Rogers-Ramanujan Identities

Abstract

The following two “sum-product” identities

$$$$

((1.1))

and

$$$$

((1.2))

where

$${\left( {a;q} \right)_n} = \prod\limits_{i = 0}^\infty {\frac{{1 - a{q^i}}}{{1 - a{q^{n + 1}}}}} {\text{ }}$$

are known as Rogers-Ramanujan identities. (Note that if n is a positive integer, then (a;q) _n = Π ^{n − 1}_{i = 0} (1 − aqⁱ); (a;q)₀ = 1.) They were first discovered by Rogers in 1894. After two decades they were rediscovered by Ramanujan and Schur, independently. MacMahon [12, Theorems 364, 365 p.291], gave the following combinatorial interpretations of (1.1) and (1-2), respectively:

• Theorem 1.1. The number of partitions of n into parts with minimal difference 2 equals the number of partitions of n into parts which are congruent to ±1 (mod 5).

• 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 which are congruent to ±2 (mod 5).