Asymptotics for the reciprocal and shifted quotient of the partition function

TL;DR

This work develops an explicit, high-order asymptotic expansion for the shifted quotient $\frac{p(n+k)}{p(n)}$ of the partition function. It builds a dual expansion framework: a precise asymptotic for $p(n+k)$ and a detailed asymptotic for $\frac{1}{p(n)}$, obtained via manipulations of the dominant Rademacher-type term and Sigma-assisted coefficient extraction, with rigorous error control. The two expansions are convolved to produce $p(n+k)/p(n)=\sum_{t=0}^{N}\frac{c_k(t)}{n^{t/2}}+O\left(n^{-(N+1)/2}\right)$, where $c_k(t)=\sum_{s=0}^{t}\omega_k^{[1]}(s)g(t-s)$ and $n_N(k)$ is an explicitly computable cutoff. The appendix develops the Sigma-based simplifications and provides sharp, uniform bounds for the auxiliary sums, enabling effective constants and making the results usable for further applications to hyperbolicity of Jensen polynomials and related partition-function inequalities.

Abstract

Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $k\in \mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest.

Theorems & Definitions (50)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Theorem 3.1