MacMahon-type $q$-series
Mircea Merca
TL;DR
This work studies MacMahon-type q-series $V_k^{\pm}(q)$ and $W_k^{\pm}(q)$ arising from nested divisor structures, introducing truncated versions $V_{k,m}^{\pm}(q)$ and $W_{k,m}^{\pm}(q)$. Using Gaussian polynomials and basic hypergeometric tools, the authors derive explicit linear-operator relations (via $H_{k,m}$ and inverse kernels $B_{k,j}$) that express these series in closed form, and then pass to the $m\to\infty$ limit to obtain product–false theta decompositions. These decompositions connect the coefficients to overpartition pairs and bipartitions with distinct odd parts, and yield representations of Ramanujan theta function squares as alternating sums of triangular-number exponents. The results establish new positivity properties for convolution-like sums and lay groundwork for further modular, combinatorial, and arithmetic investigations of MacMahon-type q-series.
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.