MacMahon-type $q$-series

TL;DR

This paper introduces new MacMahon-type $q$-series related to nested divisor structures, establishing identities, generating functions, and hypergeometric representations, with connections to overpartition pairs and bipartitions with distinct odd parts.

Contribution

It defines new MacMahon-type series $V_k(q)$ and $W_k(q)$, deriving identities, relations, and hypergeometric forms, expanding the theory of $q$-series and partition enumerants.

Findings

Established identities and relations for $V_k(q)$ and $W_k(q)$

Derived hypergeometric representations of the series

Connected the series to overpartition pairs and bipartitions

Abstract

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.