A divisor generating $q$-series and cumulants arising from random graphs

TL;DR

This paper generalizes the calculation of cumulants for a random variable related to random graphs using $q$-series, linking divisor functions and extending classical identities in number theory.

Contribution

It extends previous work by deriving the limit of the $t$-th cumulant in terms of generalized divisor functions and explores new identities related to Uchimura and Dilcher.

Findings

Limit of the $t$-th cumulant expressed via generalized divisor functions.

New limit forms for identities of Uchimura and Dilcher.

Connections established between random graph models and divisor function identities.

Abstract

Uchimura, in 1987, introduced a probability generating function for a random variable $X$ and using properties of this function he discovered an interesting $q$-series identity. He further showed that the $m$-th cumulant with respect to the random variable $X$ is nothing but the generating function for the generalized divisor function $\sigma_{m-1}(n)$. Simon, Crippa, and Collenberg, in 1993, explored the $G_{n,p}$-model of a random acyclic digraph and defined a random variable $\gamma_n^{*}(1)$. Quite interestingly, they found links between limit of its mean and the generating function for the divisor function $d(n)$. Later in 1997, Andrews, Crippa and Simon extended these results using $q$-series techniques. They calculated limit of the mean and variance of the random variable $\gamma_n^{*}(1)$ which correspond to the first and second cumulants. In this paper, we generalize the result of Andrews, Crippa and Simon by calculating limit of the $t$-th cumulant in terms of the generalized divisor function. Furthermore, we also discover limit forms for identities of Uchimura and Dilcher. This provides a fourth side to the Uchimura-Ramanujan-divisor type three way partition identities expounded by the first four authors recently.