Truncated forms of MacMahon's q-series

Abstract

In 1920, Percy Alexander MacMahon defined the partition generating functions$A_k\left(q\right):=\sum _{0<n_1<n_2<\cdots <n_k}\frac{q^{n_1+n_2+\cdots +n_k}}{{(1-q^{n_1})}^2{(1-q^{n_2})}^2\cdots {(1-q^{n_k})}^2}$ and$C_k\left(q\right):=\sum _{0<n_1<n_2<\cdots <n_k}\frac{q^{2n_1+2n_2+\cdots +2n_k-k}}{{(1-q^{2n_1-1})}^2{(1-q^{2n_2-1})}^2\cdots {(1-q^{2n_k-1})}^2}$ which have since played an important role in combinatorial mathematics. For each non-negative integer k, George E. Andrews and Simon C. F. Rose proved that $A_k\left(q\right)$ can be expressed considering the generating function of partitions where each part can be colored in one of three different colors, while $C_k\left(q\right)$ can be expressed considering the generating function of overpartitions. Recently, for each non-negative integer k, Ken Ono and Ajit Singh proved that $A_k\left(q\right)$, $A_{k+1}\left(q\right)$, $A_{k+2}\left(q\right),\dots$ give the generating function for the number of partitions of n where each part can be colored in one of three different colors, while $C_k\left(q\right)$, $C_{k+1}\left(q\right)$, $C_{k+2}\left(q\right),\dots$ give the generating function for the number of overpartitions of n. In this paper, we provide the truncated versions of these results. Some open problems are introduced in this context.