International Journal of CombinatoricsVolume 2012 (2012), Article ID 409505, 6 pagesdoi:10.1155/2012/409505

Research Article

New Partition Theoretic Interpretations of Rogers-Ramanujan Identities

A. K. Agarwal and M. Goyal

Center for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India

Received 13 January 2012; Revised 4 March 2012; Accepted 5 March 2012

Academic Editor: Toufik Mansour

Copyright © 2012 A. K. Agarwal and M. Goyal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The generating function for a restricted partition function is derived. This in conjunction with two identities of Rogers provides new partition theoretic interpretations of Rogers-Ramanujan identities.

1. Introduction, Definitions, and the Main Results

The following two “sum-product” identities are known as Rogers-Ramanujan identities where and is a rising -factorial defined by for any constant .

If is a positive integer, then obviously They were first discovered by Rogers [1] and rediscovered by Ramanujan in 1913. MacMahon [2] gave the following partition theoretic interpretations of (1.1), respectively.

Theorem 1.1. The number of partitions of into parts with the minimal difference 2 equals the number of partitions of into parts .

Theorem 1.2. The number of partitions of with minimal part 2 and minimal difference 2 equals the number of partitions of into parts .

Theorems 1.1-1.2 were generalized by Gordon [3], and Andrews [4] gave the analytic counterpart of Gordon’s generalization. Partition theoretic interpretations of many more -series identities like (1.1) have been given by several mathematicians. See, for instance, Göllnitz [5, 6], Gordon [7], Connor [8], Hirschhorn [9], Agarwal and Andrews [10], Subbarao [11], Subbarao and Agarwal [12].

Our objective in this paper is to provide new partition theoretic interpretations of identities (1.1) which will extend Theorems 1.1 and 1.2 to 3-way partition identities. In our next section, we will prove the following result.

Theorem 1.3. For a positive integer , let denote the number of partitions of such that the smallest part (or the only part) is ≡ k, and the difference between any two parts is . Then Theorem 1.3 in conjunction with the following two identities of Rogers [1, p.330] and [13, p.331] (see also Slater [14, Identities (20) and (16)]) extends Theorems 1.1 and 1.2 to the following 3-way partition identities, respectively.

Theorem 1.4. Let denote the number of partitions of into parts , let denote the number of partitions of with minimal difference 2, and let denote the number of partitions of into parts . Then where is as defined in Theorem 1.3.

Example 1.5. , since the relevant partitions are 9, , , , . , since the relevant partitions are 9, , , , .Also, Table 1 shows the relevant partitions enumerated by and for .

Table 1

Partitions enumerated by

Partitions enumerated by

0
1
empty partition
1
empty partition

1
1
1
0
—

2
0
—
1
2

3
0
—
0
—

4
1

1

5
1
5
0
—

6
0
—
2

7
0
—
0
—

8
1

2

9
2

0
—

Theorem 1.6. Let denote the number of partitions of such that the parts are ≥2, and the minimal difference is 2. Let denote the number of partitions of into parts . Then where is as defined in Theorem 1.3 and as defined in Theorem 1.4.

Example 1.7. , since the relevant partitions are 9, , . , since the relevant partitions are , , . Also, Table 2 shows the relevant partitions enumerated by for .

Table 2

Partitions enumerated by

0
1
empty partition

1
0
—

2
0
—

3
1
3

4
0
—

5
0
—

6
0
—

7
1
7

8
1

9
0
—

2. Proof of the Theorem 1.3

Let denote the number of partitions of enumerated by into parts. We shall first prove that To prove the identity (2.1), we split the partitions enumerated by into two classes: (i)those that have least part equal to ,(ii)those that have least part greater than .

We now transform the partitions in class (i) by deleting the least part and then subtracting 2 from all the remaining parts. This produces a partition of into exactly parts, each of which is ≥ (since originally the second smallest part was ≥); furthermore, since this transformation does not disturb the inequalities between the parts, we see that the transformed partition is of the type enumerated by .

Next, we transform the partitions in class (ii) by subtracting 4 from each part. This produces a partition of into parts, each of which is ≥, as in the first case; here too, the inequalities between the parts are not disturbed, we see that the transformed partition is of the type enumerated by .

The above transformations establish a bijection between the partitions enumerated by and those enumerated by .

This proves the identity (2.1).

For and , let Substituting for from (2.1) in (2.2) and then simplifying, we get Since , we may easily check by coefficient comparison in (2.3) that Now This completes the proof of Theorem 1.3.

3. Conclusion

In this paper MacMahon’s Theorems 1.1 and 1.2 have been extended to 3-way identities. The most obvious question which arises from this work is the following: Does Gordon’s generalization of Theorems 1.1 and 1.2 also admit similar extension? We must add that different partition theoretic interpretations of identities (1.6) are found in the literature (see for instance [15, 16]).

Acknowledgment

The authors thank the referees for their helpful comments. A. K. Agarwal is an Emeritus Scientist, CSIR.

References

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