Asymptotic Representations of Hypergeometric Bernoulli Polynomials of Order 2 Inside Regions 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 generating function definition given by 
$$
\frac{we^{xw}}{e^w-1}=\sum_{n=0}^{\infty}B_{ n}(x)\frac{w^n}{n!} \quad \mbox{for} , |w|<2\pi.  \ \ \ (A)
$$
As a generalization of $B_n(x)$, for any positive integer $N$, a new class of Bernoulli polynomials called Hypergeometric Bernoulli polynomials of order $N$, $B_n(N, x)$, has been introduced. For the particular case when $N=2$, these polynomials are defined by 
$$ 
\frac{w^2e^{xw}/2}{e^w-1-w}=\sum_{n=0}^{\infty}B_n(2,, x)\frac{w^n}{n!}\quad \mbox{for} , |w|<2\pi. \ \ \  (B)
$$
Previous research has established asymptotic formulas for classical Bernoulli and Euler polynomials in regions related to the roots of $\phi(w)=e^w-1$, which appears in the denominator of the classical generating function. 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 also discuss analogous asymptotic formulas for $B_n(2, , x)$ inside regions related to the roots of $\varphi(w)=e^w-1-w$, which appears in the denominator of the generating function for the hypergeometric Bernoulli polynomials of order two.