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arXiv · 2412.02257

Asymptotics for the reciprocal and shifted quotient of the partition function

Koustav Banerjee·Peter Paule·Cristian-Silviu Radu·Carsten Schneider
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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.

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Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider. 2024-12-03. Asymptotics for the reciprocal and shifted quotient of the partition function. https://arxiv.org/abs/2412.02257

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