New Partition Theoretic Interpretations of Rogers-Ramanujan Identities

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, G？llnitz [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 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