Remarks on MacMahon's q-series

In his important 1920 paper on partitions, MacMahon defined the partition generating functions {align*} A_k(q)=∑ₙ₌₁∞m(k;n)q^n&:=∑0< s_1<s_2<⋯<s_k {qs_1+s_2+⋯+s_k}{(1-qs_1)^2(1-qs_2)^2⋯(1-qs_k)^2},\\ C_k(q)=∑ₙ₌₁∞ modd(k;n)q^n&:=∑0< s_1<s_2<⋯<s_k {q2s_1+2s_2+⋯+2s_k-k}{(1-q2s_1-1)^2(1-q2s_2-1)^2⋯(1-q2s_k-1)^2}. {align*} These series give infinitely many formulas for two prominent generating functions. For each non-negative k, we prove that Aₖ(q), Aₖ₊₁(q), Aₖ₊₂(q), (resp. Cₖ(q), Cₖ₊₁(q), Cₖ₊₂(q),) give the generating function for the 3-colored partition function p₃(n) (resp. the overpartition function p̄(n)). To be precise, we have {align*} &∑ₙ₌₀∞p_3(n)qⁿ=q-k^2+k/2∑ₘ₌ₖ^∞ {2m+1}{m+k+1}Aₘ(q),\\ &∑ₙ₌₀∞ p̄(n)qⁿ=q-k^2∑ₘ₌ₖ^∞ {2m}{m+k}Cₘ(q). {align*} These formulas systematically give infinitely many formulas for the 3-colored partition function and the overpartition function in terms of MacMahon's m(•;n) and modd(•;n) partition functions.