  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.
