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.
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