Macmahon-type q-series

Abstract

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 $$ V_k(q) = \sum _{1 \le n_1 \le n_2 \le \cdots \le n_k} \frac{q^{\,n_1+n_2+\cdots +n_k}}{(1-q^{n_1})^2(1-q^{n_2})^2\cdots (1-q^{n_k})^2}, $$ V k ( q ) = ∑ 1 ≤ n 1 ≤ n 2 ≤ ⋯ ≤ n k q n 1 + n 2 + ⋯ + n k ( 1 - q n 1 ) 2 ( 1 - q n 2 ) 2 ⋯ ( 1 - q n k ) 2 , introduced recently as MacMahon-type generating functions. We further define a new MacMahon-type series $$ W_k(q) = \sum _{1 \le n_1 \le n_2 \le \cdots \le n_k} \frac{q^{\,2(n_1+n_2+\cdots +n_k)-k}}{(1-q^{2n_1-1})^2(1-q^{2n_2-1})^2\cdots (1-q^{2n_k-1})^2}, $$ W k ( q ) = ∑ 1 ≤ n 1 ≤ n 2 ≤ ⋯ ≤ n k q 2 ( n 1 + n 2 + ⋯ + n k ) - k