  In I981, Uchimura studied a divisor generating $q$-series that has
applications in probability theory and in the analysis of data structures,
called heaps. Mainly, he proved the following identity. For $|q|<1$,
\begin{equation*} \sum_{n=1}^\infty n q^n (q^{n+1})_\infty =\sum_{n=1}^{\infty}
\frac{(-1)^{n-1} q^{\frac{n(n+1)}{2} } }{(1-q^n) ( q)_n } = \sum_{n=1}^{\infty}
\frac{ q^n }{1-q^n}. \end{equation*} Over the years, this identity has been
generalized by many mathematicians in different directions. Uchimura himself in
1987, Dilcher (1995), Andrews-Crippa-Simon (1997), and recently Gupta-Kumar
(2021) found a generalization of the aforementioned identity. Any
generalization of the right most expression of the above identity, we name as
divisor-type sum, whereas a generalization of the middle expression we say
Ramanujan-type sum, and any generalization of the left most expression we refer
it as Uchimura-type sum. Quite surprisingly, Simon, Crippa and Collenberg
(1993) showed that the same divisor generating function has a connection with
random acyclic digraphs. One of the main themes of this paper is to study these
different generalizations and present a unified theory. We also discuss
applications of these generalized identities in probability theory for the
analysis of heaps and random acyclic digraphs.
