Polynomization of Sun’s conjecture

The Ramanujan Journal, vol. 70

Abstract

Abstract Let p ( n ) denote the number of partitions of a natural number n . As $$ n \rightarrow \infty $$ n → ∞ , 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 $$n\ge 6$$ n ≥ 6 : $$\begin{aligned} \root n \of {p(n)} > \root n+1 \of {p(n+1)}. \end{aligned}$$ p ( n ) n > p ( n + 1 ) n + 1 . 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 regard 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 toward k -colored 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.

Authors 2

1. Bernhard E. Heim Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University · University of Cologne

   Affiliation as printed

   Division of Mathematics, Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

   Lehrstuhl A für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

2. Markus Neuhauser corresponding Aachen Lehrstuhl für Geometrie und Analysis

   RWTH Aachen University · Kutaisi International University

   Affiliation as printed

   Kutaisi International University, 5/7, Youth Avenue, 4600, Kutaisi, Georgia

   Lehrstuhl für Geometrie und Analysis, RWTH Aachen University, 52056, Aachen, Germany