$q$-Binomial expansions of the truncated MacMahon's $q$-series
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916939507433472 |
|---|---|
| author | Liu, Ji-Cai |
| author_facet | Liu, Ji-Cai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06357 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-Binomial expansions of the truncated MacMahon's $q$-series Liu, Ji-Cai Combinatorics Number Theory 05A17, 05A19, 11P81 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. |
| title | $q$-Binomial expansions of the truncated MacMahon's $q$-series |
| topic | Combinatorics Number Theory 05A17, 05A19, 11P81 |
| url | https://arxiv.org/abs/2509.06357 |