Generating function for number of partitions with Rank and Crank

Question

The generating function for the number of partitions, $p(n)$ of a number is given by $\displaystyle{\sum_{n=1}^{\infty}p(n)q^n=\frac{1}{(q;q)_{\infty}}}$ where $(q;q)_{\infty}=\displaystyle{\lim_{n \to \infty}\prod_{k = 1}^{n}(1-q^k)}$. I understand this generating function $\frac{1}{(q;q)_{\infty}}$ how it can give us $p(n)$. But I don't understand how the number of partitions with a $Rank$ & $Crank$ generating functions emerged. $Rank$ of a partition is the largest part minus the number of parts in the partition. $Crank$ of a partition is defined as : For a partition $λ$, let $ℓ(λ)$ denote the largest part of $λ$, $ω(λ)$ denote the number of $1$'s in $λ$, and $μ(λ)$ denote the number of parts of $λ$ larger than $ω(λ)$. The $Crank$ $c(λ)$ is given by $c(\lambda)=\begin{cases} \ell(\lambda) & \text{if } \omega(\lambda)=0 \\ \mu(\lambda)-\omega(\lambda) &\text{if } \omega(\lambda)>0\end{cases}$. $N(m,n)$ and $M(m,n)$ are used to denote the number of partitions with $Rank$ $m$ and $Crank$ $m$ respectively. Then, $$\displaystyle{\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}N(m,n)z^mq^n=1+\displaystyle{\sum_{n=1}^{\infty}}}\frac{q^{n^2}}{(zq;q)_{n}(\frac{z}{q};q)_{n}}$$ and $$\displaystyle{\sum_{m=-\infty}^{\infty}\sum_{n=0}^{\infty}M(m,n)z^mq^n=\frac{(q;q)_{\infty}}{(zq;q)_{\infty}(\frac{z}{q};q)_{\infty}}}$$ I want to know how the RHSs of both of the above emerge.