Iterated integrals and Borwein-Chen-Dilcher polynomials

TL;DR

This paper investigates the zero distribution and asymptotic behavior of iterated integrals of polynomials, especially Borwein-Chen-Dilcher polynomials, and applies findings to ultraspherical polynomials.

Contribution

It provides strong asymptotics and zero distribution limits for Borwein-Chen-Dilcher polynomials and their iterated integrals, extending to ultraspherical polynomials.

Findings

Derived strong asymptotics for Borwein-Chen-Dilcher polynomials

Established zero distribution limits for iterated integrals

Applied results to ultraspherical polynomials with specific parameters

Abstract

We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein-Chen-Dilcher's polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of iterated integrals of ultraspherical polynomials with parameters $(2\alpha+1)/2$, $\alpha\in \mathbb{Z}_+$.