On a nonlinear relation for computing the overpartition function

TL;DR

This paper introduces a new, efficient integer-based method for computing the overpartition function (n) that simplifies implementation and avoids high-precision arithmetic, improving computational speed and simplicity.

Contribution

It provides a novel formula linking (n) to smaller values and combines it with a recurrence relation for fast, integer-only computation of (n).

Findings

New formula for (n) using (k) for k (n/2)

Efficient integer-based computation method

Simpler and faster than previous high-precision approaches

Abstract

In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function $\overline{p}(n)$. Computing $\overline{p}(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of $\overline{p}(n)$ that requires only the values of $\overline{p}(k)$ with $k\leqslant n/2$. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function $\overline{p}(n)$ to obtain a simple and fast computation of the value of $\overline{p}(n)$. This new method uses only (large) integer arithmetic and it is simpler to program.