Ramanujan's Partition Congruences
Based on the empirical analysis of a table of partitions Ramanujan conjectured his
famous partition congruences $$\boxed{\begin{align}p(5n + 4)&\equiv 0\pmod{5}\notag\\
p(7n + 5)&\equiv 0\pmod{7}\notag\\
p(11n + 6)&\equiv 0\pmod{11}\notag\end{align}}\tag{1}$$ and gave some of the most beautiful proofs for them (see
here). In addition to these proofs he gave the following generating functions for $p(5n + 4), p(7n + 5)$: $$\sum_{n = 0}^{\infty}p(5n + 4)q^{n} = 5\frac{\{(1 - q^{5})(1 - q^{10})(1 - q^{15})\cdots\}^{5}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{6}}\tag{2}$$ and
\begin{align}\sum_{n = 0}^{\infty}p(7n + 5)q^{n}&= 7\frac{\{(1 - q^{7})(1 - q^{14})(1 - q^{21})\cdots\}^{3}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{4}}\notag\\
&\,\,\,\,\,\,\,\,+ 49q\frac{\{(1 - q^{7})(1 - q^{14})(1 - q^{21})\cdots\}^{7}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{8}}\tag{3}\end{align} We have already established $(2)$ in
one of our posts and this post deals with the identity $(3)$ concerning generating function of partitions modulo $7$.
