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

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