2025/11/30 by Mircea Merca, Merca, Mircea
Mathematics · #11P81 11P82 05A19 05A20 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2512.00978
openalex publication_date 2025/11/30 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28
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) = ∑1 ≤ n1 ≤ n2 ≤ ⋯ ≤ nk \fracq n1+n2+⋯+nk (1-qn1)2(1-qn2)2⋯(1-qnk)2, introduced recently as MacMahon-type generating functions. We further define a new MacMahon-type series Wk(q) = ∑1 ≤ n1 ≤ n2 ≤ ⋯ ≤ nk \fracq 2(n1+n2+⋯+nk)-k (1-q2n1-1)2(1-q2n2-1)2⋯(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.