New Partition Theoretic Interpretations of Rogers-Ramanujan Identities

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^∞_𝑛=0𝑞^𝑛2𝑞;𝑞_𝑛=^∞_𝑛=11−𝑞^{5𝑛−1−1}1−𝑞^{5𝑛−4−1},_∞𝑛=0𝑞^𝑛2+𝑛𝑞;𝑞_𝑛=^∞_𝑛=11−𝑞^{5𝑛−2−1}1−𝑞^{5𝑛−3−1}, (1.1) where |𝑞|<1 and (𝑞;𝑞)_𝑛 is a rising 𝑞-factorial defined by𝑎;𝑞_𝑛=^∞_𝑖=01−𝑎𝑞^𝑖1−𝑎𝑞^𝑛+𝑖, (1.2) for any constant 𝑎.

If 𝑛 is a positive integer, then obviously𝑎;𝑞_𝑛=(1−𝑎)1−𝑎𝑞⋯1−𝑎𝑞^𝑛−1,𝑎;𝑞_∞=(1−𝑎)1−𝑎𝑞1−𝑎𝑞²⋯. (1.3)(1.4) 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 𝑛 into parts with the minimal difference 2 equals the number of partitions of 𝑛 into parts ≡±1(mod5).

Theorem 1.2. The number of partitions of 𝑛 with minimal part 2 and minimal difference 2 equals the number of partitions of 𝑛 into parts ≡±2(mod5).

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 𝑞-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 𝑘, let 𝐴_𝑘(𝑛) denote the number of partitions of 𝑛 such that the smallest part (or the only part) is ≡ k(mod4), and the difference between any two parts is ≡2(mod4). Then ^∞_𝑛=0𝐴_𝑘(𝑛)𝑞^𝑛=^∞_𝑛=0𝑞^{𝑛(𝑛+𝑘−1)}𝑞⁴;𝑞⁴_𝑛. (1.5) 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)]) ^∞_𝑛=0𝑞^𝑛2𝑞⁴;𝑞⁴_𝑛=−𝑞;𝑞²_∞𝑞²;𝑞²_∞^∞_𝑛=11−𝑞^5𝑛1−𝑞^5𝑛−21−𝑞^5𝑛−3,_∞𝑛=0𝑞^𝑛2+2𝑛𝑞⁴;𝑞⁴_𝑛=−𝑞;𝑞²_∞𝑞²;𝑞²_∞^∞_𝑛=11−𝑞^5𝑛1−𝑞^5𝑛−11−𝑞^5𝑛−4, (1.6) extends Theorems 1.1 and 1.2 to the following 3-way partition identities, respectively.

Theorem 1.4. Let 𝐵(𝑛) denote the number of partitions of 𝑛 into parts ≡2(mod4), let 𝐶(𝑛) denote the number of partitions of 𝑛 with minimal difference 2, and let 𝐷(𝑛) denote the number of partitions of 𝑛 into parts ≡±1(mod5). Then 𝐶(𝑛)=𝐷(𝑛)=^𝑛_𝑟=0𝐴₁(𝑟)𝐵(𝑛−𝑟), (1.7) where 𝐴₁(𝑟) is as defined in Theorem 1.3.

Example 1.5. 𝐶(9)=5, since the relevant partitions are 9, 8+1, 7+2, 6+3, 5+3+1. 𝐷(9)=5, since the relevant partitions are 9, 6+1+1+1, 4+4+1, 4+1+1+1+1+1, 1+1+1+1+1+1+1+1+1.
Also, ⁹_𝑟=0𝐴₁(𝑟)𝐵(9−𝑟)=5. (1.8) Table 1 shows the relevant partitions enumerated by 𝐴₁(𝜈) and 𝐵(𝜈) for 0≤𝜈≤9.

Table 1 

Theorem 1.6. Let 𝐸(𝑛) denote the number of partitions of 𝑛 such that the parts are ≥2, and the minimal difference is 2. Let 𝐹(𝑛) denote the number of partitions of 𝑛 into parts ≡±2(mod5). Then 𝐸(𝑛)=𝐹(𝑛)=^𝑛_𝑟=0𝐴₃(𝑟)𝐵(𝑛−𝑟),∀𝑛, (1.9) where 𝐴₃(𝑟) is as defined in Theorem 1.3 and 𝐵(𝑛) as defined in Theorem 1.4.

Example 1.7. 𝐸(9)=3, since the relevant partitions are 9, 7+2, 6+3. 𝐹(9)=3, since the relevant partitions are 7+2, 3+3+3, 3+2+2+2. Also, ⁹_𝑟=0𝐴₃(𝑟)𝐵(9−𝑟)=3. (1.10) Table 2 shows the relevant partitions enumerated by 𝐴₃(𝜈) for 0≤𝜈≤9.

Table 2 

2. 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 𝐴_𝑘(𝑚,𝑛)=𝐴_𝑘(𝑚−1,𝑛−𝑘−2𝑚+2)+𝐴_𝑘(𝑚,𝑛−4𝑚). (2.1) 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 (2.1).

For |𝑞|<1 and |𝑧|<|𝑞|⁻¹, let 𝑓_𝑘𝑧,𝑞=^∞∞_𝑛=0𝑚=0𝐴_𝑘(𝑚,𝑛)𝑧^𝑚𝑞^𝑛. (2.2) Substituting for 𝐴_𝑘(𝑚,𝑛) from (2.1) in (2.2) and then simplifying, we get 𝑓_𝑘𝑧,𝑞=𝑧𝑞^𝑘𝑓_𝑘𝑧𝑞²,𝑞+𝑓_𝑘𝑧𝑞⁴,𝑞. (2.3) Since 𝑓_𝑘(0,𝑞)=1, we may easily check by coefficient comparison in (2.3) that 𝑓_𝑘𝑧,𝑞=^∞_𝑛=0𝑞^{𝑛(𝑛+𝑘−1)}𝑧^𝑛𝑞⁴;𝑞⁴_𝑛. (2.4) Now ^∞_𝑛=0𝐴_𝑘(𝑛)𝑞^𝑛=^∞_𝑛=0^∞_𝑚=0𝐴_𝑘(𝑚,𝑛)𝑞^𝑛=𝑓_𝑘1,𝑞=^∞_𝑛=0𝑞^{𝑛(𝑛+𝑘−1)}𝑞⁴;𝑞⁴_𝑛. (2.5) 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.