Asymptotics and inequalities for the distinct partition function

TL;DR

The paper develops an explicit, rigorously bounded asymptotic expansion for the shifted distinct partition function $q(n+s)$, building on precise bounds for $q(n)$ via modified Bessel functions and exponential factors. By carefully expanding exponential and binomial terms and introducing coefficients $\widehat{B}_m(s)$ and $\widehat{C}_m(s)$ with explicit error controls, it proves a main asymptotic formula with computable error terms. The authors then use this expansion to establish exact thresholds for a range of inequality families (quartic invariant, double Turán, and Laguerre-type inequalities) and verify companion inequalities, thus resolving conjectures on $q(n)$-inequalities and enriching the theory of partition-related asymptotics. These results provide concrete, verifiable criteria for the asymptotic behavior of $q(n)$ and its shifted variants, reinforcing connections between partition theory, special function expansions, and Turán/Laguerre-type phenomena.

Abstract

In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function $q(n +s)$ for any nonnegative integer $s$. Then based on this refined asymptotic formula, we give the exact thresholds of $n$ for the inequalities derived from the invariants of the quartic binary form, the double Turán inequalities, the Laguerre inequalities and their corresponding companion versions.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10