Remarks on MacMahon's $q$-series

Abstract

In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k} \frac{q^{s_1+s_2+\cdots+s_k}}{(1-q^{s_1})^2(1-q^{s_2})^2\cdots(1-q^{s_k})^2},\\ C_k(q)=\sum_{n=1}^{\infty} \mathfrak{m}_{odd}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdots<s_k} \frac{q^{2s_1+2s_2+\cdots+2s_k-k}}{(1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots(1-q^{2s_k-1})^2}. \end{align*} These series give infinitely many formulas for two prominent generating functions. For each non-negative $k$, we prove that $A_k(q), A_{k+1}(q), A_{k+2}(q),\dots$ (resp. $C_k(q), C_{k+1}(q), C_{k+2}(q),\dots$) give the generating function for the 3-colored partition function $p_3(n)$ (resp. the overpartition function $\overline{p}(n)$).