SP T -FUNCTION AND ITS PROPERTIES

Transcription

1 SP T -FUNCTION AND ITS PROPERTIES a thesis submitted to the department of mathematics and the graduate school of engineering and science of bilkent university in partial fulfillment of the requirements for the degree of master of science By Oğuz Gezmiş July, 204

2 I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Asst. Prof. Dr. Hamza Yeşilyurt (Advisor I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Asst. Prof. Dr. Ahmet Muhtar Güloğlu I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Asst. Prof. Dr. Çetin Ürtiş Approved for the Graduate School of Engineering and Science: Prof. Dr. Levent Onural Director of the Graduate School ii

3 ABSTRACT SP T -FUNCTION AND ITS PROPERTIES Oğuz Gezmiş M.S. in Mathematics Supervisor: Asst. Prof. Dr. Hamza Yeşilyurt July, 204 In this thesis, we survey some properties of the spt-function. We start with providing some background information for q-series and the partition function p(n. Then we define the spt-function and study its generating function. Our aim is to prove parity result for the spt-function. We also obtain a congruence relation between spt-function and a certain mock theta function. Keywords: Partition Theory, q-series, spt-function. iii

4 ÖZET SP T -FONKSİYONU VE ÖZELLİKLERİ Oğuz Gezmiş Matematik, Yüksek Lisans Tez Yöneticisi: Yrd. Doç. Dr. Hamza Yeşilyurt Temmuz, 204 Bu tezde, spt-fonksiyonu ve onun bazı özellikleri üzerine çalışacağız. Öncelikle q-serileri ve bölüşüm fonksiyonuna değineceğiz. Daha sonra spt-fonksiyonu ve bu fonksiyonun üretici fonksiyonunu inceleyeceğiz. Amacımız spt-fonksiyonunun bir parite özelliğini ispatlamaktır. Ayrıca spt-fonksiyonunun bir mock teta fonksiyonu ile kongruans ilişkisini de ispatlayacağız. Anahtar sözcükler: Bölüşüm Teorisi, spt-fonksiyonu, q-serileri. iv

5 Acknowledgement I would like to express my sincere gratitude to my supervisor Asst. Prof. Hamza Yeşilyurt for his excellent guidance, valuable suggestions, encouragement and patience. I would also like to thank Asst. Prof. Ahmet Muhtar Güloğlu and Asst. Prof. Çetin Ürtiş for a careful reading of this thesis. I would like to thank to my family for their great support and increasing my motivation. I especially thank to my mother who calls me every day and encourages me throughout my university career. I am so grateful to İrem who has been always with me whenever I experience difficulties during my master degree studies. The thesis is supported financially by TÜBİTAK through the graduate fellowship program, namely TÜBİTAK-BİDEB 220-Yurt İçi Yüksek Lisans Burs Programı. I thank to TÜBİTAK for their support. Finally, I would like to thank my friends Abdullah, Alperen, Bekir, Burak and Recep from the department for their support and encouragement. v

6 Contents Introduction 2 Introduction to Partition Theory 3 2. q-series and The Partition Function p(n The spt-function and its Congruence Properties The Rank of a Partition Vector Partitions 2 3. Introduction to Vector Partitions Properties of the Sum N V (m, n Vector Partitions over Subset S of V Congruence Properties of N S (m, n Mock Theta Functions and Parity of spt(n Self-Conjugate Vector Partitions Introduction to Mock Theta Functions vi

7 CONTENTS 4.3 Parity of spt(n Relation Between spt(n and Mock Theta Functions

8 spt-function and Its Properties Oğuz Gezmiş July 4, 204

9 Chapter Introduction We define a partition λ of a natural number n as a set of positive integers such that sum of these positive integers is equal to n. A partition λ is denoted by λ λ + λ 2 + λ n, λ i for i, 2,, n is called a part of the partition λ. As an elementary illustration, 3 has 3 different partitions: 3, 2+, ++. In 2008, George Andrews introduced a partition function which he called the smallest part function denoted by spt-function. He defined spt(n in [2] by the sum of the total appearances of the smallest part in each partition of n. Throughout this thesis, we survey properties of the spt-function. The thesis is organized as follows. Chapter 2 begins with introducing q- series and presenting properties of the partition function p(n. We prove some identities which are helpful for further results. We focus on the smallest part function denoted by spt-function and state its congruence properties. At the end of this chapter, a related partition statistics called the rank of a partition is also introduced to give some properties of the generating function of the spt-function. In Chapter 3, we define and survey some properties of vector partitions. We

10 CHAPTER. INTRODUCTION 2 study some concepts of vector partitions in order to give the relation of these with the spt-function and its generating function. In Chapter 4, we begin with self-conjugate partitions and present parity of the generating function for self-conjugate partitions. We need self-conjugate partitions to obtain parity results for the spt-function. Lastly, we obtain a congruence relation modulo 4 between spt-function and a certain mock theta function.

11 Chapter 2 Introduction to Partition Theory In this chapter, we are concerned with q-series and partition theory. At the begining, we present some definitions and emphasize some important techniques that are used throughout this thesis such as taking limits of q-series. Chapter 2 continues with the smallest part function and properties of this function. We finish this chapter by defining the rank of a partition and stating its congruence properties. The content of this chapter is mainly taken from [], [9], [0] and []. 2. q-series and The Partition Function p(n After mentioning the definition of a partition in the previous chapter, we restrict our attention to q-series and the partition function p(n which counts number of partitions of any nonnegative integer n. We start with definition of q-series. We assume that q is a complex number with q <. We adopt the usual notation. Definition 2... Define (a 0 : (a; q 0 :, (a n (a; q n : 3 n k0 ( aq k n,

12 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 4 (a : (a; q : For < n < 0, we define (a; q n by k0 ( aq k. (a; q n : (a; q (aq n ; q. We continue by definition of the partition function p(n. Definition For any positive integer n, we define the partition f unction p(n as the number of unrestricted partitions of n. For convenience, we define p(0. The generating function for p(n is given by Theorem [] p(nq j i ( q i (q; q. Proof. [] Observe the following geometric series Therefore, by arguing informally, q k i0 q ik. i0 q ( + q + i q2 + ( + q 2 + q 4 + ( + q i + q 2i + q a +2a 2 +3a 3 +. a 0 a 2 0 a 3 0 We see that power of q gives us a partition λ such that a many a 2 many a 3 many {}}{{}}{{}}{ λ

13 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 5 generates all par- Hence the coefficients of q n in the q-series expansion of (q;q titions of the natural number n. Next, we give a formal proof. For this aim, let h n be a sequence such that h n n ( q i n. i Since h n contains finitely many absolutely convergent series. Therefore h n is itself absolutely convergent. For simplicity, let q be a real number such that 0 < q <. Thus, n j0 p(jq j n ( q i i ( q i <. Hence, n j0 p(jqj is a bounded increasing sequence in R. Therefore the series i j0 p(jq j must converge by Monotone Convergence Theorem. On the other hand, j0 p(jq j n ( q i i ( q i as n. i Thus, as desired. p(nq j ( q i (q; q An immediate corollary of Theorem 2..3 can be given by i Corollary [] Let A be a non-empty subset of N and let p A (n be the number of partitions of n such that every part of the partition of n is an element of A. Then, n p A (nq n n A q n. Observe that the generating function for the number of partitions of n with

14 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 6 distinct parts, namely p d (n, is given by p d (nq n ( q; q ( + q( + q 2 ( + q 3 ( + q k. We are now ready to state famous theorem of Euler. Proving this theorem helps us to understand importance of series manipulation which is one of the basic tools for proving any result about q-series. Theorem (Euler[, p.4] The number of partitions of positive integer n into distinct parts equals the number of partitions of n into odd parts, denoted by p 0 (n. Proof. [] As we have just stated above, the generating function for p d (n is given by p d (nq n ( q; q ( + q( + q 2 ( + q 3 ( + q k. (2. Multiply and divide (2. by (q; q, we have p d (nq n ( + q( q( + q2 ( q 2...( q k ( + q k ( q( q 2 ( q 2k ( q 2k ( q 2k+ ( q( q 3 ( q 5 ( q 2k p 0 (nq n. This section continues by congruence properties of p(n. following definition. We begin with the Definition Let f(q a n q n and g(q b n q n be power series in q. If a n b n (mod m for every integer n, we say that f(q g(q (mod m.

15 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 7 The most celebrated congruence relation in the theory of q-series are Ramanujan s relation for the partition function p(n. These congruences stated in [9], [20] are p(5n (mod 5, (2.2 p(7n (mod 7, (2.3 p(n (mod. (2.4 (2.2 and (2.3 were proved by Ramanujan in [9]. Later, Ramanujan mentioned that he also found a proof for (2.4 in [2]. In addition, Winquist gave the most elementary proof for (2.4 in [25]. In their most general Ramanujan s congruence relations can be stated by; For α ; p(5 α n + δ 5,α 0 (mod 5 α, (2.5 p(7 α n + δ 7,α 0 (mod 7 (α+2/2, (2.6 p( α n + δ,α 0 (mod α (2.7 where δ t,α is the reciprocal modulo t α of 24. (2.5 and (2.6 were proved by G.N. Watson [24] in 938 and (2.7 was proved by A.O.L. Atkin [9] in 967. We emphasize a fundamental theorem about q-series. Theorem (q-analogue of the binomial theorem[, p.8] For q, z <, (a n z n (q n (az (z. It is important to prove the following corollary of Theorem 2..7 in order to see usefulness of taking limit in q-series.

16 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 8 Corollary [, p.9] For q <, z n z < (2.8 (q n (z and ( z n q n(n 2 (z z <. (2.9 (q n Proof. [] We can prove (2.8 by letting a 0 in Theorem For (2.9, write a/b instead of a and write bz instead of z in Theorem Then we have for bz <, Letting b 0, we get (a/b n (bz n lim (a/b nb n lim ( a ( b 0 b 0 b (q n aq b (az (bz. (2.0 ( aqn Next, we put a and let b 0 in (2.0 we have ( z n q n(n /2 (z. (q n b b n ( a n q n(n 2. Since we are taking limits of an infinite series, we should justify our calculations. In other words, we have to show that (a/b n (bz n (q n converges uniformly when b 0. For this aim, choose a real number M such that q < M <. Let ɛ be given such that 0 < 2ɛ < M and b ɛ be fixed. Let N 0 be the unique positive integer such that b + a M k 2ɛ for 0 k N 0

17 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 9 and Thus, (a/b n b n (q n b + a M N 0 < 2ɛ. ( a b ( aq b ( aqn b (q n (b a (b aq (b aq n (q n ( b + a ( b + a M ( b + a M n ( M n (2ɛn N 0 (ɛ + a N 0 ( M n ( N0 ( n ɛ + a 2ɛ. 2ɛ M b n Since ɛ > 0 such that 2ɛ < M. Therefore, ( n 2ɛ <. M We can conclude that (a/b n (bz n absolutely converges by Weierstrass M-Test when b 0. In other words, we are allowed to take limits under summation sign in q-series. (q n Next we introduce a famous theorem called Jacobi Triple Product Identity. It was introduced by German mathematician Jacobi [8] in 829. For brevity, we use q-series notation of this theorem. Theorem (Jacobi Triple Product Identıty[8] For z 0 and q <, n z n q n2 ( zq; q 2 ( q/z; q 2 (q 2 ; q 2. (2.

18 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 0 A fundamental theorem which is an immediate consequence of Theorem 2..9 is given by Theorem (Jacobi s Identity[] ( n (2n + q n(n+/2 (q; q 3. Proof. [] By replacing z by z 2 q in (2., we find that n z 2n q n2 +n ( z 2 q 2 ; q 2 ( /z 2 ; q 2 (q 2 ; q 2. (2.2 By dividing both sides of (2.2 by + /z 2 we have n z2n+ q n2 +n z + /z ( z 2 q 2 ; q 2 ( q 2 /z 2 ; q 2 (q 2 ; q 2. (2.3 On the other hand it is easy to observe that ( n q n2 +n 0. (2.4 n We take limit of both sides of (2.3 as z i to deduce that lim z i n z2n+ q n2 +n z + /z n ( n (2n + q n2 +n 2 (by (2.4 and L Hospital s Rule lim z i ( z 2 q 2 ; q 2 ( q 2 /z 2 ; q 2 (q 2 ; q 2 (q 2 ; q 2 (q 2 ; q 2 (q 2 ; q 2. (2.5 Hence, we conclude from (2.5 that 2 n ( n (2n + q n(n+ (q 2 ; q 2 3. (2.6

19 CHAPTER 2. INTRODUCTION TO PARTITION THEORY If we divide the infinite sum in (2.6 into two parts, it is easy to see that n ( n (2n + q n(n+ ( n (2n + q n(n+. (2.7 Finally, replacing q 2 by q in (2.6 and using (2.7, we have ( n (2n + q n(n+/2 (q; q 3. From now on, we are concerned with smallest part function and its properties. 2.2 The spt-function and its Congruence Properties In this section, we begin with recalling definition of the smallest part function. Our main goal is to emphasize interesting congruence properties and the generating function of the smallest part function. Let n be any nonnegative integer. The smallest part function is defined by the sum of the total appearances of the smallest part in each partition of n. T he smallest part f unction is called as spt-function. We present an example to clarify the definition of spt(n. Examples The partitions of 5 are 5, 4+, 3+2, 3++, 2+++, 2+2+, The smallest parts of these partitions appear,,, 2, 3,, 5 times respectively. Therefore, sum of appearances of smallest parts is 4. That is, spt(5 4. The generating function of spt(n is given by [2] spt(nq n q n ( q n 2 (q n+ ; q. (2.8

20 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 2 Note that, we can observe the generating function of the spt-function by investigating an arbitray term of the right hand side of (2.8. First we know that (m + q nm m0 If we put (2.9 into (2.8, we have the following; ( q n 2. (2.9 q n ( q n 2 (q n+ ; q m mq nm kn+ ( q k. (2.20 Thus an arbitrary term of (2.20 is as follows: m many {}}{ mq n + + n (+q n+ +q (n++(n+ + (+q n+2 +q (n+2+(n+2 + (2.2 A typical term in (2.2 is m many {}}{ mq n + + n +(n++(n++(n+2. It is easy to observe that coefficients of the second sum in the right hand side of (2.20 give how many times the smallest part n appears in a partition which is generated by (2.2. On the other hand, the product in (2.20 provides remaining parts of the partition which contain parts bigger than n. Therefore, right hand side of (2.8 gives generating function of the spt(n for integer n. A fundamental theorem about spt-function which was proved by George Andrews in [2] is given by Theorem [2] n spt(nq n (q; q nq n q + n (q, q ( n q n(3n+ 2 ( + q n ( q n 2. (2.22 Before proving this theorem we state two lemmas and a theorem in order to use them in the proof of Theorem

21 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 3 Lemma [2] Let f(z be any function which is at least twice differentiable at z. Then, [ ] d 2 2 dz ( z( 2 z f(z z f(. Next lemma is concerned with differentiation of q-series. Lemma [2] For q <, [ ] d 2 2 dz (zq; q (z q; q 2 z (q; q 2 nq n q n. Proof. [2] Replace z by zq /2 and q by q /2 in Theorem We have ( z n q n(n /2 (z; q (q/z; q (q; q. (2.23 n Note that the formula for finite sum of geometric series is given by n k0 r k rn r. (2.24 Therefore, [ d 2 ] 2 dz (zq; q (z q; q 2 z [ d 2 ] n ( zn q n(n /2 (by (2.23 2(q; q dz 2 z z [ d 2 ( ] z ( z n q n(n+/2 2n+ 2(q; q dz 2 z z [ ] d 2 2n ( n q n(n+/2 z n+j (by (2.24 2(q; q dz 2 2(q; q ( n q n(n+/2 2n j0 j0 z ( n + j( n + j. (2.25

22 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 4 It is easy to see that, 2n j0 Thus 2 ( n + j( n + j [ d 2 dz 2 (zq; q (z q; q ] z n j n j 2 n j n 2(q; q j n j n j 2 n(n + (2n + 3. (2.26 ( n q n(n+/2 n(n + (2n + 3 (by (2.25 and (2.26 q ( n q n(n+ n(n + 2 (2n + 3(q; q 2 q d ( n (2n + q n(n+/2 3(q; q dq By using logarithmic differentiation, we can see that d dq (q; q3 3(q; q 3 q d 3(q; q dq (q; q3 (by Theorem nq n q n. Therefore, 2 [ d 2 dz 2 (zq; q (z q; q ] z q d 3(q; q dq (q; q3 (q; q 2 nq n q. n Before stating our next theorem, we need a definition. Definition T he Heine s series or the basic hypergeometric series is denoted by mφ n ( a, a 2,, a m ; q, z b, b 2,, b n : k0 (a k (a 2 k (a m k z k (b k (b 2 k (b n k (q k.

23 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 5 where z <, q < and b i q n for any nonnegative integer n. Next we state an important theorem about hypergeometric series. Theorem (q-analog of Whipple s Theorem[6] a2 q2+n bcde ( a, q a, q a, b, c, d, e, q N ; q, 8φ 7 a, a, aq, aq, aq, aq, b c d e aqn+ (aq; q N( aq de ; q N ( aq d ; q N( aq e ; q N 4φ 3 ( aq bc, d, e, q N ; q, q aq, aq, deq N b c a. Proof. See [6], page 42. Now we are ready to give the proof of Theorem Proof. [2](Proof of Theorem Let f(z (z; q n (z ; q n q n ( z( z (q; q n. We know that (q; q n (q; q (q n+ ; q (2.27

24 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 6 for any n. Therefore [ ( d 2 ] (z; q n (z ; q n q n 2 dz 2 (q; q n z [ ( ] d 2 + ( z( z (z; q n (z ; q n q n 2 dz 2 (q; q n ( z( z [ ( d 2 ] 2 dz ( z( (z; q n (z ; q n q n 2 z (q; q n ( z( z z (q; q n q n (by Lemma ( q n (q; q n q n ( q n 2 (q; q q n ( q n 2 (q n+ ; q (by (2.27. z (2.28 From (2.28 we see that [ ( d 2 2(q; q dz 2 ] (z; q n (z ; q n q n (q; q n z q n ( q n 2 (q n+ ; q. (2.29 On the other hand, by setting d e z, and then letting b, c, N and a in Theorem 2.2.6, we have (z; q n (z ; q n q n Note that (q; q n (zq; q (z q; q (q; q 2 ( + ( n q n(3n+ 2 ( + q n (z; q n (z ; q n. (q/z; q n (zq; q n (2.30 n spt(nq n q n (by (2.8 ( q n 2 (q n+ ; q [ ( d 2 ] (z; q n (z ; q n q n 2(q; q dz 2 (q; q n z (by (2.29. (2.3

25 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 7 Therefore, n 2(q; q spt(nq n [ d 2 dz 2 (zq; q (z q; q (q; q 2 (by (2.30 and (2.3 nq n (q, q q + n (q, q (by Lemma and ( + ] ( n q n(3n+ 2 ( + q n (z n (z n (q/z n (zq n ( n q n(3n+ 2 ( + q n ( q n 2 z In [2], Andrews also gave congruence properties of spt-function which are very similar to congruence properties of p(n. Theorem [2] spt(5n (mod 5, spt(7n (mod 7, spt(3n (mod 3. Proof. See [2], page 8. Examples The partitions of 4 can be stated as follows; 4, 3+, 2+2, 2++, +++. Therefore, spt(4 0 0 (mod 5. Next, we continue by defining rank of a partition. The concept of our next section was given by F. J. Dyson [2].

26 CHAPTER 2. INTRODUCTION TO PARTITION THEORY The Rank of a Partition Our aim in this section is to introduce congruence properties and generating function of the rank of a partition. We begin with presenting a definition. Definition Let λ be a partition. T he rank of a partition λ is defined as largest part of λ minus number of parts of λ. Examples A partition of 0 has rank 5 4. Let N(m, n denote the number of partitions of n with rank m. We give the generating function of N(m, n by [7] m N(m, nz m q n q n2 (zq n (z q n. (2.32 Another description for the left hand side of (2.32 can be stated as follows. Proposition [3] m N(m, nz m q n (q n ( n q n(3n+/2 ( z( z ( zq n ( z q n. Proof. [3] We know that z re iθ rcos(θ + risin(θ where r is modulus of z and angle θ is argument of z. Therefore cos(θ z + z 2. (2.33 Watson [23] gave following equation by using Theorem so that ( n ( + q n (2 2cos(θq n(3n+/2 + 2q n cos(θ + q 2n [ ] ( q r q n2 + n m (. 2qm cos(θ + q 2m r (2.34

27 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 9 Hence ( n ( + q n (2 2cos(θq n(3n+/2 + 2q n cos(θ + q 2n ( n ( + q n (2 z z q n(3n+/2 + (by (2.33 q n (z + z + q 2n ( n ( + q n ( z( z q n(3n+/2 + ( zq n ( z q n [ ] ( q r q + n2 n r m ( (by (2.34 zqm ( z q m ] q (q [ n2 + (zq n (z q n (q N(m, nz m q n (by (2.32. m (2.35 Therefore, we obtain from (2.35 and after some elementary manipulations that m ( N(m, nz m q n + (q (q n ( n ( + q n ( z( z q n(3n+/2 ( zq n ( z q n ( n q n(3n+/2 ( z( z. ( zq n ( z q n After Dyson defined the rank of a partition, he left following congruence properties of the rank of a partition as conjectures in [2]. These were proven by Atkin and Swinnerton-Dyer in [0]. Their methods for proving Dyson s conjectures are heavily dependent on the theory of modular forms. Theorem [0] Let N(m, t, n denote the number of partitions of n with rank congruent to m modulo t. Then we have, N(k, 5, 5n + 4 N(k, 7, 7n + 5 p(5n p(7n for 0 k 4, for 0 k 6.

28 CHAPTER 2. INTRODUCTION TO PARTITION THEORY 20 Let us consider partitions of 4 and rank of these partitions as follows: λ 4 rank of λ 4 3 λ rank of λ λ rank of λ λ rank of λ λ rank of λ (2.36 As an example for n 0 in Theorem we see from (2.36 that N(0, 5, 4 N(, 5, 4 N(4, 5, 4 p(4 5.

29 Chapter 3 Vector Partitions In this chapter, we introduce vector partitions. At the beginning, we give construction of vector partitions and clarify our subject by giving an example. Later, we work on a sum N V (m, n which is defined in a set V and state its congruence properties. The final topic in Chapter 3 is vector partitions over a subset S of V and a sum N S (m, n defined in S. The content of this chapter is mainly based on [7] and [4]. 3. Introduction to Vector Partitions Let π be a partition. We denote number of parts of π as #(π and sum of parts of π as σ(π. Note that #( σ( 0 for the empty partition of 0. Let D be the set of partitions into distinct parts and let P be the set of partitions with unrestricted parts. A set V is defined so that V : D P P. 2

30 CHAPTER 3. VECTOR PARTITIONS 22 In other words, { V (π, π 2, π 3 π is a partition with distinct parts } π 2 and π 3 are partitions with unrestricted parts. We call elements of V as vector partitions. For π (π, π 2, π 3 in V, define sum of parts s, weight ω, and crank r by, s(π : σ(π + σ(π 2 + σ(π 3, ω(π : ( #(π, r(π : #(π 2 #(π 3. If s(π n then we call π as a vector partition of n. Examples 3... Let π (4 + 5, , Then s(π 22, ω(π ( 2 and r(π Note that, π is a vector partition of 22. Next, we restrict our attention to introducing a sum N V (m, n which counts weight of vector partitions of n with crank m. 3.2 Properties of the Sum N V (m, n In this section, our aim is to present basic properties of the sum N V (m, n and its congruence properties. We begin with some notations and definitions. We denote the number of vector partitions of n with crank m counted according to weight ω as N V (m, n such that N V (m, n : π V s(πn r(πm ω(π.

31 CHAPTER 3. VECTOR PARTITIONS 23 Let N V (k, t, n be the number of vector partitions of n with crank congruent to k modulo t counted according to weight ω. Thus, N V (k, t, n : m N V (mt + k, n : π V s(πn r(π k (mod t ω(π. Weight of a vector partition π (π, π 2, π 3 only depends on the number of parts of π. Therefore transforming partitions π 2 and π 3 does not effect N V (m, n. Hence, N V (m, n N V ( m, n. (3. Since t m m (mod t. From (3., we can readily say that, N V (t m, t, n N V (m, t, n. The generating function for N V (m, n is given by [4] m N V (m, nz m q n ( q n ( zq n ( z q n (q; q (zq; q (z q; q. (3.2 Note that, we can observe the generating function of N V (m, n by investigating an arbitray exponent in the product (3.2 given by P art P art 2 {}}{{}}{ ( q ( q 2 ( q 3... ( + zq + z 2 q ( + z q + z 2 q (3.3 Let π (π, π 2, π 3. Part- in (3.3 generates partitions into distinct parts with minus sign for partitions containing odd number of parts and plus sign for partitions containing even number of parts. Choose a partition π π+π π n among partitions generated by Part-. Then, the coefficient of q π +π2 + +πn in Part- gives ω(π. On the other hand, Part-2 gives difference of number of parts of two partitions. That is, if we choose π 2 π2 + π π2 k and π 3 π3 + π π3 m among

32 CHAPTER 3. VECTOR PARTITIONS 24 partitions generated by (zq;q and (z q;q respectively, we see that power of z in the term from Part-2 z k m q π 2 +π πk 2 +π 3 +π πm 3 gives the difference of number of parts of π 2 and π 3 which is defined as r(π. by Another description for the generating function (3.2 of N V (m, n can be given Lemma [4] m N V (m, nz m q n (q n ( n q n(n+/2 ( z( z. ( zq n ( z q n Proof. [4] In order to prove this lemma, use the limiting form of Jackson s theorem ([22]. This theorem can be stated as zq a a 2 a 3 ( z, q z, q z, a, a 2, a 3 ; q, 6φ 5 z, z, zq a, zq a 2, zq a 3 ( zq n ( za a 2 q n ( za a 3 q n ( za 2 a 3 q n ( za q n ( za 2 q n ( za 3 q n ( za a 2 a 3 q n. It is easy to observe that lim z and (3.4 (z n ( z n lim z ( z( qz ( q n z ( z( q z ( q n z lim z ( + z 2 (3.5 (a 3 n lim a 3 a n 3 ( a 3( qa 3 ( q n a 3 a n 3 ( n q n(n /2. (3.6

33 CHAPTER 3. VECTOR PARTITIONS 25 By setting a z, a 2 z and letting z, a 3 in (3.4 we find that (z n (q z n ( q ( z n (a n (a 2 n (a 3 n ( z n ( ( ( zq zq z n a a n 2 n 2( n (q n ( q n (z n (z n q n(n+/2 + Therefore n zq a a 2 a 3 ( zq a 3 (q n n ( n (zq n (z q n (q n (by (3.5 and (3.6 ( z( z ( n q n(n+/2 ( + q n ( zq n ( z q n (q; q 2. (by (3.4 (zq; q (z q; q (3.7 m N V (m, nz m q n ( q n ( zq n ( z q n (by (3.2 On the other hand, (q; q (zq; q (z q; q [ ] ( z( z ( n q n(n+/2 ( + q n + (q; q ( zq n ( z q n (by (3.7. (3.8 + ( z( z ( n q n(n+/2 ( + q n ( zq n ( z q n Hence, we have m [ N V (m, nz m q n + (q; q (by (3.8 (q n n ( z( z ( n q n(n+/2 ( zq n ( z q n (3.9 ] ( z( z ( n q n(n+/2 ( + q n ( zq n ( z q n ( n q n(n+/2 ( z( z ( zq n ( z q n (by (3.9..

34 CHAPTER 3. VECTOR PARTITIONS 26 Next we restrict our attention to congruence properties of N V (m, n. Garvan proved following lemma which is necessary for stating congruence properties of N V (m, n. Lemma [4] N V (0, t, tn+δ t N V (, t, tn+δ t N V (t, t, tn+δ t p(tn + δ t t (3.0 for t 5, 7, where δ t is reciprocal of 24 modulo t. For t prime, (3.0 is equivalent to the coefficient of q tn+δt in ( q n ( ζ t q n ( ζ t q n being zero, where ζ t exp(2πi/t. Proof. See [4], page 54. Garvan proved his main result in [4] by using Lemma and some q-series identities. Theorem [4] N V (k, 5, 5n + 4 p(5n for 0 k 4, N V (k, 7, 7n + 5 N V (k,, n + 6 p(7n p(n + 6 for 0 k 6, for 0 k 0. Let us clarify Theorem by giving an example. Examples We can write vector partitions of 5 with crank congruent to

35 CHAPTER 3. VECTOR PARTITIONS 27 modulo 7 as follows: π (, 2 + 2, ω(π ( 0 π 2 (, + +, + ω(π 2 ( 0 π 3 (, 5, ω(π 3 ( 0 π 4 (, +, 3 ω(π 4 ( 0 π 5 (, 3 +, ω(π 5 ( 0 π 6 (, 2 +, 2 ω(π ( 0 π 7 (, 2 +, ω(π 7 ( π 8 (, 4, ω(π 8 ( π 9 (, +, 2 ω(π 9 ( π 0 (2, +, ω(π 0 ( π (2, 3, ω(π ( π 2 (3, 2, ω(π 2 ( π 3 (4,, ω(π 3 ( π 4 (2 +, 2, ω(π 4 ( 2 π 5 (3 +,, ω(π 5 ( 2. (3. We see from (3. that N V (, 7, 5 π V s(π5 r(π (mod 7 ω(π p(5 7. Another important property of N V (m, n can be stated as follows. Theorem [5] N V (m, n 0 for all (m, n (0,. Proof. See [5], page

36 CHAPTER 3. VECTOR PARTITIONS 28 Next, we are concerned with a subset S of V. 3.3 Vector Partitions over Subset S of V Our aim in this section is to introduce vector partitions defined on a subset S of V. This concept is given by Andrews, Garvan and Liang in [7]. For any partition π, let s(π be the smallest part of π. Define s( for the empty partition. A subset S of the set of vector partitions V is given by S : { π (π, π 2, π 3 V : s(π < and s(π min{ s(π 2, s(π 3 } }. Let π (π, π 2, π 3 and π i be the sum of all parts of π i for i, 2, 3. Define π : π + π 2 + π 3, ω (π : ( #(π, crank(π : #(π 2 #(π 3. The number of vector partitions of n over S with crank m counted according to weight ω is denoted by N S (m, n and it can be showed as N S (m, n : π S π n crank(πm ω (π. (3.2 Let N S (m, t, n be the number of vector partitions of n in S with crank congruent to m modulo t counted according to weight ω. We have N S (m, t, n : k N S (kt + m, n π S π n crank(π m (mod t ω (π.

37 CHAPTER 3. VECTOR PARTITIONS 29 In a similar way with crank of vector partitions over V, we have N S (m, n N S ( m, n and N S (m, t, n N S (t m, t, n. Note that there are several descriptions of generating function for N S (m, n. One of these descriptions and its immediate corallary are given below. Theorem [7] Let S(z, q m N S (m, nz m q n. (3.3 Then we have, S(z, q Corollary [7] For n, q n (q n+ ; q (zq n ; q (z q n ; q. (3.4 π S π n ω (π m N S (m, n spt(n. (3.5 Proof. [7] Note that, we can find an equation for spt(n by letting z in (3.4 so that S(, q N S (m, nq n m q n (q n+ ; q (by Theorem 3.3. (q n ; q (q n ; q q n (q n+ ; q ( q n 2 spt(nq n (by (2.8.

38 CHAPTER 3. VECTOR PARTITIONS 30 Therefore we have from (3.2 that π S π n crank(πm ω (π m N S (m, n spt(n. Another important property of N S (m, n can be given as follows. Theorem [7] N S (m, n 0 for all (m, n. Proof. See [7], page 3. Next we focus congruence properties of N S (m, n. 3.4 Congruence Properties of N S (m, n Our aim in this section is to prove congruence properties of N S (m, n. At the begining, we start with obtaining a relation between generating functions of N S (m, n, N V (m, n and N(m, n. We complete this section with stating some congruence properties of N S (m, n and giving an example. The content of this section is mainly based on [7]. Andrews, Garvan and Liang concluded following theorem by using Bailey s Lemma. Theorem [7] (q q n (q n+ ; q ( zq n ( z q n (zq n+ ; q (z q n+ ; q ( ( n q n(n+/2 ( zq n ( z q n n n ( n q n(3n+/2 ( zq n ( z q n.

39 CHAPTER 3. VECTOR PARTITIONS 3 Once we have Theorem 3.4., the relation between generating functions of N S (m, n, N V (m, n and N(m, n is given by Theorem [7] S(z, q N S (m, nz m q n (q m ( n ( z( z ( n q n(n+ 2 ( zq n ( z q n n [ N V (m, nz m q n m ( n q n(3n+ 2 ( zq n ( z q n ] N(m, nz m q n m. Proof. [7] Let us recall equalities in Lemma 3.2. and Propostion We have and m N(m, nz m q n (q n ( n q n(3n+/2 ( z( z ( zq n ( z q n (3.6 m N V (m, nz m q n (q n ( n q n(n+/2 ( z( z. (3.7 ( zq n ( z q n Hence S(z, q m (q n N S (m, nz m q n q n (q n+ ; q (by (3.3. ( zq n ( z q n (zq n+ ; q (z q n+ ; q ( ( n q n(n+ 2 ( zq n ( z q n m n (by Theorem 3.4. [ N ( z( z V (m, nz m q n Last line follows from (3.6 and (3.7. ( n q n(3n+ 2 ( zq n ( z q n m ] N(m, nz m q n.

40 CHAPTER 3. VECTOR PARTITIONS 32 We continue by presenting a lemma whose proof is analogous to proof of Lemma Lemma N S (0, t, tn + δ t N S (, t, tn + δ t N S (t, t, tn + δ t spt(tn + δ t t (3.8 for t 5, 7 where δ t is reciprocal of 24 modulo t. For t prime, (3.8 is equivalent to the coefficient of q tn+δ t S(ζ t, q ( t r0 N S (r, t, nζ r t q n in being zero, where ζ t exp(2πi/t. Proof. By replacing z with ζ t in (3.3, we have m t N S (m, nζt m q n k0 t ζt k k0 t m k (mod t ζt k k0 N S (m, nζt m q n N S (m, n q n m k (mod t N S (k, t, nq n. (3.9 From (3.9, we can see that t k0 N S (k, t, tn + δ t ζ k t is coefficient of q tn+δt in (3.3. Now suppose that (3.8 is true. Hence t k0 t N S (k, t, tn + δ t ζt k N S (0, t, tn + δ t k0 ζ k t 0 (by definition of ζ t.

41 CHAPTER 3. VECTOR PARTITIONS 33 Suppose now that coefficient of q tn+δt in is zero. That is t k0 The minimal polynomial of ζ t over Q is N S (k, t, tn + δ t ζ k t 0. (3.20 p(x + x + x x t. We have from definition of the minimal polynomial of ζ t and (3.20 that N S (0, t, tn + δ t N S (, t, tn + δ t N S (t, t, tn + δ t. (3.2 Therefore spt(tn + δ t N S (m, tn + δ t (by Corollary m t N S (k, t, tn + δ t k0 tn S (0, t, tn + δ t (by (3.2. Now we are ready to state our next theorem. Theorem [7] N S (k, 5, 5n + 4 N S (k, 7, 7n + 5 spt(5n spt(7n for 0 k 4, for 0 k 6. Proof. [7] From Lemma 3.4.3, it is enough to show that the coefficient of q tn+δ t in S(ζ t, q m N S (m, nζ m t q n is zero when t 5, 7 and ζ t exp(2πi/t. ( t r0 N S (r, t, nζ r t q n

42 CHAPTER 3. VECTOR PARTITIONS 34 Put ζ t instead of z in Theorem Thus S(ζ t, q m N S (m, t, nζ m t q n [ ( ζ t ( ζt m (by Theorem [ ( t ( ζ t ( ζt r0 ( t r0 N S (r, t, nζ r t N V (m, nζ m t q n N V (r, t, nζ r t On the other hand we know from Theorem that q n q n m ( t r0 N(m, nζ m t q n ] N(r, t, nζ r t (3.22 q n ]. N V (k, 5, 5n + 4 N V (k, 7, 7n + 5 p(5n p(7n for 0 k 4, for 0 k 6. (3.23 We also know from Theorem that N(k, 5, 5n + 4 N(k, 7, 7n + 5 p(5n p(7n for 0 k 4, for 0 k 6. (3.24 Therefore, from (3.22, (3.23 and (3.24 we have for t 5 and 7, t r0 N S (r, t, tn + δ t q tn+δt ( t ( ζ t ( ζt r0 N V (r, t, tn + δ t ζ r t t r0 N(r, t, tn + δ t ζ r t q tn+δ t 0. Examples If we look at the vector partitions of 5 with crank congruent to modulo 7 in (3. then we see that only following partitions π 7 (, 2 +,

43 CHAPTER 3. VECTOR PARTITIONS 35 π 8 (, 4, π 9 (, +, 2 π (2, 3, π 4 (2 +, 2, π 5 (3 +,, lie in the subset S of V. We have ω (π 7 ω (π 8 ω (π 9 ω (π ( ( 0 and ω (π 4 ω (π 5 ( 2. Therefore, N S (, 7, 5 π S s(π5 crank(π (mod 7 ω (π spt(5 7 2.

44 Chapter 4 Mock Theta Functions and Parity of spt(n In this chapter, our aim is to show that parity of spt-function can be determined by certain mock theta functions. At the beginning, we concern with self-conjugate S-partitions and relate it with spt-function. Finally, we state some consequences about parity of spt-function. The content of this chapter is mainly based on [8] and [23]. 4. Self-Conjugate Vector Partitions In this section, we cover properties of self-conjugate partitions which are defined in [8] and relate these partitions with spt-function. At the end, we give generating function of such partitions. Note that, results of this section are used in order to state properties of parity of spt-function. Recall the definition of the set S. S : { π (π, π 2, π 3 V : s(π < and s(π min{ s(π 2, s(π 3 } }. 36

45 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 37 Let l be a map on S given by, l(π l(π, π 2, π 3 (π, π 3, π 2. An S-partition π (π, π 2, π 3 is a fixed point for l if and only if π 2 π 3. We call these fixed points as self-conjugate S-partitions. We denote the number of self-conjugate S-partitions counted in terms of the weight ω by N SC (n so that N SC (n : π S π n l(ππ ω (π. A congruence relation between N SC (n and spt(n can be given by Proposition 4... [8] N SC (n spt(n (mod 2. Proof. By Corollary 3.3.2, we know that π S π n crank(πm ω (π m N S (m, n spt(n. Note that, for any self-conjugate S-partition π (π, π 2, π 3 of n, crank(π #(π 2 #(π 3 0. Another partition β (β, β 2, β 3 of n having crank 0 should be in the form such that #(β 2 #(β 3 but β 2 β 3. In this case, the contribution of such partitions to N S (0, n is a multiple of 2, call it 2k where k is a nonnegative integer. We have also an equation from previous chapter so that N S (m, n N S ( m, n. Therefore, spt(n 2 m N S (m, n + 2k + N SC (n.

46 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 38 Thus, N SC (n spt(n (mod 2. Andrews, Garvan and Liang gave a generating function for N SC (n in [8] by Theorem [8] SC(q : N SC (nq n q n (qn+ ; q (q 2n ; q 2 ( q; q q n ( q; q n ( q n. Proof. [8] In order to see the first equality we divide the product q n (q n+ ; q (q 2n ; q 2 into two parts. First, power of q in the part q n (q n+ ; q q n ( q n+ ( q n+2 generates a partition π into distinct parts such that n is the smallest part of π. Note also that π is an element of the set D. On the other hand, the coefficient of q π is if number of parts of π is even and if number of parts of π is odd. Secondly we have (q 2n ; q 2 ( + q (n+(n + q (n+n+(n+n + ( + q (n++(n+ + q (n++n++(n++n+... In this case, let π 2 be a partition constructed by numbers within the first paranthesis in powers of q. In the same way let π 3 be another partition constructed by numbers within the second paranthesis in powers of q. It is easy to observe that π 2 and π 3 are partitions with unrestricted parts and their smallest parts are bigger than n. In other words, partitions π 2 and π 3 lie in the set P. If we define

47 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 39 a partition π such that π (π, π 2, π 3 then we conclude that the sum q n (qn+ ; q (q 2n ; q 2 is the generating function for N SC (n. For the second equality, notice that for n (q n+ ; q (q 2n ; q 2 ( q n+ ( q n+2 ( q 2n ( q 2n+2 ( q 2n+4 ( q 2n ( + q n+ ( + q n+2 ( q n ( + q n ( + q n+ ( + q n+2 ( q n ( q n ; q ( q; q n. ( q n ( q; q Hence, q n (qn+ ; q (q 2n ; q 2 ( q; q q n ( q; q n ( q n. In the next section, we are concerned with a fundamental topic in theory of theta functions. 4.2 Introduction to Mock Theta Functions In the last letter of Ramanujan to Hardy [23], Ramanujan mentioned notion of mock theta functions. He also gave 7 examples for these functions. We are interested in some of these functions of order 3. The content can be found in [9] and [23].

48 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 40 Here is the complete list of the mock theta functions of order 3. f(q q n2 ( + q 2 ( + q 2 2 ( + q n 2 χ(q ρ(q φ(q Ψ(q ω(q v(q q n2 ( + q 2 ( + q 4 ( + q 2n q n2 ( q( q 3 ( q 2n q n2 ( q + q 2 ( q 2 + q 4 ( q n + q 2n q 2n(n+ ( q 2 ( q 3 2 ( q 2n+ 2 q n(n+ ( + q( + q 3 ( + q 2n+ q 2n(n+ ( + q + q 2 ( + q 3 + q 6 ( + q 2n+ + q 4n+2 Following identity was stated by Ramanujan and proven by Watson in [23]. 2φ( q f(q f(q + 4Ψ( q v 4 (0, q where v 4 (z, q is a theta function v 4 (z, q n ( + q r (4. r ( n e 2πinz q n2 (e 2πiz q; q 2 (e 2πiz q; q 2 (q 2 ; q 2. v 4 (z, q also satisfies that (See for example [4] v 4 (0, q m0 ( m+ q 2m+ + q 2m+. (4.2

49 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 4 Note that we know the equality Therefore (4. can be written as ( q; q (q; q 2. 2φ( q f(q f(q + 4Ψ( q v 4 (0, q(q; q 2. (4.3 This chapter continues with giving properties of parity of spt(n. 4.3 Parity of spt(n An important property about parity of spt(n is emphasized in this section. We begin with stating some preliminary lemmas. Lemma [8] ( q; q q n ( q; q n q n (q 2 ; q 2 n ((q 2n (q ( n q n2 (q; q 2 n. (4.4 We also need two theorems in order to prove equalities in Lemma Theorem [5] (q/a n (a/t n (q/t n + (t (a ( ( (t (t n (a (a n q n q + n q n t t q n tq n tq n aq n t. aq n t Proof. See [5], page Other theorem is given by the following equality.

50 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 42 Theorem [5] ( (a (b (a n(b n (q (c (q n (c n ( (b (a (c (q q n q n aq n aq n (c/b n b n. (a n ( q n Proof. See [5], page 404. We continue to give a proof of Lemma Proof. (Proof of Lemma 4.3.[8] Since ( q; q n ( q n ( + q( + q2 ( + q n ( q2 ( q 4 ( q 2n 2 q n ( q( q 2 ( q n (q2 ; q 2 n (q n. We have (q 2 ; q 2 q n ( q; q n q n (q 2 ; q 2 q n (q 2 ; q 2 n (q n. (4.5 On the other hand, we know that for n Therefore, (q 2 ; q 2 n (q 2 ; q 2 (q 2n ; q 2. (4.6 (q 2 ; q 2 q n (q 2 ; q 2 n (q n k0 k0 q n (by (4.6 (q n (q 2n ; q 2 q n q 2nk (by Theorem 2..7 (q n (q 2 ; q 2 k k0 q n(2k+ (q 2 ; q 2 k (q n ( (by Theorem (q 2 ; q 2 k (q 2k+ ; q (4.7

51 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 43 We also know that (q (q 2k+ ; q (q 2k. (4.8 If we multiply both sides of (4.5 by (q; q, we find that (q; q (q 2 ; q 2 as desired. q n ( q; q n q n (q; q (q 2 ; q 2 (q; q k0 k0 k0 (q 2 ; q 2 k q n (q 2 ; q 2 n (q n ( (q 2 ; q 2 k (q 2k+ ; q ( (q; q (q 2k+ ; q (q; q (q 2 ; q 2 k ((q; q 2k (q (by (4.8 For the second equality in our theorem, consider following limit lim a 0 (q2 /a; q 2 n (a/t n ( q2 a ( q3 a ( q2n a n Replace q by q 2, t by q and let a 0 in Theorem we find that a (by (4.7 t n ( n q n(n+ t n. ( (q; q 2 (q; q 2 n ( n q n2 (q; q 2 n + (q; q 2 q 2n. (4.9 q2n On the other hand, let q q 2 and a b c 0 in Theorem 4.3.3, we have ( (q 2 ; q 2 (q 2 ; q 2 n (q 2 ; q 2 q 2n. (4.0 q2n We know that and (q; q (q 2 ; q 2 (q; q 2, (4. (q; q 2n (q 2 ; q 2 n (q; q 2 n. (4.2

52 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 44 Hence ((q; q (q 2 ; q 2 2n (q; q n ( (q; q 2 n (q; q 2 + (q; q 2 (q; q (by (4.2. (q 2 ; q 2 n ( ( (q; q 2 n (q; q 2 + (q; q (by (4.. (q 2 ; q 2 (q 2 ; q 2 n ( n q n2 (q; q 2 q 2n (q; q 2 n q + (q; q q 2n 2n (q 2 ; q 2 q2n (by (4.9 and (4.0. ( n q n2 (q; q 2 q 2n (q; q 2 n q + (q; q 2n 2n q2 q2n (by (4. ( n q n2 (q; q 2 n. Note that the infinite sum ( n q n2 (q; q 2 n (4.3 is a mock theta function. Andrews, Dyson and Hickerson studied (4.3 in [6] and they interpreted (4.3 in terms of partitions in a following way: Consider partitions of n into odd parts with the condition that if k occurs as a part, then all positive odd numbers less than k also occur. Denote S (n be the number of such partitions with largest part congruent to 3 modulo 4 minus the number of such partitions with largest parts congruent to modulo 4. Andrews, Dyson and Hickerson gave a generating function for S (n in [6] by n S (nq n n ( n q n2 (q; q 2 n. (4.4 Explicit formula for the coefficients of (4.4 was given in [6]. In order to state the formula, we need a definition from [6]. Definition Define an arithmetic function T (m for integers which are congruent to modulo 24 as follows:

53 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 45 Let m be an integer which is congruent to modulo 24. Consider Pell s equation u 2 6v 2 m. (4.5 Notice that if (u, v is a solution of (4.5 then u is an integer which is congruent to ± modulo 6 and v is even. We call two solutions (u, v and (u, v equivalent if u + v 6 ±( r (u + v 6. for some integer r. Let T (m be the number of inequivalent solution of (4.5 with u + 3v ± (mod 2 minus the number of inequivalent solutions of (4.5 with u + 3v ±5 (mod 2. Andrews, Dyson and Hickerson proved that T (m is a multiplicative function. They also proved that Lemma [6] Let p be a prime which is congruent to modulo 6 or the negative of a prime which is congruent to 5 modulo 6, and let e. Then: (a If p is not congruent to modulo 24 and e is odd, then T (p e 0. (b If p is not congruent to modulo 24 and e is even, then if p 3 or 9 (mod 24 T (p e ( e/2 if p 7 (mod 24. (c If p (mod 24 then either T (p 2 or T (p 2. In the first case, T (p e e +. In the second case, T (p e ( e (e +. They stated a theorem about the coefficients of (4.4 as follows. Theorem [6] For n, 2S (n T ( 24n. Andrews, Dyson and Hickerson emphasized that Theorem is equivalent to a theorem given by

54 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 46 Theorem [6] n ( n q n2 (q; q 2 n n 2n ( n q n(3n ( + q 2n q j(j+/2. j0 Now, we are ready to prove a result about parity of spt(n. Theorem [8] spt(n S (n (mod 2. Proof. By Proposition 4.., we have spt(n N SC (n (mod 2. (4.6 On the other hand, by Theorem 4..2 we have the generating function of N SC (n N SC (nq n ( q; q However, in Lemma 4.3., we proved that q n ( q; q n. (4.7 ( q n N SC (nq n ( q; q q n ( q; q n ( q n ( n q n2 (q; q 2 n. (4.8 We also know that Thus S (nq n N SC (nq n ( q; q ( n q n2 (q; q 2 n. (4.9 q n ( q; q n ( q n ( n q n2 (q; q 2 n (by (4.8 S (nq n (by (4.9. (by (4.7 (4.20

55 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 47 Equating coefficients of q in (4.20 we have N SC (n S (n. Therefore, by (4.6 we get spt(n N SC (n S (n (mod 2 as desired. It now follows from Lemma 4.3.5, Theorem and Theorem that Theorem [4] (i N SC (n 0 if and only if p e 24n. (ii spt(n is odd if and only if 24n p 4a+ m 2 for some prime p 23 (mod 24 and some integers a, m, where (p, m. We should remark that parity result for the spt-function was firstly given by A. Folsom and K. Ono in [3]. However, their results were corrected as in Theorem We finish this chapter by proving one more result about connection between spt(n and a mock theta function of third order. 4.4 Relation Between spt(n and Mock Theta Functions In this section, our aim is to prove a result which relates spt(n and a mock theta function. Let us start with stating an identity from [23] so that ( f(q + 4 (q ( n q n(3n+/2. (4.2 + q n

56 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 48 where f(q is the mock theta function of order 3 defined by Ramanujan([20]. Now we are ready to state our next theorem. Theorem [8]Let Ψ(q q n2 (q; q 2 n ψ(nq n. Then spt(n ( n ψ(n (mod 4. Proof. [8] It is easy to show that We have (q ( q n 2 ( + q n + 4 q n 2 ( q 2n. ( ( n q n(3n+/2 + q n 4(q ((q f(q (by (4.2 If we consider (4.22 then we see that n (q (q spt(nq n (q nq n q n + (q nq n q + n (q Ψ( q + 4 (q; q2 v 4 (0, q 4(q (by (4.3. nq n q + n (q ( n q n(3n+/2 ( + q n ( + q n (q ( n q n(3n+/2 On the other hand, it is easy to observe that + q n (mod 4. (4.23 ( n q n(3n+/2 ( + q n ( q n 2 by (2.2.2 ( n q 3n(n+/2 ( q n 2 (4.24 (q; q 2 (q 2 ; q 2 (q; q. (4.25

57 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 49 We have, v 4 (0, q 2 (q (q ((q; q 2 (q 2 ; q 2 (q; q 2 2 (q ((q; q (q; q 2 2 (by (4.25 (q ((q (q; q 2 (q 2 ; q 2 (q; q 2 (q; q 2 v 4 (0, q(q; q 2 (by defn. of v 4 (0, q. (4.26 Therefore, (q ( n q n(3n+/2 + q n Ψ( q + 4 (q; q2 v 4 (0, q 4(q (by (4.23 Thus we have from (4.24 and (4.27 that n ( spt(nq n (q Ψ( q + (v 4 (0, q 2 (by (4.26 4(q Ψ( q + ( m+ q 2m+ (by (4.2. (q + q 2m+ m0 (4.27 nq n q + n m0 ( m+ q 2m+ Ψ( q (mod 4. + q 2m+ (4.28

58 CHAPTER 4. MOCK THETA FUNCTIONS AND PARITY OF SP T (N 50 Now, we investigate the infinite sum above in modulo nq n q + n m0 q 4n+ q 4n+ m0 ( m+ q 2m+ + q 2m+ q 4m + q 4m (mod 4 q 8n+2 q 2 8n+2 q 8n+2 q 8n+2 0 (mod 4. q 4n q + 2 4n q 8n 2 q + 2 8n 2 q 4n+2 q 4n+2 m0 q 4m+ + q 4m+ q 4n+2 q 4n+2 (mod 4 (4.29 Hence we get from (4.28 and (4.29 that n spt(nq n Ψ( q (mod 4. In other words, spt(n ( n ψ(n (mod 4 by the definition of function Ψ(q.

59 Bibliography [] G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, Massachusetts, 976. [2] G. E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008, [3] G. E. Andrews, Partitions, Durfee symbols and Atkin-Garvan moments of ranks, Invet. Math. 69 (2007, [4] G. E. Andrews, Applications of Basic Hypergeometric Functions, SIAM Rev. 6 (974, [5] G. E. Andrews, J. Jiménez-Urroz and K. Ono, q-series identities and values of certain L-functions, Duke Math. J., 08(200, [6] G. E. Andrews, F. J. Dyson and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math., 9(988, [7] G. E. Andrews, F. G. Garvan, and J.Liang, Combinatorial interpretations of congruences for the spt-function, Ramanujan J., 29(202, [8] G. E. Andrews, F. G. Garvan, and J.Liang, Self-conjugate vector partitions and the parity of the spt-function, Acta Arith., 58(203, [9] A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (967, [0] A. O. L. Atkin and P.Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3, 4(954,

60 BIBLIOGRAPHY 52 [] B. C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Rhode Island, 939. [2] F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge 8(944, 0-5. [3] A. Folsom and K. Ono, The spt-function of Andrews, Proc. Nat. Acad. Sci. USA 05(2008, [4] F. G. Garvan, New combinatorial interpretations of Ramanujan s partition congruences mod 5,7 and, Trans. Amer. Math. Soc., 305(988, [5] F. G. Garvan, Combinatorial interpretations of Ramanujan s partition congruences in Ramanujan Revisited:Proc. of the Centenary Conference, Univ. of Illinosis at Urbana-Champaign, June -5,987, Acad. Press, San Diego,988. [6] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encylc. of Math. and Its Appl.,Vol. 35, Cambridge University Press, Cambridge, 990. [7] G. H. Hardy and E. M. Wright, An Introduction to Theory of Numbers, 5th ed., Clarendon Press, Oxford, 979. [8] C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Sumtibus fratrum, 829. [9] S. Ramanujan, Some Properties of p(n, the number of partitions of n, Proc. Cambridge Philos. Soc., 9(99, [20] S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 927; reprinted by Chelsea, New York, 962; reprinted by American Mathematical Society, Providence, RI, [2] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc., 9(920, [22] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 966.

61 BIBLIOGRAPHY 53 [23] G. N. Watson, The final problem: An account of the mock theta functions, J. London. Math. Soc. 2 (2(936, [24] G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. 79 (938, [25] L. Winquist, An elementary proof of p(m (mod, J. Combin. Theory, 6(969,