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

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 $k\neq 0$.

💡 Research Summary

The paper investigates the asymptotic behaviour of shifted finite differences of the overpartition function (\overline{p}(n)). For a fixed integer shift (j\ge1) and a positive integer order (r), the operator (\Delta_j^r) is defined by applying the forward difference (\Delta_j a(n)=a(n)-a(n-j)) repeatedly (r) times. The authors’ primary goal is to obtain a full asymptotic expansion of (\Delta_j^r\overline{p}(n)) together with an explicit, effective error bound, thereby answering Problem 2.4 posed by Wang, Xie and Zhang concerning a sharp lower bound for (\Delta_r\overline{p}(n)).

The work begins with a review of Zuckerman’s Rademacher‑type series for (\overline{p}(n)) and Engel’s refined error term for the classical Hardy–Ramanujan–Rademacher formula. Using these tools, Lemma 2.2 establishes that for any non‑zero integer (k) and sufficiently large (n), \