  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.
