A divisor generating q-series identity and its applications to probability theory and random graphs

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, equation* ₍=₁^ n qⁿ (q^n+1) _ =₍=₁^ (-1) ^n-1 q^{n (n+1) {2 } } (1-qⁿ) (q) ₙ = ₍=₁^ qⁿ 1-qⁿ. 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.