$q$-Binomial expansions of the truncated MacMahon's $q$-series

Abstract: In 1920, MacMahon introduced two families of $q$-series to study divisor sums. Recent work has shown that MacMahon's $q$-series are closely connected to overpartitions and $3$-colored partitions. Merca introduced truncated forms of MacMahon's $q$-series to generalize earlier results by Andrews-Rose and Ono-Singh, and posed two conjectures regarding the $q$-binomial expansions of these truncated series. In this paper, we provide combinatorial proofs of Merca's conjectures through the combinatorial interpretation of $q$-binomial coefficients.