$q$-Binomial expansions of the truncated MacMahon's $q$-series
Combinatorics
2025-09-09 v1 Number Theory
Abstract
In 1920, MacMahon introduced two families of -series to study divisor sums. Recent work has shown that MacMahon's -series are closely connected to overpartitions and -colored partitions. Merca introduced truncated forms of MacMahon's -series to generalize earlier results by Andrews-Rose and Ono-Singh, and posed two conjectures regarding the -binomial expansions of these truncated series. In this paper, we provide combinatorial proofs of Merca's conjectures through the combinatorial interpretation of -binomial coefficients.
Keywords
Cite
@article{arxiv.2509.06357,
title = {$q$-Binomial expansions of the truncated MacMahon's $q$-series},
author = {Ji-Cai Liu},
journal= {arXiv preprint arXiv:2509.06357},
year = {2025}
}
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11 pages