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

TL;DR

This paper reviews recent and historical results on Bernoulli numbers of the second kind, highlighting rediscoveries of integral representations and asymptotic formulas, and clarifying their origins and derivations.

Contribution

It uncovers the historical roots of known integral representations and asymptotics of Bernoulli numbers of the second kind, correcting attribution errors and demonstrating derivations via complex analysis.

Findings

The integral representation was originally discovered in the 19th century by Ernst Schröder.

The same representation can be derived through complex integration techniques.

First-order asymptotics of these numbers were also rediscovered multiple times, with earlier origins than commonly credited.

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 Schr\"oder. 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.