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

TL;DR

This paper generalizes a $q$-series and a divisor function, deriving new identities and analytic properties, with implications for combinatorics and number theory, extending previous results and connecting to recent research.

Contribution

It introduces a novel generalization of the divisor function $\sigma_{s,z}(n)$ and extends existing $q$-series identities, linking them to combinatorial and number-theoretic applications.

Findings

Generalized $q$-series identities related to divisor functions.

Derived new properties of the generalized divisor function $\sigma_{s,z}(n)$.

Connected results to recent research by Bringmann and Jennings-Shaffer.

Abstract

In this article, a $q$-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries of our main theorem, we get results of Dilcher and Andrews, Crippa, and Simon. This main theorem involves a surprising new generalization of the divisor function $\sigma_s(n)$, which we denote by $\sigma_{s,z}(n)$. Analytic properties of $\sigma_{s,z}(n)$ are also studied. As a special case of one of our theorem we obtain result from a recent paper of Bringmann and Jennings-Shaffer.