Asymptotics for the reciprocal and shifted quotient of the partition function

TL;DR

This paper develops asymptotic expansions with error bounds for the shifted quotient of the partition function, extending previous results and including new expansions for related functions.

Contribution

It provides a general asymptotic expansion with error estimates for the quotient p(n+k)/p(n), generalizing prior work and including expansions for p(n+k) and 1/p(n).

Findings

Asymptotic expansion for p(n+k)/p(n) with error bounds

Asymptotic expansion for p(n+k)

Asymptotic expansion for 1/p(n)

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.