Limit Theorems for Andrews’ Restricted Overpartitions

The study of overpartitions in recent years has been used to great effect in various fields, including hypergeometric series, q-series identities, and mathematical physics. We investigate the limiting distributions of the number of parts in a family of overpartitions of n, introduced by Andrews, where parts are counted with two different weights. Using Andrews’ identities and the saddle-point method, we establish two central limit theorems (CLTs) for the number of parts as n → ∞, corresponding to these weightings. We also derive explicit formulas for the mean and variance in each case.