Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros

TL;DR

This paper studies a family of polynomials derived from partition statistics, analyzing their asymptotic behavior and the distribution of their zeros to understand their limiting properties.

Contribution

It introduces a new analysis of polynomials linked to partition statistics, focusing on their asymptotics and zero distributions, which was not previously explored.

Findings

Asymptotic behavior of the polynomials characterized

Distribution patterns of zeros identified and analyzed

Limiting densities of zeros determined

Abstract

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting behavior of their zeros as sets and densities.