q; q2)n(−q; q)2n

qn. (5.29)

Proof. We consider a = 0, b = 1 and c = q in Corollary 3.2.1 with equation (5.2). We

complete the proof by applying Theorem 2.4.2.

Corollary 5.2.17.

∑λ∈D=

3,2

(−1)µ33(λ)q|λ| −

∑λ∈D3,1

(−1)µ33(λ)q|λ| =

∞∑n=0

(−1)nqn

(−q2; q2)n(5.30)

=∞∑n=0

q2n2+n

(−q)2n+1. (5.31)

55

Proof. We consider a = 0, b = 1 and c = q followed by q → −q in Corollary 3.2.1 with

equation (5.2) and a = −1, b = ∞ and c = q in Corollary 3.2.1 with equation (5.3). To

complete the proof we apply Theorem 2.4.2.

Corollary 5.2.18.

1 + 2q∑

λ,γ,δ∈D4µ(γ)=µ(δ)

(−1)µ(λ)q|λ|+|γ|+|δ| = (1 + q)∞∑n=0

(q; q2)n(−q; q2)n

qn. (5.32)

Proof. We consider a = 0, b = 1 and c = q in Corollary 3.2.1 with equation (5.4). We

see that

∞∑j=0

(1 + q2j+1)2(−1)jq2j2

= 1 + 2q∞∑n=0

(−1)nq2n(n+1) (5.33)

= (1 + q)∞∑n=0

(q; q2)n(−q; q2)n

qn. (5.34)

We note that

∞∑j=0

(1 + q2j+1)2(−1)jq2j2

= 1 + 2q∞∑n=0

(−1)nq2n(n+1) (5.35)

= 1 + 2q∑

λ,γ,δ∈D4µ(γ)=µ(δ)

(−1)µ(λ)q|λ|+|γ|+|δ|, (5.36)

where our last step is just the application of III.2 in [19, p. 359].

56

Corollary 5.2.19.

∑λ∈D=

5,4

(−1)µ55(λ)q|λ| −

∑λ∈D5,1

(−1)µ55(λ)q|λ|

=∞∑n=0

(−1)nqn2+n

(−q)2n+1. (5.37)

Proof. We consider a = −1, b = ∞ and c = q in Corollary 3.2.1 with equation (5.2).

Corollary 5.2.20.

1 + q∑

λ∈D=12,10

(−1)µ(λ)q|λ| − q∑

λ∈D12,2

(−1)µ(λ)q|λ|

=∞∑n=0

qn

(−q; q2)n. (5.38)

Proof. Let a = 0, b = 1 and c = q in Corollary 3.2.1 with (5.5). Our result then follows

from applying Theorem 2.4.2.

We again note that the Bailey pairs used in equations (5.2) - (5.6) were chosen

so that when they matched up with our conjugate Bailey pairs the appropriate side

collapsed. It should be mentioned that those Bailey pairs were not the only candidates

that would have produced the type of results seen above. From Slater [23] we see that

B(1), F(1), and rows 1, 2, 3 and 8 from the second table on p. 468 could have also

produced similar results.

57

5.3 Indefinite Quadratic Forms

In the following section we look into double series involving an indefinite quadratic

form. By 1959, E. Hecke had studied the forms in detail [20]. Among the many results

that were studied in detail was

∑j≥2|n|

(−1)j+nqj(j+1)/2−n(3n−1)/2 = (q)2∞ (5.39)

which was originally discovered by L. J. Rogers [22].

It is with little difficulty that we can show the above identity and others with

our new conjugate Bailey pairs. Noting that our γn in most cases is already a restricted

sum of the type we are looking for, all that is left to do is find a suitable Bailey pair to

match it with. The following section will discuss some new results as well as tying in

some identities from Andrews [5] and Andrews, Dyson, and Hickerson [8].

We first return to equation (5.39).

Corollary 5.3.1.

∑j≥2|n|

(−1)j+nqj(j+1)/2−n(3n−1)/2 = (q)2∞. (5.40)

Proof. We consider a = 0, b = 1, c = −1 in Corollary 3.2.1 in (5.6) or a = 0, b = ∞, c = ∞

in Corollary 3.2.1 in (5.7).

In [5], Andrews uses complicated Bailey pairs with the implementation of Bailey

chains, as well as some clever algebra to prove many identities including the following:

58

Corollary 5.3.2.

∑j≥|n|

(−1)nqj(3j+1)/2−n2(1− q2j+1) =

(q)2∞(−q)∞

. (5.41)

Proof. We consider a = 0, b = ∞, c = −q in Corollary 3.2.1 with (5.8).

Andrews, Dyson and Hickerson then adapted the method used in [5] to prove

similar identities involving the rank of a partition [8]. The main motivation for the

paper was the function

σ(q) =∞∑n=0

qn(n+1)/2

(−q)n(5.42)

which can be found in [4]. We note that σ(q) is just the generating function for strict

partitions with odd rank subtracted from those with even rank. We find the following

corollary in [8, p. 392, eq. (1.5)].

Corollary 5.3.3.

σ(q) =∑j≥|n|

(−1)n+jqj(3j+1)/2−n2(1− q2j+1). (5.43)

Proof. We consider a = 0, b = ∞, c = q in Corollary 3.2.1 with (5.8).

Also in [8, p. 404], a similar generating function to σ(q) is defined. For n ≥ 1,

we consider partitions of n into odd parts with no gaps. Let S∗(n) be the excess of the

number of such partitions with largest part congruent to 3 modulo 4 over the number

with largest part congruent to 1 modulo 4. Andrews, Dyson and Hickerson then show

59

that [8, p.404, eq. 5.2]

∑n≥1

S∗(n)qn =∑n≥1

(−1)nqn2

(q; q2)n=∑n≥1

(−1)nqn(3n+1)(1 + q2n)2n−1∑j=0

q−j(j+1)/2. (5.44)

It is with minimal work that we can show an equivalent formula.

Corollary 5.3.4.

∑n≥0

(−1)nqn2

(q; q2)n=∑j≥|n|

(−1)jqj(3j+1)−n(2n−1)(1− q4n+2). (5.45)

Proof. We consider a = 0, b = ∞, c = q in Corollary 3.2.1 with (5.9).

The following identity can be found in [2, p. 457, (3.16)].

Corollary 5.3.5.

∑j≥2|n|

(−1)j+nqj(j+1)/2−n2= (q)∞(q2; q2)∞. (5.46)

Proof. We consider a = 0, b, c→∞ in Corollary 3.2.1 with (5.8).

The following corollaries are other indefinite quadratic forms in which our identity

simplifies down to an infinite product, each revealing an elegant identity.

Corollary 5.3.6.

∑j≥|n|

(−1)jq2j(j+1)−n(2n−1) =(q4; q4)2∞(−q; q2)∞

. (5.47)

60

Proof. We consider a = 0, b = 1, c = −1 in Corollary 3.2.1 with (5.9).

Corollary 5.3.7.

∑j≥|n|

(−1)jqj(3j+2)−n(2n−1)(1 + q2j+1) = (−q)∞(q2; q2)2∞. (5.48)

Proof. We consider a = 0, b = ∞, c = 1 in Corollary 3.2.1 with (5.9).

Corollary 5.3.8.

∑j≥|n|

(−1)jq3j2−n(2n−1)(1− q2j+1)2(1 + q2j+1) = −q(−q)∞(q2; q2)2∞. (5.49)

Proof. We consider a = 0, b = ∞, c = q2 in Corollary 3.2.1 with (5.9).

Corollary 5.3.9.

∑j≥2|n|

(−1)j+nqj(j+1)/2−2n2=

(q)2∞(−q; q2)∞

. (5.50)

Proof. We consider a = 0, b = 1, c = −q followed by q → −q in Corollary 3.2.1 with

(5.8).

Before moving on to our last three corollaries, it is neccessary to introduce a simple

identity which enables us to make the proper simplifications to obtain our results. It is

2∞∑n=0

qn(2n+1)

(q)2n=

∞∑n=0

(1 + (−1)n)qn(n+1)/2

(q)n(5.51)

= (q)∞ + (−q)∞. (5.52)

61

Corollary 5.3.10.

2∑

j≥2|n|(−1)jqj(j+1)−n(2n−1) = (q2; q2)∞ ((−q)∞ + (q)∞) . (5.53)

Proof. We consider a = 0, b = ∞, c = ∞ in Corollary 3.2.1 with (5.9).

Corollary 5.3.11.

2∑

j≥2|n|(−1)nqj(3j+2)−2n(3n−1)(1− q2j+1) = (q4; q4)∞ + (q)2∞(−q2; q2)∞. (5.54)

Proof. We consider a = 0, b = ∞, c = 1 followed by q → −q in Corollary 3.2.1 with

(5.6).

Corollary 5.3.12.

(2+q)(q4; q4)∞+q(q)2∞(−q2; q2)∞ = 2∑

j≥2|n|(−1)nq3j

2−2n(3n−1)(1+q2j+1)2(1−q2j+1).

(5.55)

Proof. We consider a = 0, b = ∞, c = q2 followed by q → −q in Corollary 3.2.1 with

(5.6).

5.4 Partitions

As in the previous chapter, we present some combinatorial interpretations of our

q-series identities from the previous sections.

62

Corollary 5.4.1. Let A(q) denote the generating function for the partitions of n with

no even gaps and the only odd parts must be the largest parts in which each part is

counted as (−1)k where k equals the difference of µ(λ) and the number of different even

parts. Then,

A(q) =∞∑n=0

(1− q2n+1)(1 + q2n+1)

qj(j+1)/2. (5.56)

Proof. Recall Corollary 5.2.1.

Corollary 5.4.2. Let A(q) denote the generating function for partitions of n into ones

and distinct even parts with no even part exceeding twice the total number of parts.

Then,

A(q) =1

(−q;−q)∞

∞∑j−∞

(−1)jq3j(j+1)/2

(1 + q2j+1). (5.57)

Proof. Recall (5.13) and let q → −q.

Corollary 5.4.3. Let A(q) denote the generating function for partitions of n into odd

parts without gaps where the largest part appears an odd number of times and each

partition is counted (−1)#d(λ). Then,

1 + 2A(q) =∞∑

j=−∞

1− q|4j+1|

1 + q|4j+1| (−1)jq2j2. (5.58)

63

Proof. Recall Corollary 5.2.6. We then note that

∞∑n=0

qn2

(−q; q2)n(1− q2n+1)

=∞∑n=0

qn2

(−q; q2)n+

∞∑n=0

q(n+1)2

(−q; q2)n(1− q2n+1)(5.59)

= 1 +∞∑n=1

qn2

(−q; q2)n+

∞∑n=1

qn2

(−q; q2)n−1(1− q2n−1)(5.60)

= 1 + 2∞∑n=1

qn2

(−q; q2)n(1− q2n−1). (5.61)

Replacing q with −q we can see that the sum in (5.61) is clearly the generating function

A(q),

1 + 2∞∑n=1

(−1)nqn2

(q; q2)n(1 + q2n−1)

= 1 + 2∞∑n=1

(−1)nq1+3+···+(2n−1)

(1− q)(1− q3) · · · (1− q2n−3)(1− q2(2n−1)). (5.62)

Corollary 5.4.4. Let A(q) denote the generating function for partitions of n with no

gaps in which each partition is counted as (−1)k where k is the product of λ1 and µλ1(λ).

Then,

A(q) =(−q; q2)∞(q2; q2)∞

∞∑j=−∞

1− q4j+1

1 + q4j+1 (−1)jq2j(3j+1). (5.63)

Proof. Recall Corollary 5.2.7.

64

Corollary 5.4.5. Let A(q) denote the generating function for partitions of n with no

odd gaps in which each partition is counted as (−1)k where k is the product of λ1 and

µλ1λ). Then,

A(q) =(−q; q2)∞(q2; q2)∞

∞∑j=−∞

1− q4j+1

1 + q4j+1 (−1)jq2j(5j+3). (5.64)

Proof. Recall Corollary 5.2.8.

Corollary 5.4.6. Let a(n) denote the number of overpartitions of n with overlined parts

either odd or congruent to 2 modulo 4, all other parts odd, and no odd part exceeding

twice the number of parts congruent to 2 modulo 4 plus 1. Let b(n) be the number of

tripartitions of n in which the first two partitions have distinct parts and each part is

congruent to 3 modulo 6 and the third partition has parts congruent to ±2 modulo 6.

Let c(n) be the number of partitions of n with parts congruent to 1, 4, 5, 7, 8 and 11

modulo 12. Then for n ≥ 1, b(n) + 2c(n− 1) = a(n).

Proof. Recall Corollary 5.2.11, let q → −q and we see that

∞∑n=0

(−q; q2)n+1(q; q2)n+1

· q2n2

(q4; q4)n=

(−q3; q6)2∞(q2, q4; q6)∞

− 2q(q, q4, q5, q7, q8, q11; q12)∞

. (5.65)

Remark 5.4.7. If we define b(n) and c(n) as in Corollary 5.4.6, then we see that

b(n) ≥ 2c(n− 1) since the left side of (5.65) only has positive terms.

65

The following corollary is a variation of Euler [3, p. 5],

∑λ∈O

q|λ| =∑λ∈D

q|λ|. (5.66)

Corollary 5.4.8. Let f(λ) be defined by

f(λ) =

1 if µ1

3(λ) > µ23(λ)

0 otherwise.(5.67)

Then ∑λ∈O

(−1)µ(λ)−bλ1/2cq|λ| =∑λ∈D

(−1)µ33(λ)+f(λ)

q|λ|. (5.68)

Proof. Recall (5.30). We interpret the right-hand side of the equation combinatorially

as the generating function for partitions into odd parts in which some of our partitions

are counted as positive and others as negative. More specifically,

∞∑n=0

(−1)nqn

(−q2; q2)n=∑λ∈O

(−1)µ(λ)−bλ1/2cq|λ|. (5.69)

We see that we can also interpret the left-hand side as

∑λ∈D=

3,2

(−1)µ33(λ)q|λ| −

∑λ∈D3,1

(−1)µ33(λ)q|λ| =

∑λ∈D

(−1)µ33(λ)+f(λ)

q|λ|. (5.70)

66

Corollary 5.4.9. Let A(q) denote the generating function for partitions of n with no

even gaps in which each partition is counted as (−1)µ(λ). Then

A(q) =∑

λ∈D=5,4

(−1)µ55(λ)q|λ| −

∑λ∈D5,1

(−1)µ55(λ)q|λ|. (5.71)

Proof. Recall Corollary 5.2.19.

5.5 Sums of Triangular Numbers

We turn our attention to the formula of Gauss,

∞∑n=0

q∆n =(q2; q2)∞(q; q2)∞

where ∆n is the nth triangular number, i.e. n(n+ 1)/2. With this identity, we are able

to make some interesting remarks about the generating functions related to the sums of

the triangular numbers.

This section will focus on the number of ways to represent a number as the sum

of triangular numbers. As is standard, we define the function,

ψ(q) =∞∑n=0

qn(n+1)/2, (5.72)

a classical theta function studied by Ramanujan [13, p.100, (5.1)]. We can then see that

ψk(q) =∞∑n=0

tk(n)qn (5.73)

67

where tk(n) counts the number of representations of n as the sum of k triangular numbers.

We note that order is important here, unlike with partitions: for example, t3(5) = 3,

since 5 = 3+1+1 = 1+3+1 = 1+1+3. While all of these remarks have been made in

previous work, none have used the conjugate Bailey pair approach presented below and

no other method has been able to encompass so many results so easily.

Corollary 5.5.1.

1. ψ2(q) =∞∑

j=−∞

(−1)jqj(j+1)

(1− q2j+1)(5.74)

2. ψ2(q2) =∞∑j=0

qj

1 + q2j+1 (5.75)

3. ψ4(q) =∞∑

j=−∞

qj

(1− q2j+1)2(5.76)

4. ψ3(q) =∑j≥|n|

(1 + q2j+1)(1− q2j+1)

q2j(j+1)−n(2n−1) (5.77)

5. ψ2(q) =∑j≥|n|

(−1)j+nqj(3j+2)−n(3n+1)(1 + q2j+1) (5.78)

6. ψ2(q) =∑j≥|n|

(−1)j+nqj(j+1)−n2(5.79)

Remark 5.5.2. We note that (5.74) can be found in [15, p. 619, (7.2)], (5.75) can be

found in [13, p. 139, ex. (iv)], (5.76) can be found in [16, p.285], (5.77) can be found in

[5, p. 114, (1.5)] and (5.79) can be found in [2, p. 452, (1.4)] with q → q2 and z = 1/q.

68

Proof. For equation (5.74), we consider b = 1, c = −q and d = −1 in Theorem 4.0.2.

(1− q)∞∑

n=−∞

(−1)nqn(n+1)

(1− q2n+1)

=(−q, q2; q2)∞(q3,−q2; q2)∞

∞∑n=0

(q,−q2,−q,−q; q2)n(q2; q2)2n(1− q2n+1)

(−1)nqn (5.80)

=(−q;−q)∞(q;−q)∞

2φ1

(q,−qq3

; q2,−q)

(5.81)

=(−q;−q)∞(q;−q)∞

· (−q2, q2; q2)∞(q3,−q; q2)∞

(5.82)

using III.2 [19, p. 359],

= (1− q)(q2; q2)2∞(q; q2)2∞

. (5.83)

For equation (5.75), we consider a, b, c→ 1,−1,−1 in Corollary 3.2.1 with equa-

tion (5.9).

For equation (5.76), we consider a, b, c → 1, 1, 1 in Corollary 3.2.1 with equation

(5.9).

For equation (5.77), we consider a, b, c → 0, 1, 1 in Corollary 3.2.1 with equation

(5.9).

For equation (5.78), we consider a, b, c→ 0,∞, 1 in Corollary 3.2.1 with equation

(5.7).

For equation (5.79), we consider a, b, c→ 0, 1,−1 in Corollary 3.2.1 with equation

(5.8).

69

Chapter 6

Infinite Families of Conjugate Bailey Pairs

6.1 A generalization of Watson’s 8φ7 transformation formula

In the proof of our general conjugate Bailey pair in Chapter 3, we use Watson’s

8φ7 transformation formula with a great deal of simplification to prove our result. It is

with this motivation that we can extend our general conjugate Bailey pairs to infinite

families of pairs with a known generalization of Watson’s formula. The transformation

formula that we seek can be found in a proceeding provided by Andrews in [10, p. 199].

Theorem 6.1.1. For k ≥ 1, N a nonnegative integer,

2k+4φ2k+3

[a, q

√a,−q

√a, b1, c1, · · · , bk, ck, q

−N√a,−

√a, aq/b1, aq/c1, · · · , aq/bk, aq/ck, aqN+1 ; q,

akqk+N

b1 · · · bkc1 · · · ck

]

=(aq)N (aq/bkck)N(aq/bk)N (aq/ck)N

∑m1,··· ,mk−1≥0

(aq/b1c1)m1· · · (aq/bk−1ck−1)mk−1

(q)m1· · · (q)mk−1

·(b2, c2; q)m1

(b3, c3; q)m1+m2· · · (bk, ck; q)m1+···+mk−1

(aq/b1, aq/c1; q)m1(aq/b2, aq/c2; q)m1+m2

· · · (aq/bk, aq/ck; q)m1+···+mk−1

·(q−N )m1+···mk−1

(bkckq−N/a)m1+···+mk−1

· (aq)mk−2+2mk−3+···+(k−2)m1qm1+m2+···+mk−1

(b2c2)m1(b3c3)m1+m2 · · · (bk−1ck−1)m1+···+mk−2.

(6.1)

70

6.2 Our Main Result

Using the above generalization of Watson’s transformation formula, we have the

following conjugate Bailey pair.

Theorem 6.2.1. Let

δn =(a, b0, c0; q)n

(aq; q)n

(q2(k+1)

b0c0b1c1 · · · bkck

)n

·∑

m1,··· ,mk≥0

(b1, c1; q)n+m1· · · (bk, ck; q)n+m1+···+mk

(q2/b0, q2/c0; q)n+m1

· · · (q2/bk−1, q2/ck−1; q)n+m1+···+mk

·(q2/b0c0; q)m1

· · · (q2/bk−1ck−1; q)mk

(q)m1· · · (q)mk

· q2(mk+2mk−1+···+km1)

(b1c1)m1 · · · (bkck)m1+···+mk(6.2)

and

γn =(q2/bk, q

2/ck; q)∞(q, q2/bkck; q)∞

· (a)n(q/a)n

(−a)−nq−(n2 )

·∑i≥n

(q/a, b0, c0, . . . , bk, ck; q)i(aq, q2/b0, q

2/c0, . . . , q2/bk, q

2/ck; q)i(1− q2i+1)

(− aq2(k+1)

b0c0 · · · bkck

)iq

(i2

).

(6.3)

Then

γn =∑j≥n

δj

(q)j−n(q)j+n(6.4)

71

Proof. We first let N →∞, a→ q2n+1, b1 → qn+1/a, c1 → q and bi → biqn, ci → ciq

n

for 2 ≤ i ≤ k in Equation (6.1).

limN→∞ 2k+4φ2k+3

q2n+1, qn+3

2 ,−qn+32 , qn+1/a, q, b2q

n, · · · , ckqn, q−N

qn+1

2 ,−qn+12 , aqn+1, q2n+1, qn+2/b2, · · · , qn+2/ck, 0

; q,q2k−2+n+N

b2c2 · · · bkck

=

(q2n+2)∞(q2/bkck)∞(qn+2/bk)∞(qn+2/ck)∞

·∑

m1, · · ·mk−1 ≥ 0

(aqn)m1(q2/b2c2)m2

· · · (q2/bk−1ck−1)mk−1

(q)m1· · · (q)mk−1

(aqn+1, q2n+1; q)m1

·(b2q

n, c2qn; q)m1

· · · (bkqn, ckq

n; q)m1+···+mk−1

(qn+2/b2, qn+2/c2; q)m1+m2

· · · (qn+2/bk, qn+2/ck; q)m1+···+mk−1

· (q2)mk−1+2mk−2+···+(k−1)m1

(b2c2)m1(b3c3)m1+m2 · · · (bkck)m1+···+mk−1

. (6.5)

We now let k → k + 2 followed by m1 → j and mi → mi−1, bi → bi−2, ci → ci−2 for

2 ≤ i ≤ k + 2 to obtain

limN→∞ 2k+7φ2k+6

qn+32 ,−qn+3

2 , qn+1/a, q, b0qn, · · · , ckq

n, q−N

qn+1

2 ,−qn+12 , aqn+1, qn+2/b0, · · · , qn+2/ck, 0

; q,aq2k+2+n+N

b0 · · · bkc0 · · · ck

=(q2n+2)∞(q2/bkck)∞

(qn+2/bk)∞(qn+2/ck)∞

∑j,m1,··· ,mk≥0

(aqn)j(q2/b0c0)m1

· · · (q2/bk−1ck−1)mk

(q)m1· · · (q)mk

(q, aqn+1, q2n+1; q)j

·(b0q

n, c0qn; q)j · · · (bkq

n, ckqn; q)j+m1+···+mk

(qn+2/b0, qn+2/c0; q)j+m1

· · · (qn+2/bk−1, qn+2/ck−1; q)j+m1+···+mk

· (q2)mk+2mk−1+···+km1+(k+1)j

(b0c0)j(b1c1)j+m1 · · · (bkck)j+m1+···+mk. (6.6)

72

We now start to simplify the right-hand side of equation (6.6),

=(q2n+2)∞(q2/bkck)∞

(qn+2/bk)∞(qn+2/ck)∞·(q)2n(aq, q2/b0, q

2/c0, · · · , q2/bk−1, q

2/ck−1; q)n(a, b0, c0, · · · , bk, ck; q)n

·

(b0 · · · bkc0 · · · ck

(q2)(k+1)

)n ∑j,m1,··· ,mk≥0

(a)n+j(q2/b0c0)m1

· · · (q2/bk−1ck−1)mk

(q)j(q)m1· · · (q)mk

·(b0, c0; q)n+j · · · (bk, ck; q)n+j+m1+···+mk

(aq)n+j(q)2n+j(q2/b0, q

2/c0; q)n+j+m1· · · (q2/bk−1, q

2/ck−1; q)n+j+m1+···+mk

· (q2)mk+2mk−1+···+km1+(k+1)(j+n)

(b0c0)n+j(b1c1)n+j+m1 · · · (bkck)n+j+m1+···+mk. (6.7)

Separating the sum over j from the mi’s in equation (6.7) we see that,

=(q)∞(q2/bkck)∞

(qn+2/bk)∞(qn+2/ck)∞·(aq, q2/b0, q

2/c0, · · · , q2/bk−1, q

2/ck−1; q)n(1− q2n+1)(a, b0, c0, · · · , bk, ck; q)n

·

(b0 · · · bkc0 · · · ck

(q2)(k+1)

)n∑j≥n

1(q)j+n(q)j−n

·(a, b0, c0; q)j

(aq)j

(q2(k+1)

b0c0 · · · bkck

)j

·∑

m1,··· ,mk≥0

(b1, c1; q)j+m1· · · (bk, ck; q)j+m1+···+mk

(q2/b0, q2/c0; q)j+m1

· · · (q2/bk−1, q2/ck−1; q)j+m1+···+mk

·(q2/b0c0)m1

· · · (q2/bk−1ck−1)mk

(q)m1· · · (q)mk

· q2(mk+2mk−1+···+km1)(b1c1)m1 · · · (bkck)m1+···+mk

. (6.8)

We can now recognize our earlier defined δj in equation (6.8),

=(q)∞(q2/bkck)∞

(qn+2/bk)∞(qn+2/ck)∞·(aq, q2/b0, q

2/c0, · · · , q2/bk−1, q

2/ck−1; q)n(1− q2n+1)(a, b0, c0, · · · , bk, ck; q)n

·

(b0 · · · bkc0 · · · ck

(q2)(k+1)

)n∑j≥n

δj

(q)j+n(q)j−n. (6.9)

73

So we have

∑j≥n

δj

(q)j+n(q)j−n

=(qn+2/bk)∞(qn+2/ck)∞

(q)∞(q2/bkck)∞

· (1− q2n+1)(a, b0, c0, · · · , bk, ck; q)n(aq, q2/b0, q

2/c0, · · · , q2/bk−1, q2/ck−1; q)n

·

((q2)(k+1)

b0 · · · bkc0 · · · ck

)n

· limN→∞ 2k+7φ2k+6

qn+32 ,−qn+3

2 , qn+1/a, q, b0qn, · · · , ckq

n, q−N

qn+1

2 ,−qn+12 , aqn+1, qn+2/b0, · · · , qn+2/ck, 0

; q,aq2k+2+n+N

b0c0 · · · bkck

.(6.10)

We now simplify the right-hand side of equation (6.10),

=(qn+2/bk)∞(qn+2/ck)∞

(q)∞(q2/bkck)∞

· (1− q2n+1)(a, b0, c0, · · · , bk, ck; q)n(aq, q2/b0, q

2/c0, · · · , q2/bk−1, q2/ck−1; q)n

·

(q2(k+1)

b0 · · · bkc0 · · · ck

)n

·∞∑i=0

(q2n+3; q2)i(qn+1/a, b0q

n, c0qn, · · · , bkq

n, ckqn; q)i

(q2n+1; q2)i(aqn+1, qn+2/b0, q

n+2/c0, · · · , qn+2/bk, qn+2/ck; q)i

·

(aq2k+2+n

b0c0 · · · bkck

)i(−1)iqi(i−1)/2. (6.11)

74

We can now work on restricting our sum on i,

=(q2/bk)∞(q2/ck)∞(q)∞(q2/bkck)∞

· (a)n(q/a)n

·∞∑i=0

(q2n+1; q2)i+1(q/a, b0, c0, · · · , bk, ck; q)n+i(q2n+1; q2)i(aq, q

2/b0, q2/c0, · · · , q2/bk, q2/ck; q)n+i

·

(q2k+2

b0c0 · · · bkck

)i+nai(−1)iqi(i−1)/2+ni (6.12)

=(q2/bk)∞(q2/ck)∞(q)∞(q2/bkck)∞

· (a)n(q/a)n

(−a)−nq−(n2 )

·∞∑i=0

(q/a, b0, c0, · · · , bk, ck; q)n+i(aq, q2/b0, q

2/c0, · · · , q2/bk, q2/ck; q)n+i· (1− q2(n+i)+1)

(−aq2k+2

b0c0 · · · bkck

)i+nq

(i+n2

). (6.13)

Shifting our sum (i ≥ 0 → i ≥ n) we see that this is just equal to our earlier defined

γn.

We now apply the above formula to exhibit new infinite families of conjugate

Bailey pairs.

6.3 Infinite families of conjugate Bailey pairs and Identities

We can consider one infinite family of conjugate Bailey pairs which generalizes

our previously mentioned conjugate Bailey pair,

δn =(q)2n

(1 + q2n+1)qn, γn = q−n

2 ∑i≥n

qi2+i. (6.14)

75

Note that the above pair can be found in Chapter 3 with a, b, c → −1, 1, q. We

now use our new theorem to define a new generalized version of the pair. Let q 7→ q2

(i.e. un = vn = 1/(q2; q2)n), followed by a = −q, bi = q, ci = q2 in Theorem 6.2.1. We

get the following infinite family of conjugate Bailey pairs.

Corollary 6.3.1. The following is a conjugate Bailey pair with un = vn = 1/(q2; q2)n,

δn =(q)2n

(1 + q2n+1)q(k+1)n

·∑

m1,··· ,mk≥0

qmk+2mk−1+···+km1(q; q2)m1· · · (q; q2)mk

(1− q2(n+m1)+1) · · · (1− q2(n+m1+···+mk)+1)(q2; q2)m1· · · (q2; q2)mk

(6.15)

and

γn = q−n2 ∑i≥n

qi2+i(k+1)

(1− q2i+1)k. (6.16)

With this infinite family we can consider the symmetric bilateral Bailey pair

αn = (−1)nqn(3n+1)/2 and βn = (q; q2)n/(q2; q2)2n which can be found in [12, p. 5,

Sec. 6, (ii)] with a = 1, b→∞ and x replaced by q. Then we have

Theorem 6.3.2.

∑m1,··· ,mk≥0

qmk+2mk−1+···+km1(q; q2)m1· · · (q; q2)mk

(1 + q2m1+1) · · · (1− q2(m1+···+mk)+1)(−q; q)2m1(q2; q2)m2

· · · (q2; q2)mk

=∞∑i=0

(−1)iq3i(i+1)/2+i(k−1)

(1− q2i+1)k−1 . (6.17)

76

We can also consider αn = (−1)nqn(n+1)/2 and βn = (−1)nqn2(q; q2)n/(q

2; q2)2n

which also comes from [12, p. 5, Sec. 6, (ii)] with a = 1, b → 0 and x replaced by q.

Then we have

Theorem 6.3.3.

∑m1,··· ,mk≥0

(−1)m1qmk+2mk−1+···+km1+m2

1(q; q2)m1· · · (q; q2)mk

(1 + q2m1+1) · · · (1− q2(m1+···+mk)+1)(−q; q)2m1(q2; q2)m2

· · · (q2; q2)mk

=∞∑i=0

(−1)iqi(i+1)/2+i(k−1)

(1− q2i+1)k−1 . (6.18)

We need not restrict ourselves to choosing examples in which a→ −1 or examples

in which bi = q and ci = q2. Considering the case q → q2, a → 0 and bi = ci = q in

Theorem 6.2.1, we get the following infinite families of conjugate Bailey pairs.

Corollary 6.3.4. The following is a conjugate Bailey pair with un = vn = 1/(q2; q2)n,

δn = (q; q2)2n

∑nk≥···≥n1≥n

q2(n+n1+···+nk)

(1− q2n1+1) · · · (1− q2nk+1)(6.19)

and

γn =(q; q2)2∞(q2; q2)2∞

q−2n2 ∑i≥n

(1 + q2i+1)

(1− q2i+1)2k+1 q2i(i+k+1). (6.20)

We can consider our new found pair with the Bailey pair, αn = q2n2+n and βn =

1/(q)2n in the symmetric bilateral Bailey transform to achieve the following identity.

77

Theorem 6.3.5.

(q; q2)2∞(q2; q2)2∞

∞∑i=0

(1 + q2i+1)

(1− q2i+1)2kq2i(i+k)

= (1− q)∑

nk≥···≥n0≥0

(q; q2)n0

(q2; q2)n0

· q2(n0+n1+···+nk)

(1− q2n1+1) · · · (1− q2nk+1). (6.21)

We have only shown three of the many possible families of identities which could

be found using our new conjugate Bailey pair. It should be clear that any of the results

stated in Chapters 4 and 5 are now capable of being generalized to an infinite family of

identities.

78

Chapter 7

Combinatorial and Partition-Related Remarks

In this section of the paper we discuss combinatorial interpretations of two iden-

tities and explain how these interpretations lead to new and more general discoveries.

The first of the two identities is the following,

Lemma 7.0.6.∞∑n=0

(q; q2)n(q2; q2)n+1

qn =1

1− q. (7.1)

The above identity can be proven in a number of ways, including conjugate Bailey

pairs as Andrews and Warnaar most recently established [9]. While it was through their

paper that I found this identity, it’s simplicity is what led me to search for a combinatorial

proof. The second identity discussed in this section can be found in Ramanujan’s Lost

Notebook [7, Entry 9.5.2, p. 239].

Lemma 7.0.7.

∞∑n=0

(q; q2)nqn =

∞∑n=0

(−1)nq3n2+2n(1 + q2n+1). (7.2)

As with the previoius identity, Andrews and Warnaar found that this could be

proven with their new conjugate Bailey pairs. Combinatorial interpretation not only

leads to a proof of this theorem but also to proofs of finite versions of the identity.

79

7.1 Some Generalizations of Fine’s Identity

The investigation and generalization of (7.1) yields an identity of Fine. In the

remainder of this section we show how the same interpretation can lead to many other

generalizations and other elegant identities. We start with finite forms and end with a

discussion of inserting an arbitrary function, An(q), into our generalized form.

7.1.1 A General Case of a Simple Bijection

While finding a combinatorial proof of (7.1), we found that it was able to be

generalized. The original proof of the identity involved a sign-changing bijection which

eliminated all but a very small set of the partitions that were being counted, leaving

a very simple q-series and completing the original identity. The theorem below, rather

than define a sign-changing bijection, keeps things in a more general sense allowing for

the generalizations that follow later in the section.

Theorem 7.1.1. Let An(q) be any function dependent on n and q that does not affect

the convergence of our series. Then

∞∑n=0

(−aq; q2)n(bq2; q2)n+1

An(q)(cq)n =∞∑n=0

An(q)(cq)n

+q2∑k,n≥0

(−aq; q2)k(bq2; q2)k+1

[bAn+k(q) + acAn+k+1(q)

](cq)n(cq3)k. (7.3)

Proof. Let E be the set of overpartitions into even parts with ones and only allowing

evens to be overlined. Let E(m) be the set of λ ∈ E such that |λ| = m. Then let En,k(m)

be a subset of E(m) with exactly n−k parts equal to one, exactly k overlined evens each

80

≤ 2n (they must be distinct), and all even parts ≤ 2n+ 2. Clearly

E(m) =⋃n≥0

0≤k≤n

En,k(m).

Let us define the following statistics. Let eu(λ) be the number of not overlined

even parts of λ, e(λ) be the number of overlined even parts of λ and e1(λ) be the number

of parts equal to one. Then we have

∞∑n=0

(−aq; q2)n(bq2; q2)n+1

An(q)(cq)n

=∑

m,n≥0

n∑k=0

∑λ∈En,k(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)(q) (7.4)

=∞∑m=0

∑λ∈E(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)(q). (7.5)

We will define a function, φ, on λ ∈ En,k(m), λ 6= (1, 1, . . . , 1) which will be one-to-one

and have order 2.

Definition of φ. We will need to define two statistics on a partition λ ∈ En,k(m).

Let l(λ) be the largest overlined part of λ (0 if there is no such part). Let le(λ) be the

largest even part of λ that is not overlined (0 if there is no such part). Notice that

the domain for our function, φ, implies that one of the values must be nonzero. Our

definition of φ has two cases:

Case i: If l(λ) ≥ le(λ). Then we remove the largest overlined part of λ and return

the result as φ(λ).

81

Case ii: If l(λ) < le(λ). Then we overline the largest not overlined part of λ and

return the result as φ(λ).

It should be clear that our map is one-to-one and that φ2(λ) = λ. Thus we have

broken up E(m) \ {(1, . . . , 1)} into two equal classes; those which require Case i when

we apply φ and those which require Case ii. Let us define E1(m) (resp. E2(m)) as those

λ ∈ E(m) \ {(1, . . . , 1)} which require Case i (resp. Case ii) when we apply φ. Then for

λ ∈ E1(m) we have the following.

e1(φ(λ)) = e1(λ), (7.6)

e(φ(λ)) = e(λ)− 1, (7.7)

eu(φ(λ)) = eu(λ) + 1. (7.8)

Clearly E(m) = E1(m) ∪ E2(m) ∪ {(1, . . . , 1)} and so we have

∞∑n=0

(−aq; q2)n(bq2; q2)n+1

An(q)(cq)n

=∞∑m=0

∑λ∈E(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)(q) (7.9)

=∞∑m=0

(cq)m +∑m≥0

λ∈E1(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)(q)

+∑m≥0

λ∈E2(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)(q). (7.10)

82

Using our map we can simply sum over partitions in E1(m),

=∞∑m=0

(cq)m +∑m≥0

λ∈E1(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)(q)

+b

ac

∑m≥0

λ∈E1(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|Ae1(λ)+e(λ)−1(q) (7.11)

=1

1− cq+

∑m≥0

λ∈E1(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|

×(Ae1(λ)+e(λ)(q) +

b

acAe1(λ)+e(λ)−1(q)

). (7.12)

We can now convert our partition identity back to q-series,

∞∑n=0

(−aq; q2)n(bq2; q2)n+1

An(q)(cq)n

=1

1− cq+

∑m≥0

λ∈E1(m)

(ac)e(λ)beu(λ)ce1(λ)q|λ|

×(Ae1(λ)+e(λ)(q) +

b

acAe1(λ)+e(λ)−1(q)

)(7.13)

=1

1− cq+

∞∑n=0

(An(q) +

b

acAn−1(q)

)(cq)n

n−1∑k=0

(−aq; q2)k(bq2; q2)k+1

aq2k+1 (7.14)

=1

1− cq+ q2

∑k,n≥0

(−aq; q2)k(bq2; q2)k+1

×(acAn+k+1(q) + bAn+k(q)

)(cq)n(cq3)k. (7.15)

We have now obtained the desired result.

83

With the proper substitutions equation (7.3) becomes

∞∑n=0

(aq)n(bq)n

An(q)tn = (1− b)∞∑n=0

An(q)tn +

∑k,n≥0

(aq)k(bq)k

[bAn+k(q)− atqAn+k+1(q)

]tn(tq)k. (7.16)

Our investigation of equation (7.1) led to the generalization which can be seen

above. But if we consider An(q) = 1 in (7.16) we have the simple identity found in Fine

[17, p. 2, (2.4)],

∞∑n=0

(aq)n(bq)n

tn =(1− b)(1− t)

+(b− atq)(1− t)

∞∑k=0

(aq)k(bq)k

(tq)k. (7.17)

This result can be iterated (and has been done in Fine’s text) to

∞∑k=0

(aq)k(bq)k

tn =(1− b)(1− t)

∞∑n=0

(atq/b)n(tq)n

bn. (7.18)

7.1.2 Finite Sums

One of the more interesting aspects of the combinatorial proof of Theorem 7.1.1

is its ability to also interpret finite sums. We have the following

Theorem 7.1.2. Let n be a positive integer. Then

(1− t)n∑k=0

(aq; q2)k(bq2; q2)k+1

tk = 1− tn+1

+qn∑k=0

(aq; q2)k(bq2; q2)k+1

(tn+1(a− bq) + tk(bq − at)

)q2k. (7.19)

84

Proof. We use the same notation as in Theorem 7.1.1. We first consider the set of

partitions

Xn =⋃

m≥00≤k≤n

{λ : λ ∈ E2(m) ∩ En,k(m)}.

We then consider the set φ(Xn) = {φ(λ) : λ ∈ Xn}. Notice that while all elements in

Xn can be found inn∑k=0

(aq; q2)k(bq2; q2)k+1

(cq)k,

not all elements of φ(Xn) will be counted in the sum. Namely, those partitions in which

e1(λ) + e(λ) = n + 1. We also note that the only partitions counted in the above sum

which are not found in Xn or φ(Xn) are when e(λ) = eu(λ) = 0. So we get the following

identity.

n∑k=0

(aq; q2)k(bq2; q2)k+1

(cq)k (7.20)

=n∑k=0

(cq)k +∑λ∈Xn

(ac)e(λ)beu(λ)ce1(λ)q|λ|

+∑

λ∈φ(Xn)

(ac)e(λ)beu(λ)ce1(λ)q|λ|

−∑

λ∈φ(Xn)e1(λ)+e(λ)=n+1

(ac)e(λ)beu(λ)ce1(λ)q|λ|. (7.21)

85

Implementing our map as we did in the infinite case we get

=n∑k=0

(cq)k +(1 +

ac

b

) ∑λ∈Xn

(ac)e(λ)beu(λ)ce1(λ)q|λ|

−∑

λ∈φ(Xn)e1(λ)+e(λ)=n+1

(ac)e(λ)beu(λ)ce1(λ)q|λ|. (7.22)

It is left now to return our partition identity to a q-series.

n∑k=0

(aq; q2)k(bq2; q2)k+1

(cq)k (7.23)

=n∑k=0

(cq)k +(1 +

ac

b

) ∑λ∈Xn

(ac)e(λ)beu(λ)ce1(λ)q|λ|

−∑

λ∈φ(Xn)e1(λ)+e(λ)=n+1

(ac)e(λ)beu(λ)ce1(λ)q|λ| (7.24)

=1− (cq)n+1

1− cq+(1 +

ac

b

) n∑k=0

(cq)kk∑i=0

(−aq; q2)i(bq2; q2)i+1

bq2i+2

−(cq)n+1n∑i=0

(−aq; q2)i(bq2; q2)i+1

aq2i+1 (7.25)

=1− (cq)n+1

1− cq+

q

1− cq

n∑i=0

(−aq; q2)i(bq2; q2)i+1

q2i(bq + acq)((cq)i − (cq)n+1)

− q

1− cq

n∑i=0

(−aq; q2)i(bq2; q2)i+1

q2ia(cq)n+1(1− cq) (7.26)

=1− (cq)n+1

1− cq+

q

1− cq

n∑i=0

(−aq; q2)i(bq2; q2)i+1

q2i

×[(cq)i(bq + acq)− (cq)n+1(bq + a)

]. (7.27)

We now replace cq with t and multiply each side by 1− t to obtain the desired result.

86

Note that our theorem has the following special case.

Corollary 7.1.3. For n a positive integer,

n∑k=0

(q; q2)k(q2; q2)k+1

(1− qn+k+2)qk = 1 + q + · · ·+ qn. (7.28)

Proof. Let a = −1, b = 1 and t = q.

The proper substitutions into equation (7.19) yield a finite version of Fine’s result.

Corollary 7.1.4.

(1− t)n∑k=0

(aq)k(bq)k

tk = (1− tn+1)(1− b)

+n∑k=0

(aq)k(bq)k

(tn+1(aq − b) + tk(b− atq)

)qk, (7.29)

which has a similar special case,

Corollary 7.1.5.

(1− q)2n∑k=0

qk

(1− qk+1)(1− qk+2)=

(1− qn+1)(1− qn+2)

. (7.30)

Proof. Consider a = 1, b = q2 and t = q in (7.29).

It should be noted that finding the above corollary can be attributed to our

combinatorial interpretation, but it can be directly and easily proven using mathematical

induction.

87

7.1.3 The Eulerian Number Triangle and the Polylogarithm Function

We note that for |q| < 1 we have

∞∑n=1

qn =∞∑n=1

(nqn − (n− 1)qn

)(7.31)

=∞∑n=1

nqn −∞∑n=0

nqn+1 (7.32)

= (1− q)∞∑n=0

nqn. (7.33)

We see that for k > 0, (1 − q)∑nkqn can be written as the sum of terms of the form∑

nmqn where m < k,

(1− q)∞∑n=0

nkqn =∞∑n=0

nkqn −∞∑n=1

(n− 1)kqn (7.34)

=∞∑n=0

k−1∑m=0

(k − 1m

)(−1)k−1−mnmqn. (7.35)

Using this method, we can show the following known identities for values of k greater

than 1. For example,

∞∑n=0

nqn =q

(1− q)2, (7.36)

∞∑n=0

n2qn =q + q2

(1− q)3, (7.37)

∞∑n=0

n3qn =q + 4q2 + q3

(1− q)4, (7.38)

∞∑n=0

n4qn =q + 11q2 + 11q3 + q4

(1− q)5. (7.39)

88

We can graphically express the coefficients of the polynomial in the numerator as

1

1 1

1 4 1

1 11 11 1

1 26 66 26 1

1 57 302 302 57 1

1 120 1191 2416 1191 120 1

......

...

We note that the discovery of these polynomials is not the first; the triangle

above is known as the Eulerian Number Triangle and can be found in [14]. We define

the Eulerian numbers as⟨nr

⟩, where n refers to the number of rows down and r refers

to the column. For example,⟨

73

⟩= 1191.

We also remark that this weighted geometric sum is a special case of the Polylog-

arithm function,

Ln(z) :=∞∑i=1

zi

in. (7.40)

When we consider the negative integer values of the Polylogarithm function, we see that

for m > 0

L−m(q) =∞∑n=0

nmqn =q · pm(q)

(1− q)m+1 (7.41)

89

where

pm(q) =m−1∑n=0

⟨mn

⟩qn. (7.42)

We note that using the method shown in the beginning of this section, it is not difficult

to show that our Eulerian coefficients can be recursively defined [21, p. 99] as⟨i0

⟩=⟨

ii

⟩= 1 and for 0 < i < j

⟨j

i

⟩= (i+ 1)

⟨j − 1i

⟩+ (j − i)

⟨j − 1i− 1

⟩. (7.43)

There are a number of results regarding these Eulerian polynomials, both com-

binatorially and analytically [14], [21], [25], [18]. In a new application, we intend to use

our new generalization with these Eulerian polynomials in order to obtain interesting

q-series identities. Before we can move on, we must first gather some simple identities

regarding this new object.

Lemma 7.1.6. For m > 0,

q

1− q

m−1∑i=0

(mi

)L−i(q) = L−m(q). (7.44)

Proof. We have

90

L−m(q) =∞∑n=0

niqn (7.45)

=∑n,k≥0

(n+ k + 1)m−1qn+k+1 (7.46)

= q∑n,k≥0

m−1∑i=0

(m− 1i

)(n+ k)iqn+k (7.47)

= q∑k≥0

m−1∑i=0

(m− 1i

)kiqk

+q∑k,n≥0

m−1∑i=0

(m− 1i

)(n+ k + 1)iqn+k+1 (7.48)

= qm−1∑i=0

(m− 1i

)∑k≥0

kiqk + qm−1∑i=0

(m− 1i

)∑k≥0

ki+1qk (7.49)

= qm−1∑i=0

(m− 1i

)L−i(q) + q

m∑i=1

(m− 1i− 1

)L−i(q) (7.50)

=q

1− q+ q

m−1∑i=1

((m− 1i

)+(m− 1i− 1

))L−i(q) + qL−m(q) (7.51)

=q

1− q+ q

m−1∑i=1

(mi

)L−i(q) + qL−m(q) (7.52)

= qm−1∑i=0

(mi

)L−i(q) + qL−m(q). (7.53)

If we now subtract qL−m(q) from each side and divide by (1 − q) we have our desired

result.

Another result which becomes useful in the following section is

91

Lemma 7.1.7.

qm∑i=1

(mi

)(1− q)i−1pm−i(q)

[(1 + k)i − ki

]=m−1∑i=0

(mi

)ki(1− q)ipm−i(q).

(7.54)

Proof. Starting with our left-hand side we have,

qm∑i=1

(mi

)(1− q)i−1pm−i(q)

[(1 + k)i − ki

](7.55)

= q

m∑i=1

(mi

) i∑j=1

(i

j − 1

)kj−1(1− q)i−1pm−i(q) (7.56)

= qm∑j=1

m∑i=j

(mi

)( i

j − 1

)kj−1(1− q)i−1pm−i(q) (7.57)

= q

m−1∑j=0

m−1∑i=j

(m

i+ 1

)(i+ 1j

)kj(1− q)ipm−i−1(q) (7.58)

= qm−1∑j=0

kj(m

j

)m−1∑i=j

(m− j

i+ 1− j

)(1− q)ipm−i−1(q) (7.59)

= q

m−1∑j=0

kj(m

j

)m−1−j∑i=0

(m− j

i+ 1

)(1− q)i+jpm−i−j−1(q) (7.60)

= (1− q)mm−1∑j=0

kj(m

j

)m−1−j∑i=0

(m− j

i+ 1

)qpm−i−j−1(q)

(1− q)m−i−j. (7.61)

92

We can now recognize our polylogarithm function,

= (1− q)mm−1∑j=0

kj(m

j

)m−1−j∑i=0

(m− j

i+ 1

)Li+j+1−m(q) (7.62)

= (1− q)mm−1∑j=0

kj(m

j

)m−1−j∑i=0

(m− j

i

)L−i(q) (7.63)

=(1− q)m+1

q

m−1∑j=0

kj(m

j

)Lj−m(q) (7.64)

=(1− q)m+1

q

m−1∑j=0

kj(m

j

)qpm−j(q)

(1− q)m−j+1 (7.65)

=m−1∑j=0

kj(m

j

)(1− q)jpm−j(q). (7.66)

7.1.4 Combining Eulerian Polynomials and our Generalization

In this section we make a choice for our function An(q). With the knowledge that

we have of Euler polynomials from the previous section we are able to discover some

rather elegant weighted identities.

Theorem 7.1.8.

(1− q)mL−m(q) =

∞∑k=0

(q; q2)k(q2; q2)k+1

qk

km(1− q)m + q2+2km−1∑i=0

(mi

)ki(1− q)ipm−i(q)

. (7.67)

93

Proof. We first allow a 7→ −1, b 7→ q2, t 7→ q and An(q) = nm for m ≥ 1 in equation

(7.3).

∞∑k=0

(q; q2)k(q2; q2)k+1

kmqk

=∞∑k=0

kmqk + q2∑k,n≥0

(q; q2)k(q2; q2)k+1

[(n+ k)m − (n+ k + 1)m

]qn+3k (7.68)

= L−m(q) + q2∑k,n≥0

(q; q2)k(q2; q2)k+1

(n+ k)m −m∑i=0

(mi

)(n+ k)i

qn+3k (7.69)

= L−m(q)− q2∑k,n≥0

(q; q2)k(q2; q2)k+1

m−1∑i=0

(mi

)(n+ k)iqn+3k (7.70)

= L−m(q)− q2∑k≥0

(q; q2)k(q2; q2)k+1

q3km−1∑i=0

(mi

) ∑n≥0

0≤j≤i

njki−j(i

j

)qn (7.71)

= L−m(q)− q3∑k≥0

(q; q2)k(q2; q2)k+1

q3km−1∑i=0

(mi

) i∑j=0

ki−j(i

j

)pj(q)

(1− q)j+1 (7.72)

= L−m(q)−∑k≥0

(q; q2)k(q2; q2)k+1

q3k+3

×m−1∑j=0

m−j−1∑i=0

(m

i+ j

)(i+ j

j

)ki

pj(q)

(1− q)j+1 (7.73)

= L−m(q)−∑k≥0

(q; q2)k(q2; q2)k+1

q3k+3

×m−1∑j=0

pj(q)

(1− q)j+1

(m

j

)m−j−1∑i=0

(m− j

i

)ki (7.74)

= L−m(q)−∑k≥0

(q; q2)k(q2; q2)k+1

q3k+3

×m−1∑j=0

pj(q)

(1− q)j+1

(m

j

)[(1 + k)m−j − km−j

](7.75)

94

= L−m(q)−∑k≥0

(q; q2)k(q2; q2)k+1

q3k+3

×m∑j=1

pm−j(q)

(1− q)m−j+1

(m

j

)[(1 + k)j − kj

]. (7.76)

Adding our second term on the right-hand side to both sides we obtain the desired

result.

While this theorem may appear to be nothing more than a jumble of mathematical

symbols, we see that it has the ability to produce interesting weighted identities with

it’s special cases. For example, we can consider m = 1 and m = 2.

Corollary 7.1.9.

∞∑k=0

(q; q2)k(q2; q2)k+1

qk(k − kq + q2k+2) =q

(1− q)(7.77)

(7.78)

and

∞∑k=0

(q; q2)k(q2; q2)k+1

qk(k2(1− q)2 + qk+2(1 + q + 2k − 2kq)

)=q(1 + q)(1− q)

. (7.79)

We can also get related results from similar substitutions into equation (7.16).

95

Theorem 7.1.10.

q · pm(q)(1− q)2

=

∞∑k=0

qk

(1− qk+1)(1− qk+2)

km(1− q)m + q2+km−1∑i=0

(mi

)ki(1− q)ipm−i(q)

.

(7.80)

For example, m = 1 and m = 2,

Corollary 7.1.11.

∞∑k=0

(k − kq + qk+2)

(1− qk+1)(1− qk+2)qk =

q

(1− q)2(7.81)

(7.82)

and

∞∑k=0

(k2(1− q)2 + qk+2(1 + q + 2k − 2kq)

)(1− qk+1)(1− qk+2)

qk =q(1 + q)(1− q)2

. (7.83)

7.1.5 Another choice for An(q)

We can also choose

An(q) = (cq; q2)n/(dq2; q2)n+1 (7.84)

in equation (7.3). Like the weighted case, our notation starts to get slightly more con-

voluted, but it once again turns out some elegant special cases.

96

Theorem 7.1.12.

(1− t)F (t) = (1− t)∞∑k=0

(bq; q2)k(dq2; q2)k+1

tk + q(cq − at)F (tq2)

+tq2(b− dq3)(a− cq)∞∑n=0

(bt/dq; q2)n(tq2; q2)n+1

(dq4)nF (tq4+2n)

(7.85)

where

F (t) =∞∑n=0

(aq, bq; q2)n(cq2, dq2; q2)n+1

(t)n. (7.86)

97

Proof. We start by using our main bijection. Thus,

F (t) =∞∑n=0

(aq, bq; q2)n(cq2, dq2; q2)n+1

tn (7.87)

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn

+ q∑k,n≥0

(aq; q2)k(cq2; q2)k+1

[cq

(bq; q2)n+k(dq2; q2)n+k+1

− at(bq; q2)n+k+1(dq2; q2)n+k+2

]tn(tq2)k (7.88)

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn

+ q∑k,n≥0

(aq; q2)k(bq; q2)k+n

(cq2; q2)k+1(dq2; q2)k+n+2

×[cq(1− dq2k+2n+4)− at(1− bq2k+2n+1)

]tn(tq2)k (7.89)

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn

+ q∑k,n≥0

(aq; q2)k(bq; q2)k+n

(cq2; q2)k+1(dq2; q2)k+n+2

[(cq − at) + q2k+2n+1(abt− cdq4)

]tn(tq2)k.

(7.90)

98

We now separate our second term and use some of our previous results to simplify them

before bringing them back together,

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn

+ q(cq − at)∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq2)k∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

tn

+ q2(abt− cdq4)∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq4)k∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

(tq2)n (7.91)

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn

+ q(cq − at)∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq2)k

×

(1

1− t+q(dq2k+3 − btq2k)

1− t

∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

(tq2)n)

+ q2(abt− cdq4)∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq4)k∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

(tq2)n (7.92)

using Theorem 7.1.1. With that simplification we are able to now work on combining

our last two terms,

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn +q(cq − at)

(1− t)F (tq2)

+q2(cq − at)(dq3 − bt)

(1− t)

∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq4)k∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

(tq2)n

+ q2(abt− cdq4)∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq4)k∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

(tq2)n. (7.93)

99

Notice that our last two terms are the same up to their coefficients. Since

q2(cq − at)(dq3 − bt)(1− t)

+ q2(abt− cdq4) =(a− cq)(b− dq3)tq2

(1− t)(7.94)

we have,

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn +q(cq − at)

(1− t)F (tq2).

+(a− cq)(b− dq3)tq2

(1− t)

∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq4)k∞∑n=0

(bq2k+1; q2)n(dq2k+4; q2)n+1

(tq2)n.

(7.95)

Finally, using (7.18) we have,

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn +q(cq − at)

(1− t)F (tq2)

+(a− cq)(b− dq3)tq2

(1− t)

∞∑k=0

(aq, bq; q2)k(cq2, dq2; q2)k+1

(tq4)k∞∑n=0

(bt/dq; q2)n(tq2; q2)n+1

(dq2k+4)n

(7.96)

=∞∑n=0

(bq; q2)n(dq2; q2)n+1

tn +q(cq − at)

(1− t)F (tq2)

+(a− cq)(b− dq3)tq2

(1− t)

∞∑n=0

(bt/dq; q2)n(tq2; q2)n+1

(dq4)nF (tq4+2n). (7.97)

Continuing to return to generalizations of Fine’s identity, with the proper substi-

tutions we obtain

100

Theorem 7.1.13.

(1− t)G(t) = (1− c)(1− d) + (1− c)(d− btq)∞∑k=0

(bq)k(dq)k

(tq)k + (c− atq)G(tq)

+tq(b− d)(aq − c)∞∑n=0

(btq/d)n(tq)n+1

(dq)nG(tq2+n).

(7.98)

where

G(t) =∞∑n=0

(aq)n(bq)n(cq)n(dq)n

(t)n. (7.99)

Some special cases of equation (7.85) and (7.98) are the following:

∞∑k=0

(q; q2)2k

(q2; q2)2k+1

qk(1− q4k+3) =1

1− q(7.100)

and∞∑n=0

qn(1− q2n+3)(1− qn+1)2(1− qn+2)2

=1

(1− q)3. (7.101)

7.2 Combinatorial Interpretations of One of Ramanujan’s Entries

We examine in detail the two infinite sums,

∞∑n=0

(aq; q2)n(bq)n and∞∑n=0

(aq; q)n(bq)n.

101

Such a study yields the following result found in Ramanujan’s Lost Notebook (Entry

9.5.2)

∞∑n=0

(q; q2)nqn =

∞∑n=0

(−1)nq3n2+2n(1 + q2n+1) (7.102)

and a very similar identity

∞∑n=0

(q; q)nqn =

∞∑n=0

(−1)nqn+3

(n+1

2

)(1 + qn+1). (7.103)

We then use our method on the finite sum

2M∑n=0

(aq; q2)n(bq)n (7.104)

and give a finite version of Ramanujan’s result.

7.2.1 Infinite Sums

We define the set of partitions, Hk(m), with the following conditions: 2k+ 1 and

even parts greater than 4k and less than or equal to 4k+2m are the only parts that can

occur, there are m total parts and only parts of size 2k + 1 may be repeated. We also

define the following subsets of Hk(m):

Hk1(m) = {λ ∈ Hk(m) : the part 4k + 2 appears in λ},

Hk2(m) = {λ ∈ Hk(m) : λ does not have a part of size 4k+2 or of size 4k+2m but does

have 2 parts of size 2k + 1},

102

Hk3(m) = {λ ∈ Hk(m) : λ does not have a part of size 4k + 2 but does have a part of

size 4k + 2m and at least one part of size 2k + 1}.

We define ei(λ) as the number of parts equal to i in λ,

Hk =⋃m≥0

Hk(m) and Hki

=⋃m≥0

Hki(m). (7.105)

With our new notation, we have the following identities,

Lemma 7.2.1.

∑λ∈Hk

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= 1 + bq2k+1 +(

1− b

a

) ∑λ∈Hk

1

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+∑λ∈Hk

3

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ| (7.106)

and

∑λ∈Hk

3

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ| = −ab2q6k+5 ∑λ∈Hk+1

(−a)µ(λ)−e2k+3(λ)bµ(λ)q|λ|.

(7.107)

Proof. It should be clear that Hk \ {Hk1 ∪Hk2 ∪H

k3} = {∅, {2k + 1}}. We now turn our

attention to the sets Hk1 and Hk2 . We define a map, φ, on all λ ∈ Hk1 ∪ Hk2 which has

two cases:

If λ ∈ Hk1 : We delete the part of size 4k + 2 and add two parts of size 2k + 1.

103

If λ ∈ Hk2 : We delete two parts of size 2k + 1 and add a part of size 4k + 2.

We note that this simple map transforms a partition in Hk1 into a partition in Hk2

and vice versa. It is also not difficult to see that φ2(λ) = λ. So applying our map, we

see that

∑λ∈Hk

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= 1 + bq2k+1 +∑λ∈Hk

1

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+∑λ∈Hk

2

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+∑λ∈Hk

3

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ| (7.108)

= 1 + bq2k+1 +(

1− b

a

) ∑λ∈Hk

1

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+∑λ∈Hk

3

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|. (7.109)

And so we have proven our first identity. To prove our second identity, we consider

the set Hk3 . Notice that a partition, λ ∈ Hk3 with m parts (m ≥ 2), can be written as a

partition in Hk+1 with m − 2 parts and an additional part of size 6k + 5. This can be

seen in Figure 7.2.1.

We now have the proper tools to show the following theorem:

104

m parts

2k + 1

A(m)

4k + 2m− 1

@@@R

m− 2 parts

A(m)

2 2k + 1

6k + 5

Fig. 7.1. Our map, φ, used in 7.2.1 illustrated above.

105

Theorem 7.2.2.

∞∑n=0

(aq; q2)n(bq)n = bq2(b− a)∞∑n=0

∞∑k=0

(−a)nb2n+kq3n(n+2)+k(2n+1)(aq2n+3; q2)k

+∞∑n=0

(−ab2)nq3n2+2n(1 + bq2n+1). (7.110)

Proof. We see that

∞∑n=0

(aq; q2)n(bq)n =∑λ∈H0

(−a)µ(λ)−e1(λ)bµ(λ)q|λ| (7.111)

using our above notation. We now implement our lemmas,

= 1 + bq +(

1− b

a

) ∑λ∈H0

1

(−a)µ(λ)−e1(λ)bµ(λ)q|λ|

+ ab2q5∑λ∈H1

(−a)µ(λ)−e3(λ)bµ(λ)q|λ|. (7.112)

We continue to iterate our last term using our lemmas

=∞∑n=0

(−ab2)nq3n2+2n

1 + bq2n+1 +(

1− b

a

) ∑λ∈Hn

1

(−a)µ(λ)−e2n+1(λ)bµ(λ)q|λ|

.

(7.113)

106

We now note that

∑λ∈Hn

1

(−a)µ(λ)−e2n+1(λ)bµ(λ)q|λ|

=∞∑k=1

(aq2n+3; q2)k−1(bq2n+1)k(−aq2n+1) (7.114)

= −abq2∞∑k=0

(aq2k+3; q2)kbkqk(2n+1)+4n. (7.115)

Substituting this into (7.113) we get the desired result.

7.2.2 Finite Sums

Using the same method of proof as in Lemma 7.2.1 we see that

− ba

∑λ∈Hk

1(m)

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ| =∑

λ∈Hk2(m+1)

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|.

(7.116)

We can now modify our Lemma to finite sums,

107

Lemma 7.2.3. For M ≥ 2,

∑λ∈Hk(i)0≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= 1 + bq2k+1 +∑

λ∈Hk1(M)

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+(

1− b

a

) ∑λ∈Hk

1(i)

1≤i≤M−1

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

− ab2q6k+5 ∑λ∈Hk+1(i)0≤i≤M−2

(−a)µ(λ)−e2k+3(λ)bµ(λ)q|λ|. (7.117)

Proof. Note that

∑λ∈Hk(i)0≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= 1 + bq2k+1 +∑

λ∈Hk1(i)

1≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+∑

λ∈Hk2(i)

2≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+∑

λ∈Hk3(i)

2≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|. (7.118)

We note that our map illustrated in Figure 7.2.1 yields

108

∑λ∈Hk

3(i)

2≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= −ab2q6k+5 ∑λ∈Hk+1(i)0≤i≤M−2

(−a)µ(λ)−e2k+3(λ)bµ(λ)q|λ|. (7.119)

If we consider this in conjunction with equation (7.116), we see that

∑λ∈Hk(i)0≤i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= 1 + bq2k+1 +∑

λ∈Hk1(i)

i≤M

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

− b

a

∑λ∈Hk

1(i)

1≤i≤M−1

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

− ab2q6k+5 ∑λ∈Hk+1(i)0≤i≤M−2

(−a)µ(λ)−e2k+3(λ)bµ(λ)q|λ|. (7.120)

We now can prove a finite version of Ramanujan’s result.

109

Theorem 7.2.4.

2M∑n=0

(aq; q2)n(bq)n

= (−ab2)M q3M2+2M +

M−1∑n=0

(−ab2)nq3n2+2n(1 + bq2n+1)

+ bq2(b− a)M−1∑k=0

(−a)kb2kq3k2+6k

2(M−k−1)∑n=0

bnqn(2k+1)(aq2k+3; q2)n

−M−1∑n=0

(−ab2)n(ab2M−2n)q2M(2n+1)−n2+2n+1(aq2n+3; q2)2(M−n)−1. (7.121)

Proof.

2M∑n=0

(aq; q2)n(bq)n

=∑

λ∈H0(i)0≤i≤2M

(−a)µ(λ)−e1(λ)bµ(λ)q|λ| (7.122)

= 1 + bq +∑

λ∈H01(2M)

(−a)µ(λ)−e1(λ)bµ(λ)q|λ|

+(

1− b

a

) ∑λ∈H0

1(i)

1≤i≤2M−1

(−a)µ(λ)−e1(λ)bµ(λ)q|λ|

− ab2q5∑

λ∈H1(i)0≤i≤2M−2

(−a)µ(λ)−e3(λ)bµ(λ)q|λ|. (7.123)

If we continue to iterate our last term until it is summing only from 0 to 0 we get that

110

=M−1∑k=0

(−ab2)kq3k2+2k

1 + bq2k+1 +∑

λ∈Hk1(2M−2k)

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+(

1− b

a

) ∑λ∈Hk

1(i)

1≤i≤2(M−k)−1

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

+ (−ab2)M q3M

2+2M ∑λ∈HM (i)

0≤i≤0

(−a)µ(λ)−e2M+1(λ)bµ(λ)q|λ|. (7.124)

We now change our combinatorial interpretation back to q-series. Note that

∑λ∈Hk

1(2M−2k)

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

= (aq2k+3; q2)2M−2k−1(−aq2k+1)(bq2k+1)2M−2k, (7.125)

and

(1− b

a

) ∑λ∈Hk

1(i)

(−a)µ(λ)−e2k+1(λ)bµ(λ)q|λ|

=(

1− b

a

) 2M−2k−1∑i=1

(aq2k+3; q2)i−1(−aq2k+1)(bq2k+1)i (7.126)

= bq4k+2 (b− a)2(M−k−1)∑

i=0(aq2k+3; q2)i(bq

2k+1)i (7.127)

111

and

∑λ∈HM (i)

0≤i≤0

(−a)µ(λ)−e2M+1(λ)bµ(λ)q|λ| = 1. (7.128)

Plugging these three identities into equation (7.124), we get our desired result.

If we consider the special case a, b → 1, we get the following finite version of

Ramanujan:

Corollary 7.2.5. For M > 0,

2M∑n=0

(q; q2)nqn

= (−1)M q3M2+2M +

M−1∑n=0

(−1)nq3n2+2n(1 + q2n+1)

−M−1∑n=0

(−1)nq2M(2n+1)−n2+2n+1(q2n+3; q2)2(M−n)−1. (7.129)

112

Chapter 8

Conclusions

With the motivation of Andrews and Warnaar [9], this thesis has been able to

expand greatly our knowledge of possible conjugate Bailey pairs and their applications.

Up to this point, they have merely played a small part in use with the Bailey transform

while the Bailey pairs have taken most of the fame. This thesis hopes to shine light

on not only introducing new identities, but also to introduce a new side of the Bailey

transform.

Chapters 3, 4 and 5 introduced new conjugate Bailey pairs as well as a number of

applications. It is first noted that in Chapter 3, the power of the main result is greatly

reduced when we choose values of parameters. While this had to be done to ensure a

thorough investigation of the conjugate Bailey pairs listed, other values for our open

parameters will be investigated and our tables increased.

We have expressed many known identities with what appear to be easy and simple

proofs using our new methods in hopes to display the true elegance and simplicity that lie

within our new realm. In addition to our known identities, we focus mainly on generalized

Lambert series, infinite products, Ramanujan-like identities, indefinite quadratic forms

and partitions. In the future, I hope to study in depth each of these aspects individually

and be able to focus more on the applications of each topic. It should also be remarked

that the only values for un and vn chosen were 1/(q2; q2)n and that we only used

113

the symmetric bilateral Bailey transform. In [9] many of the new conjugate Bailey

pairs found were pairs in relation to the asymmetric bilateral Bailey transform. Every

conjugate Bailey pair of this type, as with the previously mentioned two from the paper,

are special cases of our new conjugate Bailey pair. A similar study to Chapters 4 and

5 will be conducted with different values of un and vn and with other versions of the

Bailey transform.

Chapter 6 extends the work done in Chapter 3 to infinite families of pairs. In

previous works, infinite families of pairs and identities were restricted to Bailey pairs. Our

work introduces a new way of expanding our use of the Bailey transform. In subsequent

work I hope to study further both the infinite forms of Bailey pairs and conjugate Bailey

pairs and consider their similarities and the positive aspects of each individually.

In Chapter 7, two of the identities found in our exploration fueled by the Bailey

transform were studied combinatorially. A simple bijection was found in both cases

and with the help of some simple series manipulation, was able to be generalized into

some interesting results. One of the more interesting aspects of the chapter was the

introduction of a general function, An(q), into our simple bijection. With what appeared

to be an incredibly simple step, we were soon able to encompass Eulerian polynomials

and more complicated q-series identities. It is hoped that not only will I be able to

extend our previous results to other new identities for other values of An(q), but also

find other simple bijections in which this function can be inserted.

114

References

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Notes in Mathematics #899, Springer-Verlag, NY (1980), 10–48.

[2] , Hecke Modular Forms and the Kac-Peterson Identities, Transactions of the

American Mathematical Society 283 (1984), no. 2, 451–458.

[3] , The theory of partitions, Cambridge University Press, Cambridge, 1984.

[4] , Ramanujan’s “lost” notebook V, Euler’s partition identity, Adv. Math. 61

(1986), 113–134.

[5] , The Fifth and Seventh Order Mock Theta Functions, Transactions of the

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[6] , Umbral Calculus, Bailey Chains, and Pentagonal Number Theorems, Jour-

nal of Comb. Theory, Series A 91 (2000), 464–475.

[7] G. E. Andrews and B. C. Berndt, Ramanujan’s lost notebook, vol 1, Springer, New

York, 2005.

[8] G. E. Andrews, F. J. Dyson, and D. Hickerson, Partitions and indefinite quadratic

forms, Inventiones Mathematicae, Springer-Verlag 91 (1988), 391–407.

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and Monographs, Num. 27, American Mathematical Soc., Providence, RI, 1988.

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und indefiniten quadratischen Formen, Mathematische Werke, Vandenhoeck und

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die reine und angewandte Mathematik 94 (1883), 203–232.

Vita

Michael Jason Rowell10 McAllister Building

University Park, PA 16802

Born: 13 February 1981 in Fresno, CA

Education:

The Pennsylvania State UniversityUniversity Park, PA 16802PhD Mathematics, August 2007Thesis Advisers: Dr. George Andrews, Dr. Ae Ja Yee

Univeristy of San DiegoSan Diego, CA 92110BA Mathematics, May 2003Adviser: Dr. Jane Friedman

Research Interests:

• Partitions

• Hypergeometric q-series

• Combinatorics

• Number Theory