The asymptotic expansion of the Bernoulli polynomials of the second kind

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.