On some q-series identities related to a generalized divisor function\n and their implications

2020/12/19 by Rajat Gupta, Rahul Kumar, Gupta, Rajat +1
Mathematics · #11M35 #11P84 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11P81 #Secondary 11M06

paper · pdf · doi:10.48550/arxiv.2012.10697

openalex publication_date 2020/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, a q-series examined by Kluyver and Uchimura is\ngeneralized. This allows us to find generalization of the identities in the\nrandom acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one\nof the corollaries of our main theorem, we get results of Dilcher and Andrews,\nCrippa, and Simon. This main theorem involves a surprising new generalization\nof the divisor function \σ_s(n), which we denote by \σ_s,z(n).\nAnalytic properties of \σ_s,z(n) are also studied. As a special case of\none of our theorem we obtain result from a recent paper of Bringmann and\nJennings-Shaffer.\n

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