Polynomization of Sun's Conjecture

Abstract

Let $p(n)$ denote the number of partitions of a natural number $n$. As $ n \to \infty$, the $n$th root of $p(n)$ tends to $1$, which is related to the Cauchy--Hadamard test for power series. Andrews also discovered an elementary proof. Sun conjectured that this happens in a certain way for $n\geq 6$: \begin{equation*} \sqrt[n]{p(n)} > \sqrt[n+1]{p(n+1)}. \end{equation*} The conjecture was proved by Wang and Zhu; shortly thereafter, Chen and Zheng independently obtained a second proof. In this paper, we follow an approach by Rota. We consider $p(n)$ as special values of the D'Arcais polynomials, known as the Nekrasov--Okounkov polynomials. This identifies Sun's conjecture as a property of the largest real zero of certain polynomials. This leads to results towards $k$-coloured partitions, overpartitions, and plane partitions. Moreover, we also consider Chebyshev and Laguerre polynomials. The main purpose of this paper is to offer a uniform approach.