Asymptotic representation of hypergeometric Bernoulli polynomials of order 2 inside domains related to the roots of $e^w-1-w=0$

Abstract

Among several approaches towards the classical Bernoulli polynomials $B_n(x)$, one is the definition by the generating function \begin{equation*} \frac{we^{xw}}{e^w-1}=\sum\limits_{n=0}^{\infty}B_{ n}(x)\frac{w^n}{n!} \quad \text{for}\quad   |w|<2\pi. \end{equation*} As a generalization of $B_n(x)$, for any positive integer $N$, a new class of Bernoulli polynomials called Hypergeometric Bernoulli polynomials of order $N$, $B_n(N, x)$ was established. For the particular case  $N=2$   these polynomials are given by \begin{equation*}%\label{Defn-HBPgf2} \frac{1}{2}\frac{w^2e^{xw}}{e^w-1-w}=\sum\limits_{n=0}^{\infty}B_n(2,x) \frac{w^n}{n!}\quad \text{for}\quad |w|<2\pi. \end{equation*} Several asymptotic formulas for the Bernoulli and Euler polynomials inside different domains related to the roots of $\phi(w)=e^w-1$ were found.   In this paper, we consider an integral representation for $B_n(2,x)$ and establish a zero attractor for the re-scaled polynomials $B_n(2,nx)$ for large values of $n$. We briefly discuss some analogous asymptotic formulas of $B_n(2,x)$ inside domains related to the roots of $\varphi(w)=e^w-1-w$.