[论文解读] Restricted overpartitions and concave compositions: their modularity and asymptotics

Koustav Banerjee, Kathrin Bringmann+1 more|arXiv (Cornell University)|Jan 7, 2026

Advanced Mathematical Identities被引用 0

一句话总结

该论文将受限部分划分的生成函数与凹序列的生成函数联系起来，涉及模形式、模拟 theta 函数以及 Maass theta 函数，并推导出它们的渐近主项与秩统计。

ABSTRACT

In this paper we study restricted overpartitions and concave compositions. Using q-series transformations, we show that their generating functions are related to modular forms, mock theta functions, false theta functions, and mock Maass theta functions. Moreover, we obtain their asymptotic main terms. We also study related rank statistics.

研究动机与目标

• Motivate the study of restricted overpartitions and concave compositions and their connections to modular objects.
• Characterize generating_functions of restricted overpartitions and concave compositions via q-series transformations.
• Show that these generating functions relate to modular forms, mock theta functions, false theta functions, and mock Maass theta functions.
• Derive asymptotic main terms for the counting functions and investigate related rank statistics.

提出的方法

• Apply q-series transformations to derive representations of generating functions.
• Express generating functions as linear combinations involving modular forms, mock theta functions, false theta functions, and mock Maass theta functions.
• Use Tauberian theorems (Ingham-type) and Euler–Maclaurin summation to obtain asymptotics.
• Introduce and manipulate rank statistics through generating functions and known rank-related identities.
• Decompose and reformulate restricted overpartition generating functions to expose modular-type components.

实验结果

研究问题

• RQ1 What modular or mock modular structures underlie the generating functions for restricted overpartitions and concave compositions?
• RQ2 What are the asymptotic main terms for the counting functions of these restricted objects?
• RQ3 How do rank statistics for these restricted objects relate to known modular or mock modular forms?
• RQ4 Can the generating functions be decomposed into mixed modular and mock modular components for special parameter values?
• RQ5 What are the implications of these structures for related colored partition functions?

主要发现

• The generating function for restricted overpartitions with certain restrictions is expressed in terms of a mixed mock Maass theta function and a false theta function.
• The asymptotic main term for overpartitions with odd restrictions is p̄od(n) ~ (5π)/(48√2 n^{3/2}) e^{π√(5n/6)} as n → ∞.
• The generating function for overpartitions with even restrictions is a mixed mock modular form, yielding an asymptotic p̄ev(n) ~ e^{π√(2n/3)}/(4√3 n).
• Corollaries relate g(n) to p(n) and show g(n) ~ e^{π√(2n/3)}/(4√3 n) as n → ∞.
• Multiple corollaries establish mixed mock modular forms for various specialized generating functions, including q-derivative and two-variable generalizations.
• Connections are drawn between rank generating functions and known modular/mocking objects (e.g., R2, φ, f, μ) and identities such as the Ramanujan Corollary 1.12 (2φ−f = Θ^2/(q)∞).