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This is a Quantitative Study study.

October 3, 2025Open Access

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

Key Points

Combinatorial proofs establish the validity of Merca's conjectures regarding q-binomial expansions.
This work connects MacMahon's q-series to overpartitions, illustrated through q-binomial coefficients.
We explore the truncated forms of MacMahon's q-series, expanding on previous results in partition theory.
The findings can enhance understanding of number theory and combinatorial identities related to partitions.

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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Cite This Study

Ji‐Cai Liu (Mon,) studied this question.

synapsesocial.com/papers/68e02f3cf0e39f13e7fa26f6 https://doi.org/https://doi.org/10.48550/arxiv.2509.06357

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