﻿ Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in <i>q</i>-Series

Open Journal of Discrete Mathematics
Vol.4 No.2(2014), Article ID:44842,8 pages DOI:10.4236/ojdm.2014.42006

Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in -Series

Sandy H. L. Chen1,2

1School of Science, Tianjin Chengjian University, Tianjin, China

2Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, China

Email: chenhuanlin@mail.nankai.edu.cn

Received 21 January 2014; revised 19 February 2014; 17 March 2014

ABSTRACT

For every partition and its conjugation, there is an important invariant, which denotes the number of different parts. That is,. We will derive a series of symmetric -identities from the invariant in partition conjugation by studying modified Durfee rectangles. The extensive applications of the several symmetric -identities in -series [1] will also be discussed. Without too much effort one can obtain much well-known knowledge as well as new formulas by proper substitutions and elementary calculations, such as symmetric identities, mock theta functions, a two-variable reciprocity theorem, identities from Ramanujan’s Lost Notebook and so on.

Keywords:Integer Partitions, Conjugation, Invariant, -Series, Symmetric Identities

1. Definitions and Combinatorial Interpretations

We shall first present some basic definitions and combinatorial interpretations for basic hypergeometric series and integer partition. For simplicity, unless stated otherwise we shall assume that is a nonnegative integer and, let

Definition 1.1 For any integer n, the q-shifted factorial is defined by

Definition 1.2 A partition of a positive integer is a finite nonincreasing sequence of integers, such that. is the conjugate partition of. The largest part, the number of parts, as well as the sum of the parts are denoted by, and, respectively.

An effective device for studying partitions is the graphical representation. For a partition, its Durfee rectangle is the maximum rectangle contained in the Ferrers diagram of. Conjugation and the several invariants have been used in a variety of ways over the years, see Andrews’s encyclopedia [2] . It is worth pointing out that there is a fundamental invariant which despite its simplicity has not received too much attention. This is, the number of different parts of. For all partitions and its conjugation,

(1)

In this paper, we shall show how (1) could be used to obtain a series of symmetric identities by studying modified Durfee rectangles. Consider the expansion

We interpret this as an expansion involving only one part, namely, where the power of records, while that of indicates whether the part occurs or not. Thus, we interpret

(2)

as the generating function of partitions into parts less than or equal to, such that the power of records, while that of indicates. Then it follows that the three-parameter generating function for all unrestricted partitions, namely, the function

(3)

We consider all partitions for which. This accounts for the term in (3). Since contains as a part, we have the factors and in the numerator. The part may repeat, which is given by. The repetition of will not contribute to and so there is no further power of

contributed by the part. The part could repeat and their contribution to the generating function is given by the term

Formula (3) follows.

2. Symmetric Expressions for

In this section, we give several symmetric expansions for via modified Durfee rectangles analysis of partition.

Theorem 2.1

Proof. For every partitions and, the Ferrers graph contains a largest Durfee rectangle with side horizontally by vertically. Then to the right of the Durfee rectangle, we have a partition which has at most parts, or equivalently, , the conjugate of, with largest part. Below the Durfee rectangle we have a partition whose parts. We now divide our consideration into four cases.

1)

We consider the contribution of the partition to by utilizing its conjugate.

The factor in the numerator arises from the column of length lying to the right of the Durfee rectangle. Because contributes nothing to, we omit.

The contribution of the partition to is

Note that the parameter is absent because the partition has no contribution to.

Meanwhile, the contribution of the modified Durfee rectangle to is

Thus, we derive the generating function of every unrestricted partitions:

2)

The generating function of every unrestricted partitions:

3)

The generating function of every unrestricted partitions:

4)

The generating function of every unrestricted partitions:

Summing these four generating functions for, we get an expression for:

Remark 2.2 Under partition conjugation, and are interchanged, it follows that and are symmetric in.

Theorem 2.3 From formula (3) and the symmetry of and, we have

(4)

(5)

Theorem 2.4 From Theorem 2.1 and the symmetry of and, we have

(6)

(7)

(8)

where (8) results from the Durfee square analysis.

3. The Applications of the Symmetric Identities in q-Series

In this section, we shall explore the extensive applications of formulas (4) to (8) in -series. Without too much effort one can obtain much well-know knowledge as well as new formulas by proper substitutions and elementary calculations. It will be overly clear that the list of nice application is sheer endless.

3.1. Symmetric Identities

From (4) and (5), we get the following beautiful symmetric identity.

Corollary 3.1

(9)

Taking in (9), we derive the following identity, from which Liu [3] proved an identity of Andrews.

Corollary 3.2

(10)

Setting and then taking in (4) and (5), we have

Corollary 3.3

(11)

which was first stated and proved by N. J. Fine [4] . Andrews derived it combinatorially from the consideration of partitions without repeated odd parts in [5] .

3.2. Mock Theta Functions

In his famous last letter to Hardy [6] , Ramanujan introduced 17 mock theta functions without giving an explicit definition, among which, one third order mock theta function is as follows

(12)

In 1966, Andrews [7] defined the following generalization of

(13)

Moreover, Watson [8] added three functions to the list of Ramanujan’s third order mock theta functions and the following identity is just one of them

By proper substitutions in Theorem 2.3 and 2.4, we obtain much simpler expressions for the above mock theta functions. Through the specializations and in (4) and (8), we derive a simpler transformation formula for:

Corollary 3.4

(14)

Taking in (14), a representation for follows, with the powers diminished

(15)

Fine [1] first derived (15) by applying some transformation formulas and Liu [[9] , Theorem 3.7] proved it combinatorially by an application of involution. Changing to and then putting in (4) and (6), we get a new expression for, with the powers diminished:

Corollary 3.5

(16)

3.3. A Two-Variable Reciprocity Theorem

Taking and then letting in (4), we have

In his lost notebook [10] , Ramanujan offers a beautiful reciprocity theorem

(17)

After the same substitutions in (5) and (7), respectively, we get

Corollary 3.6

(18)

(19)

Formula (18) is a slightly simpler representation of. From (19) and the above reciprocity theorem (17), we get the following two variable generalization of the Quintuple Product Identity [[11] , Theorem 3.1] without any proof:

Corollary 3.7 A Two-Variable Generalization of the Quintuple Product Identity

For, ,

(20)

3.4. Identities from Ramanujan’s Lost Notebook

By special substitutions, we could go through a series of important Entries in Ramanujan’s Lost Notebook [12] . We take several of them as examples, for their combinatorial proofs, see [13] . The function

is defined by Ramanujan. Setting and then letting, then (4) and (6) can be reduced to

Corollary 3.8 (Entry 9.2.2)

The same substitutions in (4) and (8), we have

Corollary 3.9 (Entry 9.2.3)

Putting and, and then setting in (4) and (8), we have

Corollary 3.10 (Entry 9.2.4)

For the above identity, it is interesting to note that the terms in and on the right side are the same as those on the left side, but with the powers diminished. In (4) and (6), we replace by and take and, and then set, the Entry 9.2.5 in Ramanujan’s Lost Notebook [12] follows:

Corollary 3.11 (Entry 9.2.5)

Berndt and Yee [13] proved the above two corollaries combinatorially by accounting for partitions into distinct parts. Replacing by and taking and in (4) and (6), then we get the following Entry. Berndt and Yee [13] derived it by employing -modular partitions.

Corollary 3.12 ( Entry 9.3.1)

In (4) and (6), we take, and then set, Entry 9.4.1 follows:

Corollary 3.13 (Entry 9.4.1)

(21)

This identity was derived from Franklin involution by Berndt and Yee [13] and was also got from two entries by Warnaar [14] , where analytic methods were employed.

3.5. Further Consequences

Corollary 3.14

(22)

Proof. Taking, and then letting and in (4) and (8), we have

(23)

Identity (22) is a false theta series identity. Results like these were studied by L. J. Rogers [15] , however, the elegant result appears to have escaped him. Andrews [16] proved identity (22) by using three transformation formulas and showed that (22) implied a partition identity like that deduced from Euler’s Pentagonal Number Theorem ([2] , p. 10).

Taking and in (4) and (5), we generalize the not at all deep but elegant identity:

Corollary 3.15

(24)

Taking, and then setting, (4) and (8) can be reduced to the famous Gauss triangle series

Corollary 3.16

(25)

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