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 \[ Vk(q) = Σ1 n1 n2 ·s nk q\,n1+n2+·s+nk (1-qn1)2(1-qn2)2·s(1-qnk)2, \] introduced recently as MacMahon-type generating functions. We further define a new MacMahon-type series \[ Wk(q) = Σ1 n1 n2 ·s nk q\,2(n1+n2+·s+nk)-k (1-q2n1-1)2(1-q2n2-1)2·s(1-q2nk-1)2, \] and establish families of identities, generating function relations, and hypergeometric representations for the truncated forms of Vk(q) and Wk(q). Connections with overpartition pairs and bipartitions with distinct odd parts arise naturally in this context.
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