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 Correspondence should be addressed to A. K. Agarwal, aka@pu.ac.in Received 13 January 2012; Revised 4 March 2012; Accepted 5 March 2012 Academic Editor: Toufik Mansour Copyright q 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. 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.


Introduction, Definitions, and the Main Results
The following two "sum-product" identities are known as Rogers-Ramanujan identities for any constant a.

International Journal of Combinatorics
If n is a positive integer, then obviously a; q n 1 − a 1 − aq · · · 1 − aq n−1 , 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.  Gordon 3 , and Andrews 4 gave the analytic counterpart of Gordon's generalization. Partition theoretic interpretations of many more qseries 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. 1.5 [1, p.330] and [13, p.331] (see also Slater [14,Identities (20) and (16)

1.6
extends Theorems 1.1 and 1.2 to the following 3-way partition identities, respectively.

Proof of the Theorem 1.3
Let A k m, n denote the number of partitions of n enumerated by A k n into m parts. We shall first prove that To prove the identity 2.1 , we split the partitions enumerated by A k m, n into two classes: i those that have least part equal to k, ii those that have least part greater than k.
We now transform the partitions in class i by deleting the least part k and then subtracting 2 from all the remaining parts. This produces a partition of n − k − 2 m − 1 into exactly m−1 parts, each of which is ≥k since originally the second smallest part was ≥k 2 ; furthermore, since this transformation does not disturb the inequalities between the parts, we see that the transformed partition is of the type enumerated by A k m − 1, n − k − 2m 2 .
Next, we transform the partitions in class ii by subtracting 4 from each part. This produces a partition of n − 4m into m parts, each of which is ≥k, 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 A k m, n − 4m .
The above transformations establish a bijection between the partitions enumerated by A k m, n and those enumerated by A k m − 1, n − k − 2m 2 A k m, n − 4m .
This proves the identity 2.

2.5
This completes the proof of Theorem 1.3.

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 .