THE BAILEY TRANSFORM AND CONJUGATE BAILEY PAIRS

Transcription

1 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 Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2007

2 The thesis of Michael J. Rowell was reviewed and approved* by the following: George Andrews Evan Pugh Professor of Mathematics Thesis Co-Adviser Chair of Committee Ae Ja Yee Assistant Professor of Mathematics Thesis Co-Adviser James Sellers Associate Professor of Mathematics Donald Richards Professor of Statistics John Roe Professor of Mathematics Head of Department of Mathematics *Signatures are on file in the Graduate School.

3 iii Abstract This thesis introduces a new generalized conjuagate Bailey pair and infinite families of conjugate Bailey pairs. We discuss the applications of each in conjuction with the Bailey transform. Results range over many different applications: generalized Lambert 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.

4 iv Table of Contents List of Tables vii List of Figures viii Acknowledgments ix Chapter 1. Introduction Chapter 2. Preliminaries Partitions Different Sets of Partitions Hypergeometric q-series The Bailey Transform Umbral Methods Chapter 3. Conjugate Bailey Pairs A General Conjugate Bailey Pair Specific Conjugate Bailey Pairs Chapter 4. A Comprehensive Look into a Conjugate Bailey Pair Known Identities Generalized Lambert Series and Related Identities Infinite Products and Ramanujan-like Identities

5 v 4.4 Weighted sums Partitions Closing Remarks Chapter 5. A General Discussion of Various Conjugate Bailey Pairs Bailey Pairs and the Symmetric Bilateral Bailey Transform Lambert Series, Infinite Products and Ramanujan-like Identities Indefinite Quadratic Forms Partitions Sums of Triangular Numbers Chapter 6. Infinite Families of Conjugate Bailey Pairs A generalization of Watson s 8 φ 7 transformation formula Our Main Result Infinite families of conjugate Bailey pairs and Identities Chapter 7. Combinatorial and Partition-Related Remarks Some Generalizations of Fine s Identity A General Case of a Simple Bijection Finite Sums The Eulerian Number Triangle and the Polylogarithm Function Combining Eulerian Polynomials and our Generalization Another choice for A n (q) Combinatorial Interpretations of One of Ramanujan s Entries

6 7.2.1 Infinite Sums Finite Sums vi Chapter 8. Conclusions References

7 vii List of Tables 3.1 Conjugate Bailey Pairs for when a Conjugate Bailey Pairs for when a Conjugate Bailey Pairs for when a Conjugate Bailey Pairs for when a, b

8 viii List of Figures 2.1 A Young Diagram of the partition (4, 4, 2, 1) Our map, φ, used in illustrated above

9 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.

10 1 Chapter 1 Introduction In 1949 W.N. Bailey introduced the Bailey transform [12], and using this transform was able to give a simple proof of the Rogers-Ramanujan identities; for q < 1, q n2 1 + (1 q)(1 q 2 ) (1 q n ) = 1 (1 q)(1 q 6 ) (1 q 4 )(1 q 9 ) (1.1) and q n2 +n 1 + (1 q)(1 q 2 ) (1 q n ) = 1 (1 q 2 )(1 q 7 ) (1 q 3 )(1 q 8 ) (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,

11 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.

12 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.

13 4 Chapter 2 Preliminaries This section is intended to introduce the reader to the basic definitions and notations 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 = (λ 1,..., λ k ). We refer to each λ i 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, λ 1 the largest part, µ(λ) the number of parts, µ k (λ) the number of parts equal to k,

14 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, λ 1 µ(λ). We define P to be the set of all partitions. Example If we consider the partition λ = (10, 4, 4, 3, 2, 2, 2, 1), we have the following statistics: λ = 28, λ 1 = 10, µ(λ) = 8, µ 4 (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 A Young Diagram of the partition (4, 4, 2, 1).

15 Different Sets of Partitions In later sections, we will use partitions to help us prove identities by showing that the coefficient of q n 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 µ i (λ) as the number of parts of λ which are k congruent to i modulo k. We define the sets of partitions: 1. D k to be all partitions into distinct parts such that each part is congruent to k modulo k. 2. D i k to be all partitions into distinct parts such that each part is congruent to i, k or k i modulo k. 3. D k,i to be all partitions in D i k such that µi k 4. D = k,i to be all partitions in Di k such that µi k (λ) > µk i(λ). k (λ) µk i(λ). k 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

16 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 overpartition 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

17 often helpful to express it as a generating function which we will denote as P(q) ( q < 1), 8 P(q) = p(n)q n. (2.5) We first note that this can easily be expressed as p(n)q n = q λ 1 = 1 q n. (2.6) λ P n=1 We can see this by noting that each term on the right hand side, 1/(1 q n ) = 1 + q n + q 2n +, 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, q λ = (1 + q n ). (2.7) λ D n=1 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 aq k 1 k 1 ) = (1 aq i ) (2.8) i=0 and (2.9) (a) = (a; q) = lim k (a; q) k = (1 aq i ). (2.10) i=0

18 9 We can also combine our infinite products in the following way: (a 1 ; q) k (a 2 ; q) k (a n ; q) k = (a 1, a 2,, a n ; q) k. (2.11) We also define an n+1 φ n basic hypergeometric series as [19, p. 4] n+1 φ n [ ] a1, a n+1 ; q, z = b 1,, b n k=0 (a 1, a n+1 ; q) k (q, b 1,, b n ; q) k z k. (2.12) With our new notation we can now easily express two of our generating functions, λ P q n = 1 (q) (2.13) and q n = ( q). (2.14) λ D of q-series. It is with this notation that we can introduce some of the more classic identities Theorem (Euler). [3, p. 19] z n (q) n = 1 (z). (2.15) Theorem (q-binomial Theorem). [3, p. 17] (a) n (q) n z n = (az) (z). (2.16)

19 10 Theorem (Gauss). [3, p. 23] q n(n+1)/2 = (q2 ; q 2 ) (q; q 2 ). (2.17) Theorem (Jacobi Triple Product). [3, p. 21] ( 1) n a n q n(n 1)/2 = (a, q/a, q; q). (2.18) n= Theorem (Euler s Pentagonal Number). [3, p. 11] (q) = ( 1) k q k(3k+1)/2. (2.19) n= 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 q n(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.

20 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 n β n = α r u n r v n+r, (2.21) r=0 and γ n = δ r u r n v r+n, (2.22) r=n then α n γ n = β n δ n, (2.23) subject to conditions on the four sequences α n, β n, γ n and δ n 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 n β n = α r u n r v n+r, (2.24) r= n and γ n = r n δ r u r n v r+n, (2.25) then α n γ n = β n δ n, (2.26) n=

21 12 with the same convergent conditions on α n, β n, γ n and δ n. In either form, we refer to the series α n and β n as a Bailey pair and the series δ n and γ n 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, (a n a n ). 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].

22 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, 13 z n q n2 = ( zq, z 1 q, q 2 ; q 2 ). (2.27) n= Using the following simple application of (2.16) (with q q 2, z q/a followed by a ), z n q n2 (q 2 ; q 2 = ( zq; q 2 ) (2.28) ) n we can rewrite the triple product identity as z n q n2 = (q 2 ; q 2 z i j q i2 +j 2 ) n= (q i,j 0 2 ; q 2 ) i (q 2 ; q 2. (2.29) ) j 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, 0 for n 0 s(n) = 1 for n < 0. With this function we consider our Umbral mapping,

23 14 Theorem ( 1) s(n) z n q n2 = (q 2 ; q 2 ) n= i,j 0 i j z i j q i2 +j 2 (q 2 ; q 2 ) i (q 2 ; q 2 ) j (q 2 ; q 2 ) i,j 0 i<j z i j q i2 +j 2 (q 2 ; q 2 ) i (q 2 ; q 2 ) j. (2.30) Proof. We invoke the Umbral mapping, z n ( 1) s(n) z n, z n q n2 = (q 2 ; q 2 ( 1) s(i j) z i j q i2 +j 2 ) n= (q i,j 0 2 ; q 2 ) i (q 2 ; q 2 (2.31) ) j = (q 2 ; q 2 ) i,j 0 i j z i j q i2 +j 2 (q 2 ; q 2 ) i (q 2 ; q 2 ) j (q 2 ; q 2 ) i,j 0 i<j z i j q i2 +j 2 (q 2 ; q 2 ) i (q 2 ; q 2 ) j. (2.32) While the generating functions for such identities may not be aesthetically pleasing 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.

24 Theorem Let α n be a double-sided sequence which ensures convergence and k Z with k > 2. Then 15 α n ( 1) n q n(kn+k 2)/2 = n= λ D k 1 α µ k 1 k (λ) µ 1 k (λ)( 1)µ(λ) q λ. (2.33) Proof. Let z zq (k 2)/2 and q q k/2 in the Jacobi triple product. Then we have z n ( 1) n q n(kn+k 2)/2 = (z 1 q, zq k 1, q k ; q k ) (2.34) n= = λ D 1 k z µk 1 (λ) µ 1 k k (λ) ( 1) µ(λ) q λ. (2.35) We now invoke the umbral mapping z n α n. One of the more simple but elegant partition identities that follows from this Umbral map is the following: Corollary ( 1) n q n(3n+1)/2 (1 + q 2n+1 ) = = ( 1) n+s(n) q n(3n+1)/2 (2.36) n= ( 1) µ(λ) q λ λ D µ 2 3 (λ) µ1 3 (λ) 0 λ D µ 2 3 (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.

25 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 (Andrews, Warnaar). The identity, γ n = j n δ j u j n v j+n (3.1) holds for u n = v n = 1/(q 2 ; q 2 ) n with δ n = (q2 ; q 2 ) 2n ( q; q) 2n+1 q n, γ n = q n2 q j2 +j j n (3.2)

26 17 and δ n = (q) 2n q n, γ n = q 2n2 q j(j+1)/2. (3.3) j 2n One of the more striking observations of these new pairs is the existence of a restricted sum in γ n, 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].

27 18 Theorem γ n = δ j (3.4) (q; q) j n j n (fq; q) j+n = (efq/a, a; q) ( n 1 ) n q n(n 1)/2 (fq, fq/a; q) n a (efq n+1 /a, fq/a, b, c; q) j (eq/a; q) j n (efq j n 2 /ab, efq 2 /ac, fq n+1, eq; q) j (q; q) j n ( (1 efq 2j+1 /a) ef ) j q j(j+3)/2 (3.5) bc where ( δ n = (efq2 /abc, efq/a; q) (efq 2 /ab, efq 2 (a, b, c; q) n efq 2 ) n. (3.6) /ac; q) (eq; q) n abc

28 19 Proof. Our proof is an application of Watsons 8 φ 7 transformation: γ n = j n δ j (q; q) j n (fq; q) j+n (3.7) = (efq2 /abc, efq/a; q) (efq 2 /ab, efq 2 /ac; q) = (efq2 /abc, efq/a; q) (efq 2 /ab, efq 2 /ac; q) = (efq2 /abc, efq/a; q) (efq 2 /ab, efq 2 /ac; q) (a, b, c; q) j (eq; q) j n j (q; q) j n (fq; q) j+n ( (a, b, c; q) n (eq; q) n (fq; q) 2n ( (a, b, c; q) n (eq; q) n (fq; q) 2n efq 2n+3 efq 2n+1 lim d 8φ a, efq 2n+3 a, 7 efq 2n+1 a, efq 2n+1 a a ( efq 2 ) n ( abc 3 φ 2 efq 2 ) j (3.8) abc aq n, bq n, cq n efq2 eq n+1, fq2n+1 ; q, abc ) (3.9) efq 2 ) n (efqn+2 /ab, efq n+2 /ac; q) abc (efq 2n+2 /a, efq 2 /abc; q), fqn+1 a, eq a, bqn, cq n, d, eq n+1, fq 2n+1, efqn+2 ab, efqn+2 ac, 0 ; q, efq2+n bcd (3.10) by eq.(iii.17) [19, p. 360] with a = efq 2n+1 /a, b = fq n+1 /a, c = eq/a, d, e = bq n and f = cq n. After some simplification we see that this = (efq/a, a; q) ( n 1 ) n q n(n 1)/2 (fq, fq/a; q) n a (efq n+1 ( /a, fq/a, b, c; q) j (eq/a; q) j n (efq j n 2 /ab, efq 2 /ac, fq n+1 (1 efq 2j+1 /a) ef ) j q j(j+3)/2., eq; q) j (q; q) j n bc (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

29 20 conjugate Bailey pair used by Bailey [12] and Slater [24] in their work, δ n = (y) n (z) n xn y n z n, γ n = (x/y) (x/z) (y) n (z) n x n (x) (x/yz) (x/y) n (x/z) n y n z n (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 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 q 2 followed by e = a, f = 1, a aq, b bq and c cq in Theorem Corollary γ n = j n δ j (q 2 ; q 2 ) j n (q 2 ; q 2 ) j+n (3.13) = (aq; q2 ( ) n (q/a; q 2 1 ) n q n2 (q/a, bq, cq; q 2 ) j ) n a (q j n 3 /b, q 3 /c, aq 3 ; q 2 (1 q 4j+2 ( ) a ) j q j(j+2) ) j bc (3.14)

30 21 where δ n = (q2 /bc, q 2 ; q 2 ( ) (q 3 /b, q 3 /c; q 2 (1 aq)(bq, cq; q2 ) n q 2 ) n ) (1 aq 2n+1. (3.15) ) bc All pairs written in the following tables are special cases of Corollary 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.

31 22 a b c γ n δ n ( 1) n q n2 j n ( 1) n q n2 j n 1 1 q ( 1) n q n2 j n 1 1 q ( 1) n q n2 j n ( 1) n q n2 j n (1+q 2j+1 ) ( 1) j q j2 +2j (1 q 2j+1 ) 2 q j2 +2j (1 q 2j+1 ) (1+q 2j+1 ) (1 q 2j+1 ) ( 1)j q j2 +j (1+q 2j+1 ) +j (1 q 2j+1 ) qj2 ( 1) j q j2 +2j (1+q 2j+1 ) ( q) 2 (q2 ; q 2 ) 2 (q;q 2 ) 2 n (1 q 2n+1 ) q2n (q 2 ;q 4 ) n ( q 2 ; q 2 ) (q 4 ; q 4 ) (1 q 2n+1 ) ( 1)n q 2n (q) 2n (1 q 2n+1 ) qn (q; q) 2n ( q; q 2 ) 2 (q2 ; q 2 ) (1 q 2n+1 ) ( 1)n (q) n (q) 2 ( q2 ; q 2 ) 2 ( q;q 2 ) 2 n (1 q 2n+1 ) q2n 1 1 q ( 1) n q n2 j n qj2 +j ( q; q) 2n (1 q 2n+1 ) ( 1)n q n 1 1 q ( 1) n q n2 j n ( 1)j q j2 +j 1 1 ( 1) n q n2 j n (1+q 2j+1 ) +2j (1 q 2j+1 ) q2j2 1 1 ( 1) n q n2 j n ( 1)j q 2j2 +2j (q) (q; q 2 ) ( q) 2n (1 q 2n+1 ) qn (q;q 2 ) n ( q) (q 2 ; q 2 ) (1 q 2n+1 ) ( 1)n q n2 +2n (q) ( q 2 ; q 2 ) ( q;q 2 ) n +2n (1 q 2n+1 ) qn2 1 q ( 1) n q n2 j 2n qj(j+1)/2 (q2 ;q 2 ) n (1 q 2n+1 ) ( 1)n q n2 +n 1 q ( 1) n q n2 j 2n ( 1) j/2 q j(j+1)/2 (q) ( q; q 2 ) ( q 2 ;q 2 ) n (1 q 2n+1 ) qn2 +n 1 q 2 ( 1) n q n2 j n (1 + q2j+1 ) 2 ( 1) j q 2j2 (q) ( q 2 ; q 2 ) ( q;q 2 ) n+1 (1 q 2n+1 ) qn2 1 ( 1) n q n2 j 3n ( 1) j 3 q j(j+2)/3 (q 2 ; q 2 q ) 2n2 +2n j 0,1 mod 3 (1 q 2n+1 ) Table 3.1. Conjugate Bailey Pairs for when a 1.

32 23 a b c γ n δ n q n2 j n q n2 j n q j2 +2j (1 q 2j+1 ) ( 1) j q j2 +2j (1+q 2j+1 ) ( q) 2 (q2 ; q 2 ) 2 (q;q 2 ) 2 n (1+q 2n+1 ) q2n (q 2 ;q 4 ) n ( q 2 ; q 2 ) (q 4 ; q 4 ) (1+q 2n+1 ) ( 1)n q 2n 1 1 q q n2 j n qj2 +j (q) 2n (1+q 2n+1 ) (q)n 1 1 q q n2 j n ( 1)j q j2 +j ( q,q 2 ;q 2 ) (q; q) 2n (q; q) (1+q 2n+1 ) ( 1)n (q) n q n2 j n 1 1 q q n2 j n 1 1 q q n2 j n (1 q 2j+1 ) q j2 +2j (1+q 2j+1 ) 2 (1 q 2j+1 ) (1+q 2j+1 ) ( 1)j q j2 +j (1 q 2j+1 ) +j (1+q 2j+1 ) qj2 (q) 2 ( q2 ; q 2 ) 2 ( q;q 2 ) 2 n (1+q 2n+1 ) q2n ( q; q) 2n (1+q 2n+1 ) ( 1)n q n (q) ( q) 2n ( q) (1+q 2n+1 ) qn 1 1 q n2 j n ( 1)j q 2j2 +2j (q 2 ;q 2 ) (q;q 2 ) n (q;q 2 ) (1+q 2n+1 ) ( 1)n q n2 +2n 1 1 q n2 j n (1 q 2j+1 ) +2j (1+q 2j+1 ) q2j2 (q 2 ;q 2 ) ( q;q 2 ) n +2n ( q;q 2 ) (1+q 2n+1 ) qn2 1 q q n2 j n (1 q2j+1 )( 1) j q 2j2 +j (q 2 ;q 2 ) n (1+q 2n+1 ) ( 1)n q n2 +n 1 q q n2 (1 j n q2j+1 )q 2j2 +j (q 2 ;q 2 ) ( q2 ;q 2 ) n +n ( q 2 ;q 2 ) (1+q 2n+1 ) qn2 1 q 2 q n2 j n (1 q2j+1 ) 2 ( 1) j q 2j2 (q 2 ;q 2 ) (q;q 2 ) (q;q 2 )n+1 (1+q 2n+1 ) ( 1)n q n2 1 q n2 j 3n ( 1) j 3 j 3 q j(j+2)/3 (q 2 ; q 2 q ) 2n2 +2n j 0,1 mod 3 (1+q 2n+1 ) Table 3.2. Conjugate Bailey Pairs for when a 1.

33 24 a b c γ n δ n q 2n2 j n (1+q 2j+1 ) (1 q 2j+1 ) q2j(j+1) ( q) 2 (q2 ; q 2 ) 2 (q; q2 ) 2 n q2n q 2n2 j n ( 1)j q 2j(j+1) (q 2 ; q 2 ) ( q 2 ; q 2 ) 2 (q2 ; q 4 ) n ( 1) n q 2n 0 1 q q 2n2 j 2n qj(j+1)/2 (q) 2n q n 0 1 q q 2n2 j 2n ( 1) j/2 q j(j+1)/2 (q 2 ; q 2 ) ( q; q 2 ) 2 (q; q) 2n( 1) n q n 0 1 q 2n2 j 3n ( 1) j 3 q j(j+2)/3 j 0,1 mod 3 (q2 ;q 2 ) (q;q 2 ) n (q;q 2 ) ( 1) n q n2 +2n 0 q q 2n2 j 3n ( 1) j+1 3 q j(j+1)/3 (q 2 ; q 2 ) n ( 1) n q n2 +n j 0,1 mod 3 0 q q 2n2 j+1 j+ j 3n ( 1) 3 q j(j+1)/3 (q2 ;q 2 ) ( q 2 ;q 2 ) n q n2 +n j 0,1 mod 3 ( q 2 ;q 2 ) 0 q 2 q 2n2 j n (1 q2j+1 )(1 q 4j+2 )( 1) j q 3j2 ( q) (q 2 ; q 2 ) (q; q 2 ) n+1 ( 1) n q n2 0 q 2n2 j 2n ( 1)j q j(j+1) (q 2 ; q 2 ) q 2n2 +2n Table 3.3. Conjugate Bailey Pairs for when a 0. a b c γ n δ n 1 ( 1) n q n2 (q 2 ;q 2 ) (q;q 2 ) (q; q 2 ) n ( 1) n q n2 1 q n2 (q 2 ;q 2 ) ( q;q 2 ) ( q; q 2 ) n q n2 q (1 q 2n )( 1) n q n2 n (q 2 ; q 2 ) n ( 1) n q n2 n q (1 + q 2n )q n2 n (q 2 ;q 2 ) ( q 2 ;q 2 ) ( q 2 ; q 2 ) n q n2 n q 2 (1 q 2n 1 )(1 q 2n+1 )( 1) n q n2 2n (q 2 ;q 2 ) (q;q 2 ) (q; q 2 ) n+1 ( 1) n q n2 2n q 2 (1 + q 2n 1 )(1 + q 2n+1 )q n2 2n (q 2 ;q 2 ) ( q;q 2 ) ( q; q 2 ) n+1 q n2 2n q 2n2 (q 2 ; q 2 ) q 2n2 Table 3.4. Conjugate Bailey Pairs for when a, b.

34 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 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 1 into Corollary In such a case our theorem reduces to γ n = ( 1) n q n2 j n (bq, cq; q 2 ( ) j (q 3 /b, q 3 /c; q 2 (1 + q 2j+1 ) 1 j q ) j bc) j(j+2) (4.1) where δ n = (q2 /bc, q 2 ; q 2 ) (bq, cq; q 2 ( ) n q 2 ) n (q 3 /b, q 3 /c; q 2 ) (1 q 2n+1. (4.2) ) bc 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: α n = ( 1) n d n q n2 β n = (dq; q2 ) n (q/d; q 2 ) n (q 2 ; q 2 ) 2n (4.3) found in [3, p. 49, ex. 1]. Combining our two pairs we get the following:

35 26 Theorem (q 2 /bc, q 2 ; q 2 ) (q 3 /b, q 3 /c; q 2 ) (bq, cq, dq, q/d; q 2 ( ) n (q 2 ; q 2 ) 2n (1 q 2n+1 ) (bq, cq; q 2 ) j (1 d 2j+1 ( ) = (q j=0 3 /b, q 3 /c; q 2 ) j (1 d) (1 + q2j+1 ) 1 ) j q j(j+2). (4.4) bcd q 2 bc ) n Proof. Using the Bailey tranform we get the following: j=0 (bq, cq; q 2 ) j (1 d 2j+1 ( ) (q 3 /b, q 3 /c; q 2 ) j (1 d) (1 + q2j+1 ) 1 ) j q j(j+2) bcd = d n (bq, cq; q 2 ( ) j ) n= (q 3 /b, q 3 /c; q 2 (1 + q 2j+1 ) 1 j q ) j n j bc) j(j+2) (4.5) = α n γ n (4.6) n= = β n δ n (4.7) = (q2 /bc, q 2 ; q 2 ) (bq, cq, dq, q/d; q 2 ( ) n q 2 ) n (q 3 /b, q 3 /c; q 2 ) (q 2 ; q 2 ) 2n (1 q 2n+1. (4.8) ) bc Notice the Bailey pair above was chosen so that when combined with our pair, the term α n γ n 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.

36 27 Lemma n ( 1) j q j(j+1)/2 = ( 1) n q n(n+1)/2 (4.9) j= n We can now present the next two identities. Theorem We have the following, (q 2 /bc, q 2 ; q 2 ) (q 3 /b, q 3 /c; q 2 ) (bq, cq; q 2 ) n (q) 2n+1 ( q 2 ) n bc (bq, cq; q 2 ( ) ) j = (q j=0 3 /b, q 3 /c; q 2 (1 + q 2j+1 1 j ) q j(2j+3), (4.10) ) j bc and (q 2 /bc, q 2 ; q 2 ) (q 3 /b, q 3 /c; q 2 ) (bq, cq; q 2 ( ) n (q, q, q 2 ; q 2 ) n (1 q 2n+1 ) (bq, cq; q 2 ( ) j = (q j=0 3 /b, q 3 /c; q 2 (1 + q 2j+1 ) 1 ) j ( 1) 2 j q j(j+2)+2 2 j ( j ) ) j bc q 2 bc ) n (4.11) Proof. To prove (4.10) we use our Conjugate Bailey pair defined with a = 1 in Corollary in the Bailey transform with the Bailey pair α n = q 2n2 +n βn = 1 (q; q) 2n (4.12)

37 28 found in [23, H(3)]. Thus, j=0 (bq, cq; q 2 ( ) ) j (q 3 /b, q 3 /c; q 2 (1 + q 2j+1 1 j ) q j(2j+3) ) j bc (bq, cq; q 2 ( ) j = (q j=0 3 /b, q 3 /c; q 2 (1 + q 2j+1 ) 1 ) j j q j(j+2) ( 1) n q n(n+1) (4.13) ) j bc n= j = ( 1) n q n(n+1) (bq, cq; q 2 ( ) j n= (q 3 /b, q 3 /c; q 2 (1 + q 2j+1 ) 1 j q ) j n j bc) j(j+2) (4.14) = α n γ n. (4.15) n= Using our Bailey transform, = β n δ n (4.16) = (q2 /bc, q 2 ; q 2 ) (bq, cq; q 2 ( ) n q 2 ) n (q 3 /b, q 3 /c; q 2. (4.17) ) (q) 2n+1 bc 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)] α 2n = ( 1) n q 6n2 +2n, α2n+1 = 0 β n = 1 (q 2 ; q 4 ) n (q 2 ; q 2 ) 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

38 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, q 2 q]: Corollary (Jacobi). (2j + 1)( 1) j q j(j+1)/2 = (q) 3. (4.19) j=0

39 30 Proof. We consider b = 1, c = q and d = 1 in Theorem Thus, (1 + q) (2j + 1)( 1) j q j(j+1) (4.20) j=0 = (q, q2 ; q 2 ) ( q 3, q 2 ; q 2 ) = = (q) ( q 2 ) (1 q) 2φ 1 ( q, q 2, q, q; q 2 ) n (q 2 ; q 2 ) 2n (1 q 2n+1 ) qn (4.21) ( q, q q 3 ; q2 ; q ) (4.22) (q) ( q 2 ) (1 q) (q2, q 2 ; q 2 ) (q, q 3 ; q 2 ) (4.23) Using III.2 [19, p. 359] = (1 + q)(q 2 ; q 2 ) 3. (4.24) Our final result is obtained from dividing both sides by (1+q) and mapping q q 1/2. 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 ( Slater (27)). ( q, q 5, q 6 ; q 6 ) = (q 2 ; q 2 ) ( q; q 2 ) 2 n q2n(n+1) (q 2 ; q 2 ) 2n (1 q 2n+1 ). (4.25) Proof. We first note that with the triple product identity, ( q, q 5, q 6 ; q 6 ) = = q j(3j+2) (4.26) n= (1 + q 2n+1 )q n(3n+2). (4.27)

40 31 We now consider b, c and d = 1 in Theorem Thus, (1 + q 2n+1 )q n(3n+2) = (q 2 ; q 2 ( q; q 2 ) 2 ) n q2n(n+1) (q 2 ; q 2 ) 2n (1 q 2n+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, q n (q) 2n+1 = (1 ( 1) n ) 2 q(n 1)/2 (q) n. (4.29) Corollary (Slater (38)). ( q, q 7, q 8 ; q 8 ) = (q) q n (q) 2n+1. (4.30) Proof. As with our previous corollary, we again use the triple product identity to see that ( q, q 7, q 8 ; q 8 ) = = q n(4n+3) (4.31) n= (1 + q 2n+1 )q n(4n+3). (4.32)

41 32 We then consider b, c in Theorem 4.11, (1 + q 2n+1 )q n(4n+3) = (q 2 ; q 2 ) = (q2 ; q 2 ) (1 q) = (q2 ; q 2 ) (1 q) q 2n(n+1) (q) 2n+1 (4.33) lim a 2φ 1 lim a ( ) aq, aq q 3 ; q2 ; q2 a 2 (q; q 2 ( ) ) q 2 /a, q 2 /a (q 2 /a 2 ; q 2 2 ) φ 1 q 3 ; q 2 ; q (4.34) (4.35) using III.3 [19, p. 359], = (q) q n (q) 2n+1. (4.36) 4.2 Generalized Lambert Series and Related Identities We define the following as a generalized Lambert series, n= a n q n(n+1)/2 1 bz n. (4.37) It has been shown that Lambert series can be useful in furthering our understanding 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

42 with theta functions. We present a number of generalized Lambert series which follow 33 from Theorems and Corollary ( 1) j q 2j(j+1) j= (1 q 2j+1 ) = (q) ( q; q 2 ) ( q; q) 2n (q; q) 2n+1 q n. (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 is its ability to create weighted identities when allowing d 1. The following identity is an example of one such identity. Corollary (2j + 1)q j(j+1) j= (1 + q 2j+1 ) = (q) ( q) ( q; q 2 ) 2 n (q) 2n (1 + q 2n+1 ) qn. (4.39) Proof. We first note that (2j + 1)q j(j+1) j= (1 q 2j+1 ) = j=0 (1 + q 2j+1 ) (1 q 2j+1 ) (2j + 1)qj(j+1). (4.40)

43 34 We then consider b = 1, c = q and d = 1 in Theorem (1 q)(1 + q 2n+1 ) (1 q 2n+1 (2n + 1)q n(n+1) ) = ( q, q2 ; q 2 ) (q 3, q 2 ; q 2 ) = ( q; q) ( q 2 ; q) (q, q 2, q, q; q 2 ) n (q 2 ; q 2 ) 2n (1 q 2n+1 ) ( q)n (4.41) (q; q 2 ) 2 n ( q; q) 2n (1 q 2n+1 ) ( q)n. (4.42) We divide each side by (1 q) followed by allowing q q to obtain our result. Corollary (2j + 1)q 2j(j+1) j= (1 q 2j+1 ) = (q2 ; q 2 ) (q; q 2 ) (q; q 2 ) 3 n (q 2 ; q 2 ) 2n (1 q 2n+1 ) ( 1)n q n2 +2n. (4.43) Proof. We first note that (2j + 1)q 2j(j+1) j= (1 q 2j+1 ) = j=0 (1 + q 2j+1 ) (1 q 2j+1 ) (2j + 1)q2j(j+1). (4.44) We then consider b and c = d = 1 in Theorem We also consider series in which our sum is only one-sided. Corollary j=0 q j(j+2) 1 + q 2j+1 = (q4 ; q 4 ) ( q 2 ; q 4 ) (q; q 2 ) 3 n ( q; q2 ) n (q 2 ; q 2 ) 2n (1 + q 2n+1 ) ( 1)n q 2n (4.45)

44 35 and (2j + 1) ( 1)j q j(j+2) 1 + q j=0 2j+1 = ( q 2 ; q 2 ) (q 4 ; q 4 ) ( q; q 2 ) 3 n (q; q2 ) n (q 2 ; q 2 ) 2n (1 + q 2n+1 ) q2n (4.46) = (q) 2 ( q2 ; q 2 ) 2 (q 2 ; q 4 ) 2 n (q 2 ; q 2 ) 2n (1 q 2n+1 ) q2n. (4.47) Proof. To prove (4.45), we consider b = 1, c = 1 and d = 1 in Theorem 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 j=0 (1 + q 2j+1 ) (1 q 2j+1 ) qj(j+1) = (q 2 ; q 4 ) n (q; q) 2n+1 q n (4.48) and j=0 (1 + q 2j+1 ) (1 q 2j+1 ) (2j + 1)( 1)j q j(j+1) = (q; q 2 ) 2 n ( q; q) 2n (1 q 2n+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.

45 36 Corollary j=0 q j(2j+3) 1 + q 2j+1 = (q) ( q; q 2 ) 2 ( q 2 ; q 2 ) 2 n (q; q) 2n+1 q n (4.50) and ( 1) j q 3j(j+1) 1 q j= 2j+1 = (q 2 ; q 2 q n ) (q; q 2. (4.51) ) n+1 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 q j(2j+3) (1 q j= 2j+1 ) 2 = ( q) (q 2 ; q 2 ) (q 2 ; q 2 ) 2 n qn (q; q) 2n+1, (4.52) j= q j(j+2) (1 q 2j+1 ) 2 = (q 2 ; q 2 ) 2 ( q; q 2 ) 2 (q 2 ; q 4 ) n q 2n (q 4 ; q 4 ) n (1 q 2n+1 ) (4.53) and j= (2j + 1)( 1) j q j(j+2) (1 q 2j+1 ) 2 = (q 2 ; q 2 ) 2 ( q; q 2 ) 2 (q; q 2 ) 4 n q2n (q 2 ; q 2 ) 2n (1 q 2n+1 ). (4.54)

46 Proof. Our first identity is a consequence of (4.10). We allow b = 1 and c = 1. For the last two, we note that 37 j= q j(j+2) (1 q 2j+1 ) 2 = (1 + q 2n+1 ) (1 q 2n+1 ) 2 qn(n+2). (4.55) and j= (2j + 1)( 1) j q j(j+2) (1 q 2j+1 ) 2 = (1 + q 2n+1 ) (1 q 2n+1 ) 2 (2n + 1)( 1)n q n(n+2). (4.56) We then apply Theorem with b = c = 1 and d = 1 and b = c = d = Infinite Products and Ramanujan-like Identities In many cases, our α n γ n can be reduced to an infinite product using Jacobi s triple product. We are not the first to realize this application of the Bailey Transform. 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 (q 4, q 8, q 8 ; q 8 ) = (q) ( q 2 ; q 2 ( q; q 2 ) 3 ) n qn2 +2n (q 2 ; q 2 ) 2n (1 q 2n+1 ). (4.57) Proof. We then consider c = d = 1 and b in Theorem

47 38 Corollary ( q 2, q 2, q 4 ; q 4 ) + 2q( q 4, q 4, q 4 ; q 4 ) = (q) ( q 2 ; q 2 ) (1 + q 2n+1 )( q; q 2 ) 3 n (1 q 2n+1 )(q 2 ; q 2 ) 2n q n2. (4.58) Proof. We consider c = q 2, d = 1 and b in Theorem (q 2 ; q 2 ) (1 + q 2n+1 )( q; q 2 ) 3 (1 + q)( q; q 2 n ) (1 q 2n+1 )(q 2 ; q 2 q n2 (4.59) ) 2n = q = q = q (1 + q 2n+1 ) 2 q 2n2 (4.60) (1 + q 2n+1 )q 2n2 (4.61) n= [( q 2, q 2, q 4 ; q 4 ) + 2q( q 4, q 4, q 4 ; q 4 ] ) (4.62) where the last step taken was an application of the Jacobi triple product. Corollary ( q 4, q 8, q 8 ; q 8 ) = (q) ( q) ( q; q) 2n (q; q) 2n+1 q n. (4.63) Proof. We consider b = 1 and c = q in (4.10). Corollary ( q 6, q 12, q 12 ; q 12 ) = (q) ( q 2 ; q 2 ) n ( q; q 2 q n. (4.64) ) (q; q) 2n+1

48 39 Proof. We consider b and c = 1 in (4.10). Corollary ( q 2, q 4, q 6 ; q 6 ) + 2q( q 6, q 6, q 6 ; q 6 ) = (q) ( 1; q 2 ) n ( q 3 ; q 2 q n. (4.65) ) (q; q) 2n+1 Proof. We consider b and c = q 2 in (4.10). Corollary (q 4, q 16, q 20 ; q 20 ) = (q) (q 2 ; q 4 ) q n(n+2) (q) 2n+1. (4.66) Proof. We consider b =, c = 1 in Theorem Corollary (q 4 ; q 4 ) 3 = (q) ( q2 ; q 2 ) (q; q 2 ) 2 n ( q; q2 ) n (q 2 ; q 2 ) 2n (1 q 2n+1 ) qn2 +2n. (4.67) Proof. We consider c = 1, d = 1 and b in Theorem 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 ( q k+i, q k i, q 2k ; q 2k ) = (1 + q (k+i)(2n+1) )q kn2 in. (4.68)

49 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 ( q k+i, q k i, q 2k ; q 2k ) = λ D 2k k+i ( 1) µ2k 2k (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 (1 q (k+i)(2n+1) )q kn2 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 ( 1) µ(λ) q λ ( 1) µ(λ) q λ λ D 12,10 = λ D 12,2 = ( 1) n q n (q; q 2. (4.71) ) n+1 Proof. We first allow b = 1 and c = q into (4.11). Our final result can be obtained by applying Theorem

50 41 Corollary ( 1) µ(λ) q λ ( 1) µ(λ) q λ λ D 20,14 = λ D 20,6 +q ( 1) µ(λ) q λ q ( 1) µ(λ) q λ λ D 20,18 = λ D 20,2 (4.72) (4.73) ( 1) n q n(n+1) = (q 2 ; q 4 ) n (1 q 2n+1 ). (4.74) Proof. We first allow b and c = q into (4.11) to get (1 + q 4j+1 + q 8j+3 + q 12j+6 )( 1) j q 2j(5j+2) ( 1) n q n(n+1) = (q j=0 2 ; q 4 ) n (1 q 2n+1 ). (4.75) Our final result can be obtained by applying Theorem Weighted sums As previously stated, Theorem is capable of producing weighted q-series identities. The following section presents more identities of this type. Corollary (2j + 1)(1 + q 2j+1 )q j(2j+1) (q; q 2 ) = n ( q; q) j=0 2n (1 q 2n+1 ) ( 1)n q n2 +n. (4.76) Proof. We consider Theorem with c = q, d = 1 and b.

51 42 Corollary (2j + 1)( 1) j q j(2j+1) j= = (q) ( q; q 2 ) (q; q 2 ) n q n2 +n ( q; q) 2n (1 q 2n+1 ). (4.77) Proof. We note that (2j + 1)( 1) j q j(2j+1) = (2j + 1)( 1) j (1 + q 2j+1 )q j(2j+1). (4.78) j= j=0 We then consider Theorem with c = q, d = 1 and b. Corollary (2j + 1)( 1) j q j(3j+2) = (q 2 ; q 2 (q; q 2 ) 2 ) n q2n(n+1) (q j= 2 ; q 2 ) 2n (1 q 2n+1 ). (4.79) Proof. We note that (2j + 1)( 1) j q j(3j+2) = (2j + 1)( 1) j (1 + q 2j+1 )q j(3j+2). (4.80) j= j=0 We then consider Theorem with d = 1 and b, c.

52 43 Corollary (1 + q 2j+1 )(2j + 1)( 1) j q 2j2 (4.81) j= = (q) ( q 2 ; q 2 ) (1 + q 2n+1 )(q; q 2 ) n (1 q 2n+1 )(q 4 ; q 4 ) n q n2. (4.82) Proof. We note that (1 + q 2j+1 )(2j + 1)( 1) j q 2j2 = (1 + q 2j+1 ) 2 (2j + 1)( 1) j q 2j2. (4.83) j= j= We then consider Theorem with c = q 2, d = 1 and b. 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 Let a(n) denote the number of ways of choosing a not overlined part, λ i, in any overpartition of n such that no overlined part exceeds 2λ i and no other part exceeds 2λ i + 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).

53 44 Proof. Recall Corollary We then see that ( q; q) 2n (q; q) 2n+1 q n = ( q) (q8 ; q 8 ) (q) ( q 4 ; q 4 ). (4.84) It is left to observe that a(n) and b(n) are the coefficients of q n in (4.84). Corollary Let a(n) denote the number of ways of choosing a not overlined part, λ i, in any overpartition of n with all overlined parts even such that no overlined part exceeds 2λ i and no other part exceeds 2λ i + 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 We then see that ( q 2 ; q 2 ) n (q; q) 2n+1 q n = ( q) ( q 2, q 4, q 8, q 10, q 12 ; q 12 ) (q24 ; q 24 ) (q). (4.85) It is left to observe that a(n) and b(n) are the coefficients of q n in (4.85). Corollary Let a(n) denote the number of ways of choosing a part, λ i, not overlined in any overpartition of n with overlined parts even (zero allowed) such that no overlined part exceeds 2λ i 2 and no other part exceeds 2λ i + 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).

54 45 Proof. Recall Corollary We then see that (q 6 ; q 6 ) ( q 2 ) (q) ( q 6 ; q 6 + 2q (q12 ; q 12 ) ( q 2 ) ) (q) ( q 2, q 4 ; q 6 ) ( 1; q 2 ) = n q n. (4.86) (q; q) 2n+1 It is left to observe that a(n), b(n) + 2c(n 1) are the coefficients of q n. Corollary 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) = ( 1) µ(λ) q λ ( 1) µ(λ) q λ. (4.87) λ D 6,5 = λ D 6,1 Proof. We first allow b and c = q into (4.10) to get (1 + q 2j+1 )( 1) j q j(3j+2) (4.88) j=0 = (q; q 2 ) (q; q 2 ) n (q; q) 2n+1 q n (4.89) = (q; q 2 q n ) (q ; q 2 (1 + q 2n+1 + q 2(2n+1) + ) ) n (4.90) ( = (q; q 2 1 q q 2 ) ) (q; q 2 + ) (q 3 ; q 2 + ) (q 5 ; q 2 + ) (4.91) = (q; q 2 ) n q n. (4.92)

55 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 we see that (1 + q 2j+1 )( 1) j q j(3j+2) = ( 1) µ(λ) q λ ( 1) µ(λ) q λ. (4.93) j=0 λ D 6,5 = λ D 6,1 Remark 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], (q; q 2 ) n q n = ( 1) n q 3n2 +2n (1 + q 2n+1 ). (4.94) More will be said about other combinatorial interpretations of the identity in Chapter 7. Corollary 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 We can then see that, 1 (q; q 2 ) (q 8, q 12 ; q 20 ) = q n(n+2) (q) 2n+1. (4.95) It is left to observe that a(n) and b(n) are the coefficients of q n.

56 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, α n = ( 1) n q 2n2 +n, βn = 1 ( q; q) 2n (4.96) and α n = q n, β n = q n (q) 2n (4.97) found in [23, H(2) and F(3)].

57 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 theorems when used in the Symmetric Bailey Transform. Theorem If γ n = j n δ n (q 2 ; q 2 ) j n (q 2 ; q 2 ) j+n (5.1)

58 49 then we have (q; q 2 ) n (q 2 ; q 2 ) 2n δ n = (q; q 2 ) n (q 2 ; q 2 ) 2n ( 1) n q n2 δ n = δ n ( q; q) 2n = δ n (q 2 ; q 2 ) n (q 2 ; q 4 ) n = q n(n 1) δ n (q 2 ; q 2 ) n (q 2 ; q 4 ) n = ( 1) n q n(n+1) δ n (q 2 ; q 2 ) n = ( 1) n δ n (q 4 ; q 4 ) n = q n δ n (q; q) 2n = ( 1) n q n(3n+1)/2 γ n (5.2) n= ( 1) n q n(n+1)/2 γ n (5.3) n= ( 1) n q n(2n+1) γ n (5.4) n= ( 1) n q 2n(3n+1) γ 2n (5.5) n= ( 1) n q 2n(n+1) γ 2n (5.6) n= ( 1) n q n(n+1) γ n (5.7) n= ( 1) n γ n (5.8) n= q n γ n. (5.9) n= 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 q 2 followed by q q. Equations (5.5) and (5.6) follow from C(1) and C(5) with q q 2. Equations (5.7) and (5.8) follow from the fourth and seventh row of the second table on p. 468, respectively, with q q 2. Equation (5.9) follows from F(3) with q q 2.