The asymptotic expansion of the Bernoulli polynomials of the second kind

TL;DR

This paper derives asymptotic expansions for Bernoulli polynomials of the second kind, providing insights into their behavior for large degrees and extending recent mathematical research with numerical validation.

Contribution

It introduces new asymptotic formulas for scaled Bernoulli polynomials of the second kind, expanding understanding of their large-degree behavior.

Findings

Asymptotic expansions valid for real and complex parameters

Numerical validation of the derived asymptotic formulas

Extension of recent theoretical work on Bernoulli polynomials

Abstract

We consider the Bernoulli polynomials of the second kind, which can be related to the generalised Bernoulli polynomials $B_n^{(n)}(z)$. The asymptotic expansions of the scaled polynomials $B_n^{(n)}(nz)$ are obtained as $n\to\infty$ when (i) $z$ is real and (ii) $z$ is complex bounded away from $[0,1]$. These results complement recent work of \v{S}tampach [{\it J. Approx. Theory}, {\bf 262} (2021) 105517]. Numerical results are presented to illustrate the accuracy of the different expansions obtained.