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

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 σ 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 γ n ∗ ( 1 ) . Quite interestingly, they found links between the 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 the limit of the mean and the variance of the random variable γ n ∗ ( 1 ) which correspond to the first and second cumulants. In this paper, we generalize the result of Andrews, Crippa and Simon by calculating the 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.