Remarks on MacMahon's q-series

Abstract

In his important 1920 paper on partitions, MacMahon defined the partition generating functions$A_k\left(q\right)=\sum _{n=1}^{\infty }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\left(q\right)=\sum _{n=1}^{\infty }{m}_{\mathrm{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}.$These series give infinitely many formulas for two prominent generating functions. For each non-negative k, we prove that $A_k\left(q\right),A_{k+1}\left(q\right),A_{k+2}\left(q\right),\dots$ (resp. $C_k\left(q\right),C_{k+1}\left(q\right),C_{k+2}\left(q\right),\dots$) give the generating function for the 3-colored partition function $p_3\left(n\right)$ (resp. the overpartition function $\bar{p}\left(n\right)$).