Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang

TL;DR

This work derives sharp asymptotic expansions for the shifted finite differences of the overpartition function $\overline{p}(n)$ by first obtaining an asymptotic expansion of the shifted function $\overline{p}(n+k)$ for nonzero integers $k$, introducing a coefficient sequence $A_k(t)$, and then translating this into an expansion for the $j$‑shifted, $r$‑fold difference $\Delta_j^r(\overline{p})(n-j)$ with explicit coefficients $A_j(t,r)$. The leading term is shown to be $\Delta^r_j(\overline{p})(n) \sim (\pi j/2)^r e^{\pi\sqrt{n}}/(8 n^{r/2+1})$ as $n\to\infty$, and the case $j=1$ yields $\Delta^r(\overline{p})(n) \sim (\pi/2)^r e^{\pi\sqrt{n}}/(8 n^{r/2+1})$. These results address a problem of Wang, Xie, and Zhang by providing precise asymptotics and effective error terms, and they motivate further questions on positivity thresholds and connections to plane partitions and related combinatorial structures.

Abstract

Let $\overline{p}(n)$ denote the overpartition function, and for $j\in \mathbb{N}$, $Δ^r_j$ denote the $r$-fold applications of the shifted difference operator $Δ_j$ defined by $Δ_j(a)(n):=a(n)-a(n-j)$. The main goal of this paper is to derive an asymptotic expansion of $Δ^r_j(\overline{p})(n)$ with an effective error bound which subsequently gives an answer to a problem of Wang, Xie, and Zhang. In order to get the asymptotics of $Δ^r_j(\overline{p})(n)$, we derive an asymptotic expansion of the shifted overpartition function $\overline{p}(n+k)$ for any integer $k\neq 0$.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Corollary 1.7
• Lemma 2.1
• proof
• Lemma 2.2
• proof