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.

翻译：令$p(n)$表示分拆函数。本文的主要目标是推导分拆函数平移商$p(n+k)/p(n)$（其中$k\in \mathbb{N}$）的$N$阶渐近展开式（对任意固定正整数$N$）及其误差界估计，该结果推广了Gomez、Males和Rolen的结论。为此，我们推导了平移形式$p(n+k)$与乘法逆元$1/p(n)$的带误差界的渐近展开式，这些结果本身也具有独立的研究价值。