MacMahon-type $q$-series
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908683603017728 |
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| author | Merca, Mircea |
| author_facet | Merca, Mircea |
| contents | Motivated by earlier work of P.~A.~MacMahon and recent contributions of T.~Amdeberhan, G.~E.~Andrews, K.~Ono, A.~Singh, and R.~Tauraso on higher-order partition enumerants, we study a class of $q$-series arising from nested divisor structures. In particular, we consider the $q$-series \[ V_k(q) = \sum_{1 \le n_1 \le n_2 \le \cdots \le n_k} \frac{q^{\,n_1+n_2+\cdots+n_k}} {(1-q^{n_1})^2(1-q^{n_2})^2\cdots(1-q^{n_k})^2}, \] introduced recently as MacMahon-type generating functions. We further define a new MacMahon-type series \[ W_k(q) = \sum_{1 \le n_1 \le n_2 \le \cdots \le n_k} \frac{q^{\,2(n_1+n_2+\cdots+n_k)-k}} {(1-q^{2n_1-1})^2(1-q^{2n_2-1})^2\cdots(1-q^{2n_k-1})^2}, \] and establish families of identities, generating function relations, and hypergeometric representations for the truncated forms of $V_k(q)$ and $W_k(q)$. Connections with overpartition pairs and bipartitions with distinct odd parts arise naturally in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | MacMahon-type $q$-series Merca, Mircea Combinatorics 11P81 11P82 05A19 05A20 Motivated by earlier work of P.~A.~MacMahon and recent contributions of T.~Amdeberhan, G.~E.~Andrews, K.~Ono, A.~Singh, and R.~Tauraso on higher-order partition enumerants, we study a class of $q$-series arising from nested divisor structures. In particular, we consider the $q$-series \[ V_k(q) = \sum_{1 \le n_1 \le n_2 \le \cdots \le n_k} \frac{q^{\,n_1+n_2+\cdots+n_k}} {(1-q^{n_1})^2(1-q^{n_2})^2\cdots(1-q^{n_k})^2}, \] introduced recently as MacMahon-type generating functions. We further define a new MacMahon-type series \[ W_k(q) = \sum_{1 \le n_1 \le n_2 \le \cdots \le n_k} \frac{q^{\,2(n_1+n_2+\cdots+n_k)-k}} {(1-q^{2n_1-1})^2(1-q^{2n_2-1})^2\cdots(1-q^{2n_k-1})^2}, \] and establish families of identities, generating function relations, and hypergeometric representations for the truncated forms of $V_k(q)$ and $W_k(q)$. Connections with overpartition pairs and bipartitions with distinct odd parts arise naturally in this context. |
| title | MacMahon-type $q$-series |
| topic | Combinatorics 11P81 11P82 05A19 05A20 |
| url | https://arxiv.org/abs/2512.00978 |