Limit theorems for Andrews’ restricted overpartitions

Abstract 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 $$n\rightarrow \infty ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> corresponding to these weightings. We also derive explicit formulas for the mean and variance in each case.