A further look at the truncated pentagonal number theorem

TL;DR

This paper investigates the asymptotic properties of a function derived from the truncated pentagonal number theorem, providing deeper insights into its behavior and implications in number theory.

Contribution

It offers a detailed analysis of the asymptotic behavior of a specific function related to the truncated pentagonal number theorem, extending previous work.

Findings

Derived asymptotic formulas for the function M_k(n)

Enhanced understanding of the truncated pentagonal number theorem

Potential applications in partition theory

Abstract

In this paper, we study the asymptotic behavior of the following function $$M_k(n):=(-1)^{k-1} \sum_{j=0}^{k-1}\big(p(n-j(3j+1)/2)-p(n-j(3j+5)/2-1)\big),$$ which arises from Andrews and Merca's truncated pentagonal number theorem.