Rogers-Ramanudjan Identities

Lemma 8  

$$\frac 1 {1-q^2} \frac 1 {1-q^3} \frac 1 {1-q^7} \frac 1 {1-q^8} \dots=1+\sum_{n=1}^\infty \frac{q^{n(n+1)}}{(1-q)(1-q^2)\dots(1-q^n)},$$

$$\frac 1 {1-q^1} \frac 1 {1-q^4} \frac 1 {1-q^6} \frac 1 {1-q^9} \dots=1+\sum_{n=1}^\infty \frac{q^{n^2}}{(1-q)(1-q^2)\dots(1-q^n)}.$$

Here in LHS of the first identity we have powers of $q$ which have remainders 2 or 3 mod 5 and in LHS of the second identity the powers have remainders 1 or 4 mod 5.

Exercise:Reformulate Rogers-Ramanudjan identities in the language of partitions. $\;\Box$