CLASSICAL PARTITION IDENTITIES AND BASIC HYPERGEOMETRIC SERIES

Transcription

1 UNIVERSITÀ DEGLI STUDI DI LECCE DIPARTIMENTO DI MATEMATICA "Ennio De Giorgi" Wenchang Chu Leontina Di Claudio Dipartimento di Matematica Università di Lecce Lecce - Italia CLASSICAL PARTITION IDENTITIES AND BASIC HYPERGEOMETRIC SERIES Q UADERNO 6/2004 Edizioni del Grifo

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3 UNIVERSITÀ DEGLI STUDI DI LECCE DIPARTIMENTO DI MATEMATICA De Giorgi" Wenchang Chu Dipartimento di Matematica Università di Lecce Lecce - Italia Leontina Di Claudio CLASSICAL PARTITION IDENTITIES AND BASIC HYPERGEOMETRIC SERIES QUADERNO 6/2004

4 QUADERNI DI MATEMATICA Una pubblicazione a cura del DIPARTIMENTO DI MATEMATICA "ENNIO DE GIORGI" UNIVERSITÀ DEGLI STUDI DI LECCE Commitato di Redazione Giuseppe De Cecco (Direttore) Lorenzo Barone Wenchang Chu (Segretario) I QUADERNI del Dipartimento di Matematica dell'università degli Studi di Lecce documentano gli aspetti di rilievo dell'attività di ricerca e didattica del Dipartimento Nei Quaderni sono pubblicati articoli di carattere matematico che siano: (A) lavori di rassegna e monografie su argomenti di ricerca; (B) testi di seminari di interesse generale, tenuti da docenti o ricercatori del Dipartimento o esterni; (C) lavori di specifico interesse didattico La pubblicazione dei lavori è soggetta all'approvazione del Comitato di Redazione, che decide tenendo conto del parere di un referee, nominato di volta in volta sulla base delle competenze specifiche ISBN

5 Classical Partition Identities and Basic Hypergeometric Series CRU Wenchang and DI CLAUDIO Leontina DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DEGLI STUDI DI LECCE LECCE-ARNESANO P O Box LECCE, ITALIA chu wenchang@unile it leontina diclaudio@unileit

6 2000 Mathematics Subject Classification 05AIO, 05A5, 05A7, 05A9, 05A30 B65, E25, P8, P82, P83 33B5, 33C20, 33D05, 33D5, 33E05 K ey words and phrases Partition, Ferrers diagram Jacobi triple product identity Rogers-Ramanujan identities Basic hypergeometric Series Lagrange four square theorem Congruence of partition function

7 Preface Basic hypergeometric series,often shortened as q-series,has developed rapidly during the past two decades Its increasing importance to theoretical physics, computer science and classical mathematics (algebra, analysis and combinatorics) has widely been understood and accepted Nowadays q-series is fully in its flourishing period and there is indeed a necessity to have an introduction boo on the topic This boo originates from the teaching experience of the first author In the spring of 2000, a series of lectures entitled Classical Partitions and Rogers-Ramanujan Identities was delivered by the first author at Lecce University (Italy) The same program was then replayed in the summer of 200 at Dalian University of Technology (China) In the spring of 2002 and 2004, these lectures have been extended to a course for PhD students again at Lecce University under the cover-title Teoria dei Numeri The second author is one among the participants of these lectures The main purpose of the boo is to present a brief introduction to basic hypergeometric series and applications to partition enumeration and number theory As a short account to the theory of partitions, the first part (Chapters A-B-C) covers the algebraic aspects (basic structures: partially ordered sets and lattices), combinatorial aspects (generating functions and Durfee rectangles), and analytic aspects (the Jacobi triple products and Rogers-Ramanujan identities) Further development toward basic hypergeometric series and bilateral counterparts is dealt with in the second part (Chapters D-E-F) Applications to the representations of natural numbers by square sums and Ramanujan s congruences on partition function are presented in the third and the last part (Chapters G-H) Lecce, November st, 2004 CHU Wenchang DI CLAUDIO Leontina

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9 Contents Preface v Chapter A Partitions and Algebraic Structures Al Partitions and representations A2 Ferrers diagrams of partitions 2 A3 Addition on partitions 6 A4 Multiplication on partitions 6 A5 Dominance partial ordering 7 Chapter B Generating Functions of Partitions Il BI Basic generating functions of partitions B2 Classical partitions and the Gauss formula 5 B3 Partitions into distinct parts and the Euler formula 9 B4 Partitions and the Gauss q-binomial coefficients 24 B5 Partitions into distinct parts and finite q-differences 27 Chapter C Durfee Rectangles and Classical Partition Identities 29 Cl q-series identities of Cauchy and Kummer: Unification 29 C2 q-binomial convolutions and the Jacobi triple product 30 C3 The finite form of Euler's pentagon number theorem 42 Chapter D The Carlitz Inversions and Rogers-Ramanujan Identities 45 DI Combinatorial inversions and series transformations 47 D2 Finite q-differences and further transformation 5 D3 Rogers-Ramanujan identities and their finite forms 52 Chapter E Basic Hypergeometric Series 57 El Introduction and notation 57 E2 The q-gauss summation formula 60 E3 Transformations of Heine and Jacson 63 E4 The q-pfaff-saalschiitz summation theorem 7 E5 The terminating q-dougall-dixon formula 73 E6 The Sears balanced transformations 74 E7 Watson's q-whipple transformation 79 vii

10 viii CHU Wenchang and DI CLAUDIO Leontina Chapter F Bilateral Basic Hypergeornetric Series Fl Definition and notation F2 Rarnanujan's bilateral -series identity F3 Bailey's bilateral identity F4 Bilateral q-analogue of Dixon's theorern F5 Partial fraction decornposition rnethod Chapter G The Lagrange Four Square Theorern G Representations by two square surns G2 Representations by four square surns G3 Representations by six square surns G4 Representations by eight square surns G5 Jacobi's identity and q-difference equations Chapter H Congruence Properties of Partition Function Hl Proof of p(5n + 4) O (rnod 5) H2 Generating function for p(5n + 4) H3 Proof of p(7n + 5) O (rnod 7) H4 Generating function for p(7n + 6) H5 Proof of p(ln + 6) O (rnod ) Appendix: Tannery's Lirniting Theorern Bibliography

11 CHAPTER A Partitions and Algebraic Structures In this chapter, we introduce partitions of natural numbers and the Ferrers diagrams The algebraic structures of partitions such as addition, multiplication and ordering will be studied A Partitions and representations A partition is any (finite or infinite) sequence λ (λ,λ 2,,λ, ) of non-negative integers in decreasing order: λ λ 2 λ and containing only finitely many non-zero terms The non-zero λ in λ are called the parts of λ The number of parts of λ is the length of λ, denoted by l(λ); and the sum of parts is the weight of λ, denoted by λ : λ λ λ + λ 2 + If n λ we say that λ is a partition of n, denoted by n λ The set of all partitions of n is denoted by P n In particular, P 0 consists of a single element, the unique partition of zero, which we denote by 0 Sometimes it is convenient to use a notation which indicates the number of times each integer occurs as a part: λ ( m 2 m2 m ) means that exactly m copies of the parts of λ are equal to The number m m (λ) Card { } i : λ i is called the multiplicity of in λ

12 2 CHU Wenchang and DI CLAUDIO Leontina A2 Ferrers diagrams of partitions The diagram of a partition λ may be formally defined as the set of points (or unit squares) { } λ (i, j) j λ i, i l(λ) drawn with the convention as matrices For example, the diagram of the partition λ (5442) is shown as follows: We shall usually denote the diagram of a partition λ by the same symbol The conjugate of a partition λ is the partition λ whose diagram is the transpose of the diagram λ, ie, the diagram obtained by reflection in the main diagonal For example, the conjugate of (5442) is (4433) Hence λ is the number of the nodes in the -th column of λ, or equivalently λ Card{ i : λ i } In particular, λ l(λ) and λ l(λ ) Obviously, we also have λ λ and m λ λ + Therefore we can dually express the Ferrers diagram of λ as { } λ (i, j) i λ j, j l(λ ) A2 Euler s theorem The number of partitions of n into distinct odd parts is equal to the number of self-conjugate partitions of n Proof Let S be the set of partitions of n into distinct odd parts and T the set of self-conjugate partitions of n, the mapping f : S T λ μ

13 Classical Partition Identities and Basic Hypergeometric Series 3 defined by μ i μ i : λ i + i where for all i, 2,,l(λ) 2 Obviously, μ is a selfconjugate partition with diagonal length equal to l(λ) and the weight equal to λ, which can be justified as follows: l(λ) μ + l 2 (λ) (μ i + μ i) λ + (2i ) λ + l 2 (λ) i From the Ferrers diagrams, we see that f is a bijection between S and T Therefore they have the same cardinality S T, which completes the proof l(λ) i For example, the image of partition λ (73) under f reads as μ (433) This can be illustrated as follows: λ (73) μ (433) A22 Theorem on permutations Let λ be a partition with m λ and n λ Then the m + n numbers λ i + n i ( i n) and n +j λ j ( j m) are a permutation of { 0,, 2,,m+ n } Proof Define three subsets of non-negative integers: U : { λ i + n i i n } V : { n +j λ j j m } W : { 0 m + n }

14 4 CHU Wenchang and DI CLAUDIO Leontina In order to prove the theorem, it suffices to show the following (A) U W and V W; (B) The elements of U are distinct; (C) The elements of V are distinct; (D) U V It is clearly true (A) Suppose that there exist i and j with i<j n such that λ i + n i λ j + n j Keeping in mind of the partition λ, we see that it is absurd for λ i λ j and n i>n j This proves (B) We can prove (C) similarly in view of the conjugate partition λ There remains only (D) to be confirmed Observe that the Ferrers diagram of λ is contained in the Ferrers diagram of (m n ), which is an n m rectangle We can identify the partition λ with the points inside its Ferrers diagram If the point with coordinate (i, j) is inside λ, then we have λ i i and j λ j, which are equivalent to the inequality λ i i 0 j λ j λ i + n i>n +j λ j This means that for (i, j) inside the Ferrers diagram λ with i n and j m, the corresponding λ i + n i and n +j λ j can not be the common element in U V Instead if the point with coordinate (i, j) lies outside λ, then we have λ i <i and j>λ j, which are equivalent to another inequality λ i i<0 j λ j λ i + n i<n +j λ j This implies that for (i, j) outside the Ferrers diagram λ with i n and j m, the corresponding λ i + n i and n +j λ j can not be again the common element in U V In any case, we have verified that U and V have no common elements, which confirms (D) The proof of Theorem A22 is hence completed A23 The hoolength formula Let λ be a partition The hoolength of λ at (i, j) λ is defined to be h(i, j) +λ i + λ j i j

15 Classical Partition Identities and Basic Hypergeometric Series 5 If the diagram of λ is contained in the diagram of (m n ), define ν λ + n ( n) Then the theorem on {m + n}-permutations can be used to demonstrate the following hoolength formulae: νi ( q h(i,j) i j ) ( qj ) i<j ( qν i ν j) i h(i, j) ν i! i<j (ν i ν j ) (i,j) λ (i,j) λ Proof Interchanging λ and λ in permutation Theorem A22 and then putting m λ and λ n, we see that m + λ j j ( j m) and m +j λ j ( j n) constitute a permutation of {0,, 2,,m+n } Therefore we have a disjoint union: {q λ +λ j j} λ j } {q λ +j λ j {q nj j} λ +n j0 According to the definition of the hoolength of λ, the identity can be reformulated as follows: {q h(,j)} λ j {q ν ν j } nj2 {q j} ν j Writing down this identity for the partition (λ i,λ i+, ): {q h(i,j)} λ i j {q ν i ν j } nj+i {q j} ν i j and then summing them over i, 2,,l(λ), we obtain (i,j) λ q h(i,j) + i<j q ν i ν j i ν i j q j Instead of summation, the multiplication leads us consequently to the following: νi ( q h(i,j) i j ) ( qj ) i<j ( qν i ν j) (i,j)λ In particular dividing both sides by ( q) λ and then setting q,we find that i h(i, j) ν i! i<j (ν i ν j ) (i,j) λ This completes the proof of the hoolength formula

16 6 CHU Wenchang and DI CLAUDIO Leontina A3 Addition on partitions Let λ and μ be partitions We define λ + μ to be the sum of the sequences λ and μ: (λ + μ) λ + μ Also we define λ μ to be the partition whose parts are those of λ and μ, arranged in descending order A3 Proposition The operations + and are dual each other (λ μ) λ + μ (λ + μ) λ μ Proof The diagram of λ μ is obtained by taing the rows of the diagrams of λ and μ and reassembling them in decreasing order Hence the length of the -th column of λ μ is the sum of lengths of the -th columns of λ and of μ, ie (λ μ) {i λ i } + {j μ j } λ + μ The converse follows from duality A32 Examples For two symmetric partitions given by λ (32) and μ (2), we then have λ + μ (53) and λ μ (322) Similarly, we consider a non-symmetric example If λ (33) and μ (2), then it is easy to compute λ (322) and μ (2) Therefore λ + μ (54) and λ μ (332) and (λ μ) (532) λ + μ (λ + μ) (3222) λ μ A4 Multiplication on partitions Next, we define λ μ to be the component-wise product of the sequences λ and μ: (λ μ) λ μ

17 Classical Partition Identities and Basic Hypergeometric Series 7 Also we define λ μ to be the partition whose parts are min(λ i,μ j ) for all (i, j) with i l(λ) and j l(μ), arranged in descending order A4 Proposition For the operations and, we have the dual relation: (λ μ) λ μ (λ μ) λ μ Proof By definition of λ μ, we can write { (λ μ) (i, j) :λ i and μ j i l(λ) and j l(μ)} { i : λ i i l(λ)} { j : μ j j l(μ)} It reads equivalently as (λ μ) λ μ (λ μ ) (λ μ) λ μ Another relation is a consequence of the dual property A42 Examples Consider the same partitions in the examples illustrated in A32 For λ (32) and μ (2), we have λ μ (62) and λ μ (22) The non-symmetric example with λ (33) and μ (2) yields λ μ (63) and λ μ (22) Moreover λ (322) and μ (2) and so we have (λ μ) (62) λ μ (λ μ) (222) λ μ A5 Dominance partial ordering A5 Young s lattice Let P be the set of partitions of all non-negative integers Order P component-wise; that is, (λ,λ 2, ) (μ,μ 2, ) λ μ, Then P is a partially ordered set For two partitions λ, μ, wehave λ μ sup(λ, μ) where (λ μ) max(λ,μ ) λ μ inf(λ, μ) where (λ μ) min(λ,μ ) Therefore P is a lattice, nown as Young s lattice

18 8 CHU Wenchang and DI CLAUDIO Leontina A52 Total orderings Let L n denote the reverse lexicographic ordering on the set P n of partitions of n: that is to say, L n is the subset of P n P n consisting of all (λ, μ) such that either λ μ or the first non-vanishing difference λ μ is positive L n is a total ordering Another total ordering on P n is L n, the set of all (λ, μ) such that either λ μ or else the first non-vanishing difference λ μ is negative, where λ λ +n For example, when n 5,L 5 and L 5 arrange P 5 in the sequence L 5 L 5 (5), (4), (2 3), (2 2 ), ( 3 2), ( 5 ) However the orderings L n and L n are distinct as soon as n>5 This can be exemplified from two partitions λ (3 3 ) and μ (2 3 ) as well as their orderings (λ, μ) L 6 and (μ, λ) L 6 In general, for λ, μ P n, there holds (λ, μ) L n (μ,λ ) L n Proof Suppose that (λ, μ) L n and λ μ Then for some integer we have λ μ > 0 and λ i μ i for i< If we put l λ and consider the diagrams of λ and μ, we see immediately that λ i μ i for l<i n, and that λ l >μ l, so that (μ,λ ) L n The converse can be proved analogously A53 Dominance partial ordering An ordering is more important than either L n or L n is the natural (partial) ordering N n on P n (also called the dominance partial ordering), which is defined through the partial sums as follows: (λ, μ) N n λ + λ λ μ + μ μ, However, N n is not a total ordering as soon as n>5 For example, (3 3 ) and (2 3 ) are incomparable to N 6 as their partial sums are (3456) and (2466) respectively We shall write λ μ in place of (λ, μ) N n A54 Proposition Let λ, μ P n Then (A) λ μ (λ, μ) L n L n (B) λ μ μ λ Proof We prove (A) and (B) separately

19 Classical Partition Identities and Basic Hypergeometric Series 9 (A) Suppose that λ μ Then either λ >μ, in which case (λ, μ) L n, or else λ μ In this case either λ 2 >μ 2, in which case again (λ, μ) L n, or else λ 2 μ 2 Continuing in this way, we see that (λ, μ) L n Also, for each i, we have λ i+ + λ i+2 + n (λ + + λ i ) n (μ + + μ i ) μ i+ + μ i+2 + Hence the same reasoning as before shows that (λ, μ) L n (B) Clearly it is enough to prove one implication Suppose that μ λ Then for some, we have λ + + λ i μ + + μ i, i< and λ + + λ >μ + + μ which implies that λ >μ Let u λ, v μ Now that λ and μ are partitions of the same number n, it follows that λ + + λ +2 + <μ + + μ +2 + Recalling that λ + + λ +2 + is equal to the number of nodes in the diagram of λ which lie to the right of the th column, we have λ + + λ +2 + u (λ i ) i Liewise Hence we have v μ + + μ +2 + (μ i ) i v u v (μ i ) > (λ i ) (λ i ) i i i in which the right-hand inequality holds because u>vand λ i i u So we have for μ + + μ v >λ + + λ v and therefore λ μ, which contradicts to the condition λ μ

20 0 CHU Wenchang and DI CLAUDIO Leontina A55 Theorem The set P n of partitions of n is a lattice with respect to the natural ordering, which is confirmed by the following important theorem Each pair of partitions λ, μ of n has a greatest lower bound τ inf(λ, μ), defined by ( ) τ : τ i min λ i, μ i for each i i i and a least upper bound σ sup(λ, μ) defined by σ inf(λ,μ ) Proof Let ν P n with λ ν and μ ν We see that for, 2,,n, there hold λ + λ λ ν + ν ν μ + μ μ ν + ν ν which is equivalent to ν τ inf(λ, μ) in accordance with the definition of inf Now suppose that ν P n with ν λ and ν μ By means of Proposition A54, we have ν λ λ ν ν μ μ ν which read as ν σ inf(λ,μ ) ν σ sup(λ, μ) This complete the proof of the theorem The example with λ ( 3 3), μ(2 3 ) and σ (32) shows that it is not always true that ( σ : σ i max λ i, μ i ), i even we would have desired it i In fact, the partial sums of λ and μ read respectively as (3456) and (2466), whose minimum is given by (2456) Therefore we have inf(λ, μ) ( ) Similarly, for the conjugate partitions λ ( 2 4) and μ (3 2 ), the corresponding partial sums are given respectively by (456) and (366) Their minimum reads as (356) and hence inf(λ,μ ) (32) which leads us to sup(λ, μ) (32) However the maximum between the partial sums of λ and μ is (346) It corresponds to the partial sums of the sequence (32), which is even not a partition i

21 CHAPTER B Generating Functions of Partitions For a complex sequence {α n n 0,, 2, }, its generating function with a complex variable q is defined by A(q) : n0 α n q n α n q n A(q) When the sequence has finite non-zero terms, the generating function reduces to a polynomial Otherwise, it becomes an infinite series In that case, we suppose in general q < from now on B Basic generating functions of partitions Given three complex indeterminates x, q and n with q <, the shifted factorial is defined by (x; q) (x; q) n ( xq ) 0 (x; q) (q n x; q) When n is a natural number in particular, it reduces to n (x; q) 0 and (x; q) n ( q x) for n, 2, 0 We shall frequently use the following abbreviated notation for shifted factorial fraction: a, b,, c q (a; q) n(b; q) n (c; q) n α, β,, γ n (α; q) n (β; q) n (γ; q) n

22 2 CHU Wenchang and DI CLAUDIO Leontina B Partitions with parts in S We first investigate the generating functions of partitions with parts in S, where the basic set S N with N being the set of natural numbers Let S be a set of natural numbers and p(n S) denote the number of partitions of n into elements of S (or in other words, the parts of partitions belong to S) Then the univariate generating function is given by n0 p(n S) q n S q (Ba) If we denote further by p l (n S) the number of partitions with exactly l-parts in S, then the bivariate generating function is p l (n S) x l q n xq (Bb) l,n 0 S Proof For q <, we can expand the right member of the equation (Ba) according to the geometric series S q S m 0 q m m 0 S q S m Extracting the coefficient of q n from both sides, we obtain q n q qn q S m S m 0 S S m n m 0: S The last sum is equal to the number of solutions of the Diophantine equation m n S which enumerates the partitions { m, 2 m 2,,n m n} of n into parts in S This completes the proof of (Ba) (Bb) can be verified similarly The bivariate generating function In fact, consider the formal power series expansion S xq S m 0 x m q m m 0 S x S m q S m

23 Classical Partition Identities and Basic Hypergeometric Series 3 in which the coefficient of x l q n reads as x l q n xq x l q n x S m q S m S m 0 S S m l S m n m 0: S } The last sum enumerates the solutions of the system of Diophantine equations m l S m n S which are the number of partitions { m, 2 m 2,,n n} m of n with exactly l-parts in S B2 Partitions into distinct parts in S Next we study the generating functions of partitions into distinct parts in S If we denote by Q(n S) and Q l (n S) the corresponding partition numbers with distinct parts from S, then their generating functions read respectively as Q(n S) q n ( +q ) (B2a) n0 S ( ) +xq (B2b) l,n 0 Q l (n S) x l q n S Proof For the first identity, observing that +q m 0, q m we can reformulate the product on the right hand side as ( ) +q q m q S m S S m 0, m 0, S Extracting the coefficient of q n, we obtain q n S( ) +q q n q S m m 0, S S m n m 0,: S

24 4 CHU Wenchang and DI CLAUDIO Leontina The last sum enumerates the solutions of Diophantine equation m n with m 0, S which is equal to Q(n S), the number of partitions of n into distinct parts in S Instead, we can proceed similarly for the second formula as follows: S( + xq ) S m 0, x m q m m 0, S x S m q S m The coefficient of x l q n leads us to the following x l q n S( + xq ) x l q n m 0, S S m l S m n m 0,: S x S m q S m } The last sum equals the number of solutions of the system of Diophantine equations m l S m n S with m 0, which correspond to the partitions { m, 2 m 2,,n n} m of n with exactly l distinct parts in S

25 Classical Partition Identities and Basic Hypergeometric Series 5 B3 Classical generating functions When S N, the set of natural numbers, the corresponding generating functions may be displayed, respectively, as (q; q) (qx; q) ( q; q) ( qx; q) m m q m p(n) q n n0 xq m ( + q m ) m m l,n 0 p l (n) x l q n Q(n) q n n0 ( + xq m ) Q l (n) x l q n l,n 0 (B3a) (B3b) (B3c) (B3d) Manipulating the generating function of the partitions into odd numbers in the following manner q 2 q q 2 q ( q 2 ) ( + q ) we see that it results in the generating function of the partitions into distinct parts We have therefore proved the following theorem due to Euler The number of partitions of n into odd numbers equals to the number of partitions of n into distinct parts B2 Classical partitions and the Gauss formula B2 Proposition Let p m (n) be the number of partitions into exactly m parts (or dually, partitions with the largest part equal to m) Its generating function reads as p m (n) q n n0 q m ( q)( q 2 ) ( q m ) (B2)

26 6 CHU Wenchang and DI CLAUDIO Leontina Proof For S N, the generating function of {p l (n N)} reads as l (n) x l,n 0p l q n x l p l (n)q n l 0 n 0 xq (qx; q) Extracting the coefficient of x m, we get p m (n) q n x m (qx; q) n0 For q <, the function /(qx; q) is analytic at x 0 We can therefore expand it in MacLaurin series: A l (q)x l (B22) (qx; q) where the coefficients { A l (q) } are independent of x to be determined Performing the replacement x x/q, we can restate the expansion just displayed as A l (q)x l q l (x; q) (B23) l0 It is evident that (B22) equals ( x) times (B23), which results in the functional equation A l (q)x l ( x) A l (q)x l q l l0 Extracting the coefficient of x m from both expansions, we get l0 l0 A m (q) A m (q)q m A m (q)q m which is equivalent to the following recurrence relation q A m (q) q m A m (q) where m, 2, Iterating this recursion for m-times, we find that q m A 0 (q) A m (q) ( q m )( q m ) ( q) qm A 0 (q) (q; q) m Noting that A 0 (q), we get finally p m (n) q n x m qm (qx; q) (q; q) m n0 This completes the proof of Proposition B2

27 Classical Partition Identities and Basic Hypergeometric Series 7 A combinatorial proof Let p(n λ m) be the number of partitions of n with the first part λ equal to m Then p m (n) p(n λ m) because the partitions enumerated by p m (n) are conjugate with those enumerated by p(n λ m) Therefore they have the same generating functions: p m (n) q n n0 p(n λ m) q n n0 All the partition of n enumerated by p(n λ m) have the first part λ m in common and the remaining parts constitute the partitions of n m with each part m Therefore we have p(n λ m) q n n0 p(n m λ m) q n q m p(n λ m) q n nm q m n0 n0 p(n {, 2,,m}) q n qm (q; q) m where the first line is justified by replacement n n + m on summation index, while the second is a consequence of (Ba) This confirms again the generating function (B2) B22 Proposition Let p m (n) be the number of partitions into m parts (or dually, partitions into parts m) Then we have the generating function p m (n) q n n0 which yields a finite summation formula ( q)( q 2 ) ( q m ) m ( q)( q 2 ) ( q m ) + q ( q)( q 2 ) ( q ) (B24) Proof Notice that p m (n), the number of partitions into m parts is equal to the number of partitions into parts m in view of conjugate partitions We get immediately from (Ba) the generating function (B24) The classification of the partitions of n into m parts with respect to the number of parts yields p m (n) p 0 (n)+p (n)+p 2 (n)+ + p m (n)

28 8 CHU Wenchang and DI CLAUDIO Leontina The corresponding generating function results in p m (n) q n n0 m p (n) q n 0 n0 m 0 q (q; q) Recalling the first generating function expression (B24), we get the second formula from the last relation B23 Gauss classical partition identity n0 xq n + m x m ( q)( q 2 ) ( q m ) (B25) Proof In fact, we have already established this identity from the demonstration of the last theorem, where it has been displayed explicitly in (B23) Alternatively, classifying all the partitions with respect to the number of parts, we can manipulate the bivariate generating function (xq; q) l,n0 l0 p l (n)x l q n l0 x l n0 x l q l ( q)( q 2 ) ( q l ) which is equivalent to Gauss classical partition identity p l (n)q n B24 Theorem Let p l (n m) be the number of partitions of n with exactly l-parts m Then we have its generating function l,n0 p l (n m) x l q n ( qx)( q 2 x) ( q m x) The classification with respect to the maximum part of partitions produces another identity m ( qx)( q 2 x) ( q m x) +x q ( qx)( q 2 x) ( q x) Proof The first generating function follows from (Bb)

29 Classical Partition Identities and Basic Hypergeometric Series 9 From the first generating function, we see that the bivariate generating function of partitions into parts reads as l,n0 p l (n ) x l q n (qx; q) Putting an extra part λ with enumerator xq over the partitions enumerated by the last generating function, we therefore derive the bivariate generating function of partitions into l parts with the first one λ as follows: p l (n λ ) x l q n xq (qx; q) l,n0 Classifying the partitions of n into exactly l parts with each parts m according to the first part λ, we get the following expression l,n0 p l (n m) x l q n m 0 l,n0 +x m p l (n λ )x l q n q (qx; q) which is the second identity B3 Partitions into distinct parts and the Euler formula B3 Theorem Let Q m (n) be the number of partitions into exactly m distinct parts Its generating function reads as Q m (n) q n n0 q (+m 2 ) ( q)( q 2 ) ( q m ) (B3) Proof Let λ (λ >λ 2 >λ m > 0) be a partition enumerated by Q m (n) Based on λ, define another partition μ (μ μ 2 μ m 0) by μ : λ (m + ) for, 2,,m (B32)

30 20 CHU Wenchang and DI CLAUDIO Leontina It is obvious that μ is a partition of λ ( +m 2 ) into m parts As an example, the following figures show this correspondence between two partitions λ (9743) and μ (43) λ (9743) μ (43) It is not difficult to verify that the mapping (B32) is a bijection between the partitions of n with exactly m distinct parts and the partitions of n ( +m 2 ) with m parts Therefore the generating function of {Q m (n)} n is equal to that of {p m( n ( +m 2 ) ) } n, the number of partitions of n ( +m 2 ) with the number of parts m: Q m (n) q n n0 n0 q (+m 2 ) p m (n) q n n0 p m( n ( +m 2 ) ) q n q(+m 2 ) (q; q) m thans for the generating function displayed in (B24) This completes the proof of Theorem B3 Instead of the ordinary Ferrers diagram, we can draw a shifted diagram of λ as follows (see the figure) Under the first row of λ squares, we put λ 2 squares lined up vertically from the second column For the third row, we

31 Classical Partition Identities and Basic Hypergeometric Series 2 put λ 3 squares beginning from the third column Continuing in this way, the last row of λ m squares will be lined up vertically from the m-th column The shifted diagram of partition λ (9743) From the shifted diagram of λ, we see that all the partitions enumerated by Q m (n) have one common triangle on the left whose weight is ( +m 2 ) The remaining parts right to the triangle are partitions of n ( m+ 2 ) with m parts This reduces the problem of computing the generating function to the case just explained B32 Classifying all the partitions with distinct parts according to the number of parts, we get Euler s classical partition identity ( xq n ) + n0 m ( ) m x m q (m 2 ) ( q)( q 2 ) ( q m ) (B33) which can also be verified through the correspondence between partitions into distinct odd parts and self-conjugate partitions Proof Considering the bivariate generating function of Q m (n), we have ( + xq ) m0 x m n0 Q m (n)q n

32 22 CHU Wenchang and DI CLAUDIO Leontina Recalling (B3) and then noting that Q 0 (n) δ 0,n, we deduce that ( + xq ) + m x m q (+m 2 ) (q; q) m which becomes the Euler identity under parameter replacement x x/q In view of Euler s Theorem A2, we have a bijection between the partitions into distinct odd parts and the self-conjugate partitions The self-conjugate partition λ (65322) with the Durfee square 3 3 For a self-conjugate partition with the main diagonal length equal to m (which corresponds exactly to the length of partitions into distinct odd parts), it consists of three pieces: the first piece is the square of m m on the top-left with bivariate enumerator x m q m2, the second piece right to the square is a partition with m parts enumerated by /(q; q) m and the third piece under the square is in effect the conjugate of the second one Therefore the partitions right to the square and under the square m m are altogether enumerated by /(q 2 ; q 2 ) m Classifying the self-conjugate partitions according to the main diagonal length m, multiplying both generating functions together and summing m

33 Classical Partition Identities and Basic Hypergeometric Series 23 over 0 m, we find the following identity: ( + xq +2n ) n0 m0 x m q m2 (q 2 ; q 2 ) m where the left hand side is the bivariate generating function of the partitions into odd distinct parts It is trivial to verify that under replacements x xq /2 and q q /2 the last formula is exactly the identity displayed in (B33) Unfortunately, there does not exist the closed form for the generating function of Q m (n), numbers of partitions into m distinct parts B33 Dually, if we classify the partitions into distinct parts m according to their maximum part Then we can derive the following finite and infinite series identities m ( + q j x) +x j j m ( + q j x) +x ( + q i x) (B34a) q i ( + q i x) (B34b) q i Proof For the partitions into distinct parts with the maximum part equal to, their bivariate generating function is given by q x ( + q i x) which reduces to for 0 i Classifying the partitions into distinct parts m according to their maximum part with 0 m, weget ( qx; q) m +x m q ( qx; q) The second identity follows from the first one with m

34 24 CHU Wenchang and DI CLAUDIO Leontina B4 Partitions and the Gauss q-binomial coefficients B4 Lemma Let p l (n m) and p l (n m) be the numbers of partitions of n into l and l parts, respectively, with each part m Wehavethe generating functions: l,n 0 l,n 0 p l (n m) x l q n p l (n m) x l q n ( xq)( xq 2 ) ( xq m ) (B4a) ( x)( xq) ( xq m ) (B4b) The first identity (B4a) is a special case of the generating function shown in (Bb) On account of the length of partitions, we have p l (n m) p 0 (n m)+p (n m)+ + p l (n m) Manipulating the triple sum and then applying the geometric series, we can calculate the corresponding generating function as follows: l,n 0 p l (n m) x l q n l,n n0 x l p (n m)x l q n p (n m)q n 0 n0 x l l p (n m)x q n The last expression leads us immediately to the second bivariate generating function (B4b) in view of the first generating function (B4a) B42 The Gauss q-binomial coefficients as generating functions Let p l (n m) and p l (n m) be as in Lemma B4 The corresponding univariate generating functions read respectively as l + m p l (n m) q n q l (B42a) m n 0 l + m p l (n m) q n (B42b) m n 0

35 Classical Partition Identities and Basic Hypergeometric Series 25 where the q-gauss binomial coefficient is defined by m + n (q; q) m+n m (q; q) m (q; q) n q Proof For these two formulae, it is sufficient to prove only one identity because p l (n m) p l (n m) p l (n m) In fact, supposing that (B42b) is true, then (B42a) follows in this manner: p l (n m) q n p l (n m) q n p l (n m) q n n 0 n 0 n 0 l + m l +m l + m q l m m m q q q Now we should prove (B42b) Extracting the coefficient of x l from the generation function (B4b), we get p l (n m) q n x l (x; q) m+ n0 Observing that the function /(x; q) m+ is analytic at x 0 for q <, we can expand it into MacLaurin series: B (q)x (x; q) m+ where the coefficients { B (q) } are independent of x to be determinated Reformulating it under replacement x qx as B (q)x q (qx; q) m+ and then noting further that both fractions just displayed differ in factors ( x) and ( xq m+ ), we have accordingly the following: ( x) B (q)x ( xq m+ ) B (q)x q Extracting the coefficient of x l from both sides we get B l (q) B l (q) q l B l (q) q m+l B l (q) which is equivalent to the following recurrence relation B l (q) B l (q) qm+l q l for l, 2,

36 26 CHU Wenchang and DI CLAUDIO Leontina Iterating this relation l-times, we find that B l (q) B 0 (q) (qm+ ; q) l (q; q) l m + l l q where B 0 (q) follows from setting x 0 in the generating function (xq; q) m+ B (q)x 0 Therefore we conclude the proof B43 Theorem Classifying the partitions according to the number of parts, we derive immediately two q-binomial identities (finite and infinite): l + m q l m l + m x l m l0 l0 m + n + n (B43a) m xq (B43b) 0 Proof In view of (B42a) and (B42b), the univariate generating functions for the partitions into parts m + with the lengths equal to l and l + m n are respectively given by the q-binomial coefficients q l and m m + n + Classifying the partitions enumerated by the latter according to the number of parts l with 0 l n, we establish the first n identity By means of (B4b), we have m 0 xq x l l0 n0 p l (n m) q n l0 m + l x l l which is the second q-binomial identity

37 Classical Partition Identities and Basic Hypergeometric Series 27 B5 Partitions into distinct parts and finite q-differences Similarly, let Q l (n m) be the number of partitions of n into exactly l distinct parts with each part m Then we have generating functions m q (+l 2 ) Q l (n m) q n (B5) l n 0 m ( + xq ) Q l (n m) x l q n l,n 0 (B52) whose combination leads us to Euler s finite q-differences (x; q) n n ( xq l ) l0 n ( ) q 2) ( x 0 (B53) Following the second proof of Theorem B3, we can chec without difficulty that the shifted Ferrers diagrams of the partitions into l-parts m are unions of the same triangle of length l enumerated by q (l+ 2 ) and the ordinary partitions into parts m l with length l whose generating function m reads as the q-binomial coefficient The product of them gives the l generating function for {Q l (n m)} n The second formula is a particular case of (B2b) Its combination with the univariate generating function just proved leads us to the following: m m m ( qx; q) m Q l (n m) q n x l x l q (+l 2 ) l l0 n 0 Replacing x by x/q in the above, we get Euler s q-difference formula: (x; q) m m m ( ) l l l0 l0 q (l 2) x l Remar The last formula is called the Euler q-difference formula because if we put x : q n, the finite sum results in m m ( ) l l l0 q (l 2) ln (q n ; q) m just lie the ordinary finite differences of polynomials { 0, 0 n<m ( ) n q (n+ 2 ) (q; q)n, n m

38 28 CHU Wenchang and DI CLAUDIO Leontina Keep in mind of the q-binomial limit n (q+n ; q) as n (q; q) (q; q) Letting n in Euler s q-finite differences, we recover again the Euler classical partition identity ( ) x (x; q) q ( 2) (q; q) 0 where Tannery s theorem has been applied for the limiting process

39 CHAPTER C Durfee Rectangles and Classical Partition Identities For a partition λ, its Durfee square is the maximum square contained in the Ferrers diagram of λ It can be generalized similarly to the Durfee rectangles They will be used, in this chapter, to classify partitions and establish classical partition identities C q-series identities of Cauchy and Kummer: Unification C Theorem For the partitions into parts n, classify them with respect to the Durfee rectangles of ( + τ) for a fixed τ We can derive the following (qx; q) n n τ 0 n τ q (+τ) (qx; q) +τ x (C) Proof The partitions into parts n with Durfee rectangles of ( + τ) for a fixed τ are composed by three pieces One of them is the Durfee rectangle ( + τ) in common with enumerator x q (+τ) Another is the piece right to Durfee rectangle which are partitions of length with n τ parts n τ, whose univariate generating function is in view of (B42b) (only the univariate function is considered because the length of partitions has been counted by the Durfee rectangle) The last piece corresponds to the partitions with parts + τ whose bivariate generating function is /(qx; q) +τ Classifying the partitions into parts n with respect to Durfee rectangles of ( + τ) with 0 n τ, we find n τ n τ x q (+τ) (qx; q) n (qx; q) +τ 0 which is exactly the identity required in the theorem

40 30 CHU Wenchang and DI CLAUDIO Leontina Partition λ (986653) with Durfee rectangle ( + τ) 6 4 C2 Corollary The formula just established contains the following nown results as special cases: The finite version of Kummer s theorem (τ 0) n x q 2 (C2) (qx; q) n (qx; q) 0 The identity due to Gordon and Houten 968 (n ) x q (+τ) (qx; q) (q; q) (qx; q) +τ 0 which reduces further to the Cauchy formula with τ 0 (C3) C2 q-binomial convolutions and the Jacobi triple product C2 Theorem For the partitions into parts n, with at most α+γ n parts, classify them according to the Durfee rectangles of (n ) (α ) We obtain the first q-vandermonde convolution formula α + γ n 0 α γ n q (α )(n ) (C2)

41 Classical Partition Identities and Basic Hypergeometric Series 3 Proof The univariate generating function of the partitions into parts n α + γ with at most α + γ n parts is equal to by (B42b) Fixing n the Durfee rectangle of (n ) (α ) we see that the corresponding partitions into parts n with at most α + γ n parts consist of three pieces The first piece is the rectangle of (n ) (α ) on the topleft with univariate enumerator q (α )(n ) The second piece right to the rectangle is a partition into parts with at most α parts enumerated α by The third and the last piece under the rectangle is a partition into parts n with at most γ n + (α + γ n) (α ) parts γ enumerated by Classifying the partitions according to the Durfee n rectangles of (n ) (α ) and summing the product of three generating functions over 0 n, we find the following identity: α + γ n 0 α γ q (α )(n ) n Its limiting case q reduces to ( ) α + γ n 0 ( )( ) α γ n which is the well-nown Chu-Vandermonde convolution formula

42 32 CHU Wenchang and DI CLAUDIO Leontina Partition λ (65322) with Durfee rectangle (n ) (α ) 5 2 where n 6,α3, C22 Proposition Instead, considering the Durfee rectangle of (γ n) for the same partitions, we derive the second q-vandermonde convolution formula α + γ α + γ q (γ n) (C22) n n 0 Proof The univariate generating function of the partitions into parts n α + γ with at most α + γ n parts is equal to by (B42b) For a fixed n Durfee rectangle of (γ n) the corresponding partition into parts n with at most α + γ n parts consists of three pieces: the first piece is the rectangle of (γ n) on the top-left with univariate enumerator q (γ n), the second piece right to the rectangle is a partition into parts n with γ at most γ n parts enumerated by, where we can easily n justify that the partition length can not be γ n, otherwise, we would have a larger Durfee rectangle ( +) (γ n), and the third part under the rectangle is a partition into parts with at most α parts enumerated

43 Classical Partition Identities and Basic Hypergeometric Series 33 α + by Classifying the partitions with respect to Durfee rectangles of (γ n) and then summing the product of three generating functions over 0 n, we find the following identity: α + γ α + γ q (γ n) n n 0 For q, the limiting case reads as ( ) α + γ ( )( ) α + γ n n 0 which is another binomial convolution formula Partition λ (6532) with Durfee rectangle (γ n) 2 4 where n 6,γ 0,2 C23 Corollary Given the diagram of (m τ) (n+τ), consider the partitions contained in it The classification with respect to Durfee rectangles of ( + τ) leads us to the following finite summation formula m + n n + τ 0 m n q (+τ) + τ (C23) which is a special case of the first q-chu-vandermonde convolution formula

44 34 CHU Wenchang and DI CLAUDIO Leontina Proof For the partitions into parts m τ with at most n + τ parts, m + n the univariate generating function is equal to by (B42b) Fixing n + τ a Durfee rectangle of ( + τ), we observe that the partitions into parts m τ with at most n + τ parts consist of three pieces The first piece is the rectangle of ( + τ) on the top-left with univariate enumerator q (+τ) The second piece right to the rectangle is a partition into parts m m τ with at most + τ parts enumerated by and the + τ third one under the rectangle is a partition into parts with at most n n parts enumerated by Classifying the partitions according to the Durfee rectangles of ( + τ) for 0 n and then summing the product of three generating functions over 0 n, we find the following identity: m + n n + τ 0 m n q (+τ) + τ which is exactly the identity stated in the theorem We remar that this identity is a special case of the first q-vandermonde convolution formula stated in Theorem C2 In fact replacing n with l, we can state the reversal of the q-vandermonde convolution formula in Theorem C2 as follows: α + γ l l α γ q (α+ l) l 0 Performing parameter replacements α m, γ n and l m τ we obtain immediately the identity stated in Corollary C23

45 Classical Partition Identities and Basic Hypergeometric Series 35 Partition λ ( ) with Durfee rectangle ( + τ) 3 5 where 3,τ 2 C24 The Jacobi-triple product identity From the last q-binomial convolution identity, we can derive the following bilateral summation formula m (x; q) m (q/x; q) n ( ) q ( 2) m + n x (C24) n + n It can be considered as a finite form of the well-nown Jacobi triple product identity (q; q) (x; q) (q/x; q) + n ( ) n q (n 2) x n (C25) whose limiting case x reads as the cubic form of the triple product (Jacobi): (q; q) 3 ( ) n {+2n}q (+n 2 ) (C26) n0

46 36 CHU Wenchang and DI CLAUDIO Leontina Proof According to the Euler q-finite differences (B53), we have two finite expansions (x, q) m (q/x, q) n m m ( ) i i i0 n ( ) j j j0 Their product reads as the following double sum (x, q) m (q/x, q) n m i0 j0 m ( ) i+j i m ( ) x n n j0 q (i 2) x i q (+j 2 ) x j n q (i 2)+( +j 2 ) x i j j m n q (+j 2 )+( +j 2 ) + j j where the last line is justified by the replacement i j Observe that ( ) ( ) ( ) ( ) ( ) ( ) + j +j j +j j + + j(j + ) Reformulating the double sum and then applying the convolution formula stated in Corollary C23, we derive the finite bilateral summation formula (C24) (x, q) m (q/x, q) n m ( ) q ( 2) x n m ( ) q 2) ( n m n q j(j+) + j j j0 m + n n + x When m and n tend to infinity, the limit of q-binomial coefficient reads as m + n n + (q; q) m+n (q; q) n+ (q; q) m (q; q) Applying the Tannery Theorem, we therefore have (x, q) (q/x, q) + 2) x ( ) q( (q; q) which is equivalent to the Jacobi-triple product identity (C25)

47 Classical Partition Identities and Basic Hypergeometric Series 37 In order to prove (C26), we rewrite the Jacobi triple product identity as (q; q) (x; q) (q/x; q) + + n + n0 + n ( ) n q (+n 2 ) x n ( ) n q (+n 2 ) x n ( ) n q ( n 2 ) x n Replacing the summation index n by + m in the last sum: ( ) n q (n 2) x n n m0 ( ) m q (+m 2 ) x m+ we can combine two sums into one unilateral sum (q; q) (x; q) (q/x; q) ( ) n q (+n 2 ) { x n x n+} Dividing both sides by x, we get (q; q) (qx; q) (q/x; q) n0 ( ) n q (+n 2 ) x n x n+ x n0 Applying L Hôspital s rule for the limit, we have x n x n+ lim 2n + x x Considering that the series is uniformly convergent and then evaluating the limit x term by term, we establish (q; q) 3 n0 which is the cubic form of triple product ( ) n{ 2n + } q (+n 2 ) Remar The shortest proof of the Jacobi triple product identity is due to Cauchy (843) and Gauss (866) It can be reproduced in the sequel Recall the q-binomial theorem (finite q-differences) displayed in (B53) (x; q) l l l ( ) q 2) ( x 0

48 38 CHU Wenchang and DI CLAUDIO Leontina Replacing l by m + n and x by xq n respectively, and then noting the relation (q n x; q) m+n (q n x; q) n (x; q) m ( ) n q (+n 2 ) x n (q/x; q) n (x; q) m we can reformulate the q-binomial theorem as (x; q) m (q/x; q) n m+n 0 ( ) n m + n q ( n 2 ) x n which becomes, under summation index substitution n +, the following finite form of the Jacobi triple product identity m m + n (x; q) m (q/x; q) n ( ) q 2) ( x n + n This is exactly the finite form (C24) of the Jacobi triple product identity C25 Corollary From Jacobi s triple product identity, we may further derive the following infinite series identities: Triangle number theorem (Gauss) (q 2 ; q 2 ) (q; q 2 ) Pentagon number theorem (Euler) n0 q (+n 2 ) (q; q) + n ( ) n q n 2 (3n+) Proof Reformulate the factorial fraction in this way: (q 2 ; q 2 ) (q; q 2 ) (q; q) ( q; q) (q; q 2 ) (q 2 ; q 2 ) ( q; q) (q; q) ( q; q) ( q; q) 2 (q; q) ( ; q) ( q; q) Applying the Jacobi triple product identity, we have (q 2 ; q 2 ) (q; q 2 ) 2 + n q (n 2) { + + q (n 2) n { + n0 q (+n n0 2 ) + + n0 q ( n 2 ) } q (n+ 2 ) }

49 Classical Partition Identities and Basic Hypergeometric Series 39 where the substitution n +n has been made for the first sum and ( n 2 ) ( +n 2 ) for the second sum Canceling the factor /2 by two times of the same sum, we have the triangle number theorem Now, we prove pentagon number theorem Classifying the factors of product (q; q) according to the residues of the indices modulo 3, we have (q; q) (q 3 ; q 3 ) (q; q 3 ) (q 2 ; q 3 ) Then the Jacobi triple product identity (C25) yields (q; q) n ( ) n q 3(n 2)+n n ( ) n q n 2 (3n+) which is Euler s pentagon number theorem C26 The quintuple product identity Furthermore, we can derive the quintuple product identity q, z, q/z; q qz 2,q/z 2 ; q 2 and its limit form + n + n (q; q) 3 (q; q2 ) 2 + n { zq n } q 3(n 2) ( qz 3) n { z +6n } q 3(n 2) ( q 2 /z 3) n {+6n}q n 2 (3n+) C27 Proof Multiplying two copies of the Jacobi triple products q, z, q/z; q + i ( ) i q (i 2) z i q 2,qz 2,q/z 2 ; q 2 + ( ) j q j2 z 2j j we have the double sum expression q, z, q/z; q q 2,qz 2,q/z 2 ; q 2 + ( ) i+j q (i 2)+j 2 z i+2j i, j

50 40 CHU Wenchang and DI CLAUDIO Leontina Defining a new summation index i+2j and then rearranging the double sum, we can write q, z, q/z; q q 2,qz 2,q/z 2 ; q 2 + ( ) z + j ( ) j q ( 2j 2 )+j 2 Noting the binomial relation ( ) ( ) ( ) ( ) 2j 2j + + 2j +2j 2 + j 2j we find that q, z, q/z; q q 2,qz 2,q/z 2 ; q 2 + ( ) q 2) ( z + j ( ) j q 3j2 +j 2j Applying the Jacobi product identity to the inner sum, we get + j ( ) j q 3j2 +j 2j + j ( ) j q 6(j 2)+2(2 )j q 6,q 2+2,q 4 2,q 6 This product can be simplified according to the residues of modulo 3 3m with m Z: q 6,q 2+2,q 4 2,q 6 q 6,q 2+6m,q 4 6m,q 6 +3m with m Z: (q4 6m ; q 6 ) m (q 2 ; q 6 ) m q 6,q 2,q 4,q 6 ( ) m (q 2 ; q 2 ) q m 3m2 q 6,q 2+2,q 4 2,q 6 q 6,q 4+6m,q 2 6m,q 6 2+3m with m Z: (q2 6m ; q 6 ) m (q 4 ; q 6 ) m q 6,q 2,q 4,q 6 ( ) m (q 2 ; q 2 ) q m 3m2 q 6,q 2+2,q 4 2,q 6 q 6,q 6+6m,q 6m,q 6 0

51 Classical Partition Identities and Basic Hypergeometric Series 4 because of the presence of zero-factor: (q 6m ; q) 0, m 0 (q 6+6m ; q) 0, m < 0 Substituting these results into the infinity series expression, we obtain q, z, q/z; q q 2,qz 2,q/z 2 ; q 2 + (q 2 ; q 2 ) + ( ) q ( 2) z q 6,q 2+2,q 4 2,q 6 m (q 2 ; q 2 ) + m (q 2 ; q 2 ) + m q (3m 2 )+m 3m 2 z 3m q (+3m 2 ) m 3m 2 z +3m q 3m2 m { } 2 zq m z 3m Dividing both sides by (q 2 ; q 2 ), we get the quintuple product identity: q, z, q/z; q qz 2,q/z 2 ; q 2 + m q 3(m 2 ) { zq m} (qz 3 ) m Splitting the last sum into two and then reverse the first sum, we have q, z, q/z; q qz 2,q/z 2 ; q 2 + m + m + n + n q 3(m 2 ) { zq m} (qz 3 ) m q 3(m 2 )+m z 3m q 3(n 2)+2n z 3n + m + m q 3(n 2) { z +6n}( q 2 /z 3) n q 3(m 2 )+2m z +3m q 3(m 2 )+2m z +3m which is exactly the second version of the quintuple product identity

52 42 CHU Wenchang and DI CLAUDIO Leontina Finally, dividing both sides by z q, qz, q/z; q qz 2,q/z 2 ; q 2 + n q 3(n 2) z+6n ( q 2 /z 3) n z and then letting z, we get the limiting case of the quintuple product identity (q; q) 3 (q; q 2 ) 2 + n {+6n} q n 2 (3n+) C3 The finite form of Euler s pentagon number theorem C3 Theorem The classification of partitions enumerated by ( qx; q) n with respect to the Durfee rectangles of ( + ɛ) leads us to the following finite form of the Euler pentagon number theorem Denote by θ the integral part of real number θ Then there holds ( qx; q) n n ɛ 2 0 q (+ɛ)+( 2) n ɛ ( qx; q) +ɛ +xq2+ɛ q +n ɛ ( + xq +ɛ ) ( + xq +ɛ )( q +n 2 ɛ ) x C32 Proof For the partitions into distinct parts n enumerated by ( qx; q) n, they are divided by the Durfee rectangles of ( +ɛ) into three pieces: A: the Durfee rectangle ( + ɛ) itself with enumerator x q (+ɛ) B: the piece of partitions right to the Durfee rectangle counted by n ɛ q (+ 2 ), with parts, n ɛ q 2) (, with parts C: the piece of partitions below the Durfee rectangle enumerated by { ( qx; q) +ɛ, when B has parts, ( qx; q) +ɛ, when B has parts

53 Classical Partition Identities and Basic Hypergeometric Series 43 Therefore for the fixed Durfee rectangle A, the enumerator for the rest of partitions is given by the combination of B and C as follows q (+ 2 ) n ɛ ( qx; q) +ɛ + q 2) n ɛ ( ( qx; q) +ɛ q ( 2) n ɛ +xq ɛ+2 q +n ɛ (+xq +ɛ ) ( + xq +ɛ )( q +n 2 ɛ ( qx; q) +ɛ ) Summing the last expression over 0 (n ɛ)/2, we get the identity stated in Theorem C3 Partition λ (86542) with Durfee rectangle ( + ɛ) 4 3 where ɛ,3 C33 Corollary This formula contains the following well-nown results as special cases: The limiting version with two parameters (n ) ( qx; q) n0 q n(n+ɛ)+(n 2) +xq2n+ɛ +xq n+ɛ ( qx; q) n+ɛ (q; q) n x n The Sylvester formula (ɛ,x y/q and n ) (y; q) ( y) n { yq 2n } (y; q) n q 3n2 n (q; q) n n0 2