Asymptotic behavior and zeros of the Bernoulli polynomials of the second kind

TL;DR

This paper investigates the asymptotic behavior and zeros of Bernoulli polynomials of the second kind, establishing their real and simple zeros and providing detailed asymptotic expansions for different zero regimes.

Contribution

It provides a comprehensive analysis of the zeros of Bernoulli polynomials of the second kind, including their realness, simplicity, and detailed asymptotic expansions.

Findings

Zeros are all real and simple

Asymptotic expansions for small, large, and middle zeros

Zeros are thoroughly characterized in various regimes

Abstract

The main aim of this article is a careful investigation of the asymptotic behavior of zeros of Bernoulli polynomials of the second kind. It is shown that the zeros are all real and simple. The asymptotic expansions for the small, large, and the middle zeros are computed in more detail. The analysis is based on the asymptotic expansions of the Bernoulli polynomials of the second kind in various regimes.