A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind

Abstract

In a recent issue of the Bulletin of the Korean Mathematical Society, Qi and Zhang discovered an interesting integral representation for the Bernoulli numbers of the second kind (also known as Gregory's coefficients, Cauchy numbers of the first kind, and the reciprocal logarithmic numbers). The same representation also appears in many other sources, either with no references to its author, or with references to various modern researchers. In this short note, we show that this representation is a rediscovery of an old result obtained in the XIXth century by Ernst Schröder. We also demonstrate that the same integral representation may be readily derived by means of complex integration. Moreover, we discovered that the asymptotics of these numbers were also the subject of several rediscoveries, including very recent ones. In particular, the first-order asymptotics, which are usually (and erroneously) credited to Johan F. Steffensen, actually date back to the mid-XIXth century, and probably were known even earlier.

References in corpus (2)

• Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to ${\pi }^{-1}$
• Expansions of generalized Euler's constants into the series of polynomials in ${\pi }^{-2}$ and into the formal enveloping series with rational coefficients only