Asymptotic structure of derivatives of partition-statistic generating functions

Determine the recursive and asymptotic structure of the derivatives \(\partial_z S(z;q)\) for broad families of partition statistics \(s:\mathcal P\to\mathbb C\) with two-variable generating function \(S(z;q)=\sum_{\lambda\in\mathcal P}z^{s(\lambda)}q^{|\lambda|}\), and determine what this recursive structure implies for the limiting distribution laws of the statistics.

Background

The paper observes that moments of a partition statistic can be obtained by applying the differential operator $\partial_z=z\frac{\partial}{\partial z}$ repeatedly to its two-variable generating function and then setting $z=1$. Differentiating products of factors such as $(zq^a;q^b)_\infty^{-1}$ naturally produces Lambert-series-type expressions and suggests a recursive relationship among families of generating functions and their moments.

The authors ask both how these derivatives behave asymptotically and how that behavior controls limiting distributions. This extends the paper’s treatment of the alternating sum of parts, whose asymptotically normal distribution is established in the main theorem, to substantially broader classes of partition statistics.