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\begin{center}
\vskip 1cm{\LARGE\bf 
A Note on Some Recent Results for the \\
\vskip .1in
Bernoulli Numbers of the Second Kind 
}
\vskip 1cm
\large
Iaroslav V.\ Blagouchine \\  
SeaTech\\ 
University of Toulon\\  
Avenue de l'Universit\'e \\ 
83957 La Garde \\ 
France\\  
\href{mailto:iaroslav.blagouchine@univ-tln.fr}{\nolinkurl{iaroslav.blagouchine@univ-tln.fr}} \\ 
\href{mailto:iaroslav.blagouchine@centrale-marseille.fr}{\nolinkurl{iaroslav.blagouchine@centrale-marseille.fr}} \\ 
and \\
Steklov Institute of Mathematics at St.\ Petersburg \\ 
27 Fontanka \\ 
191023 St.\ Petersburg \\ 
Russia \\   
\href{mailto:iaroslav.blagouchine@pdmi.ras.ru}{\nolinkurl{iaroslav.blagouchine@pdmi.ras.ru}} \\ 
\end{center}

\vskip .2 in
 
\begin{abstract} 
In a recent issue of {\it 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 {\it Gregory's coefficients}, {\it Cauchy numbers of the first kind},
and the {\it 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 19th 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-19th century, and probably were known even earlier. 
\end{abstract} 
 
 
\vskip 0.1in  
 
\section{Rediscovery of Schr\"oder's integral formula} 
In a recent article in {\it the Bulletin of the Korean Mathematical So\-ci\-ety} \cite{qi_02},  
several results concerning the Bernoulli numbers of the second kind were presented.  
 
We recall that these numbers (OEIS \seqnum{A002206} and \seqnum{A002207}),  
which we denote below by $G_n$, are rational 
\begin{equation}\notag 
\begin{array}{llll} 
\displaystyle  
G_1\,=\,+\dfrac{1}{2} \,,\quad& G_2\,=\,-\dfrac{1}{12} \,,\quad& G_3\,=\,+\dfrac{1}{24} \,,\quad& G_4\,=\,-\dfrac{19}{720}\,, \\[5mm] 
G_5\,=\,+\dfrac{3}{160} \,,\quad& G_6\,=\,-\dfrac{863}{60480} \,,\quad& G_7\,=\,+\dfrac{275}{24192} \,,\quad& G_8\,=\,-\dfrac{33953}{3628800} \,,\quad \ldots 
\end{array} 
\end{equation} 
\vskip 0.07in  
\noindent and were introduced by the Scottish mathematician  
and astronomer James Gregory in 1670 in the context of area's interpolation formula.  
Subsequently, they were rediscovered by many famous mathematicians, including  
Gregorio Fontana, Lorenzo Mascheroni, Pierre-Simon Laplace, Augustin-Louis Cauchy, Jacques Binet, 
Ernst Schr\"oder, Oskar Schl\"omilch, Charles Hermite and many others. 
Because of numerous rediscoveries these numbers do not have a standard name, 
and in the literature they are also referred to as {\it Gregory's coefficients},  
{\it (reciprocal) logarithmic numbers}, {\it Bernoulli numbers of 
the second kind}, normalized {\it generalized Bernoulli numbers} $B_n^{(n-1)}$ and normalized {\it Cauchy numbers 
of the first kind} $C_{1,n}$. Usually, these numbers are defined either via their generating function 
\begin{eqnarray} 
\label{eq32} 
\frac{u}{\ln(1+u)} = 1+\sum 
_{n=1}^\infty G_n \, u^n ,\qquad|u|<1\,, 
\end{eqnarray} 
or explicitly 
\begin{eqnarray}\notag 
G_n\,=\,\frac{C_{1,n}}{n!}\,=\lim_{s\to n}\frac{-B_s^{(s-1)}}{\,(s-1)\,s!\,}= 
\,\frac{1}{n!}\! \int\limits_0^1\! x\,(x-1)\,(x-2)\cdots(x-n+1)\, dx\,,\qquad n\in\mathbb{N}\,. 
\end{eqnarray} 
It is well known that $G_n$ are alternating $\,G_n=(-1)^{n-1}|G_n|\,$ and decreasing in absolute value; 
they behave as $\,\big(n\ln^2 n\big)^{-1}\,$ at $n\to\infty$ and may be bounded from below and from above accordingly 
to formulas (55)--(56) from \cite{iaroslav_08}.  
For more information about these important numbers, see \cite[pp.\ 410--415]{iaroslav_08}, \cite[p.\ 379]{iaroslav_09}, 
and the literature given therein (nearly 50 references). 
 
 
 
\begin{figure}[!t]    
\centering 
\includegraphics[width=0.83\textwidth]{Schroder-integral-formula.eps} 
\caption{A fragment of p.\ 112 from Schr\"oder's paper \cite{schroder_01}. Schr\"oder's $C_n^{(-1)}$ are exactly our $G_n$.} 
\label{hkgcwuih} 
\end{figure} 
Now, the first main result of \cite[p.\ 987]{qi_02} is Theorem 1.\footnote{Our $G_n$ are exactly $b_n$ from \cite{qi_02} 
and $\frac{c_{n,1}^{(1)}}{n!}$ from \cite[Sect.\ 5]{chikhi_01}. Despite  
a venerable history, these numbers still lack a standard notation and various authors may use different notation for them.} It states:  
{\it the Bernoulli numbers of the second kind may be represented as follows} 
\begin{equation} 
G_n\,=\,(-1)^{n+1}\!\int\limits_1^\infty \!\!\frac{dt}{\big(\ln^2 (t-1)+\pi^2\big)\,t^n}\,,\qquad n\in\mathbb{N}\,. 
\end{equation} 
The same representation appears in a slightly different form\footnote{Put $t=1+u$.} 
\begin{equation}\label{jm23nd2d} 
G_n\,=\,(-1)^{n+1}\!\int\limits_0^\infty \!\!\frac{du}{\big(\ln^2 u +\pi^2\big)\,(u+1)^n}\,,\qquad n\in\mathbb{N}\,, 
\end{equation} 
in \cite[pp.\ 473--474]{coffey_02} and \cite[Sect.\ 5]{chikhi_01},
and is called {\it Knessl's representation}  
and {\it the Qi integral representation} respectively. 
Furthermore, various internet sources 
provide the same (or equivalent) formula, either with no references to its author or with references to different  
modern writers and/or their papers. 
However, the integral representation in question is not novel and is not due to Knessl nor to Qi and Zhang; in fact, 
this representation is a rediscovery of an old result. 
In a little-known paper of the German mathematician Ernst Schr\"oder \cite{schroder_01}, written in 1879, 
one may easily find exactly the same integral representation on p.\ 112;
see Fig.\ \ref{hkgcwuih}.  
Moreover, since this result is not difficult to obtain, it is possible that the same integral representation  
was obtained even earlier.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
\section{Simple derivation of Schr\"oder's integral formula by means of the complex integration} 
Schr\"oder's integral formula \cite[p.\ 112]{schroder_01} may, of course, be derived in various ways. 
Below, we propose a simple derivation of this formula based on the method of contour integration. 
 
If we set $u=-z-1$, then equality \eqref{eq32} may be written as 
\begin{eqnarray}\notag 
\frac{z+1}{\ln z - \pi i} = -1+\sum 
_{n=1}^\infty \big|G_n\big| \, (z+1)^n ,\qquad|z+1|<1\,. 
\end{eqnarray} 
Now considering the following line integral along a  
contour $C$ (see Fig.\ \ref{kc30jfd}),
\begin{figure}[!t]    
\centering 
\includegraphics[width=0.35\textwidth]{contour.eps} 
\caption{Integration contour $C$ ($r$ and $R$ are radii of the small and big circles respectively, where 
$r\ll1$ and $R\gg1$).} 
\label{kc30jfd} 
\end{figure} 
where $n\in\mathbb{N}$, and then letting $R\to\infty\,$, $r\to0$, we have by the residue theorem 
\begin{eqnarray} 
&& \displaystyle  
\ointctrclockwise\limits_{C} \frac{\,dz\,}{\,(1+z)^n \, (\ln z - \pi i)\,}\,=\, 
\int\limits_{r}^R \! \ldots \, dz \, + \int\limits_{C_R} \!\ldots \, dz \,  
+ \int\limits_{R}^r \!\ldots \, dz \, + \int\limits_{C_r} \! \ldots \, dz \,  
\stackrel{\substack{R\to\infty \\ r\to0}}{=}   \notag\\[2mm] 
&& \displaystyle \quad 
= \, \int\limits_0^\infty \!\left\{\frac{1}{\ln x - \pi i} - \frac{1}{\ln x + \pi i} \right\}\cdot\frac{dx}{(1+x)^n} \, 
=\,2\pi i \! \int\limits_0^\infty \!\! \frac{1}{\,(1+x)^n} \cdot\frac{dx}{\,\ln^2 x +\pi^2\,}\,= \qquad\notag\\[3mm] 
&& \displaystyle  \quad  \notag 
=\,2\pi i \! \res\limits_{z=-1}\! \frac{1}{\,(1+z)^n \, (\ln z - \pi i)\,} \,=\,\frac{2\pi i}{n!}\cdot  
\left.\frac{d^n}{dz^n}\frac{z+1}{\,\ln z - \pi i\,}\right|_{z=-1} \!\!\! = \,2\pi i \, \big|G_n\big|\,, 
\end{eqnarray} 
since 
\begin{eqnarray} 
& \displaystyle  
\left| \,\int\limits_{C_R} \! \frac{\,dz\,}{\,(1+z)^n \, (\ln z - \pi i)\,} \, \right|  
\,=\, O\left(\!\frac{1}{\,R^{n-1}\ln R\,}\!\right)=o(1)\,, \qquad  & R\to\infty\,,\qquad n\geq1\,,\notag \\[5mm] 
& \displaystyle  
\left| \,\int\limits_{C_r} \! \frac{\,dz\,}{\,(1+z)^n \, (\ln z - \pi i)\,} \, \right|  
\,=\, O\left(\!\frac{r}{\,\ln r\,}\!\right)=o(1)\,, \qquad & 	r\to0\,,    \notag 
\end{eqnarray} 
and because at $z=-1$ the integrand of the contour integral has a pole of the $(n+1)$th order. This completes the proof. 
Note that above derivations are valid only for $n\geq1$, and so is Schr\"oder's integral formula, which 
may also be regarded as one of the generalizations of $G_n$ to the continuous values of $n.$ 
 
 
 
 
 
 
 
 
\section{Several remarks on the asymptotics for the Bernoulli numbers of the second kind} 
The first-order asymptotics $\,|G_n|\sim\big(n\ln^2 n\big)^{-1}\,$ at $n\to\infty$ are usually credited to  
Johan F.\ Steffensen \cite[pp.\ 2--4]{steffensen_01}, 
\cite[pp.\ 106--107]{steffensen_02}, \cite[p.\ 29]{norlund_01}, \cite[p.\ 14, Eq.\ (14)]{davis_02},  
\cite{nemes_01}, who found it in 1924.\footnote{The same first-order asymptotics also appear in \cite[p.\ 294]{comtet_01}, but without 
the source of the formula.}  
However, in our recent work \cite[p.\ 415]{iaroslav_08} we noted that exactly the same result appears in Schr\"oder's 
work written 45 years earlier, see Fig.\ \ref{hg28g3dc}, and the order of the magnitude of the closely related numbers  
is contained in a work of  
Jacques Binet dating back to 1839 \cite{binet_01}.\footnote{By the ``closely related numbers'' 
we mean the so-called {\it Cauchy numbers of the second kind} (OEIS 
\seqnum{A002657} and \seqnum{A002790}), and numbers $I'(k)$, see \cite[pp.\ 410--415, 428--429]{iaroslav_08}. The comment 
going just after Eq.\ (93) \cite[p.\ 429]{iaroslav_08} is based on the statements from \cite[pp.\ 231, 339]{binet_01}. 
The Cauchy numbers of the second kind $C_{2,n}$ and Gregory's coefficients $G_n$ are related to each other  
via the relationship $\,nC_{2,n-1}-C_{2,n}=n!\,|G_n|\,$, see \cite[p.\ 430]{iaroslav_08}.} In 1957 Davis \cite[p.\ 14, Eq.\ (14)]{davis_02} improved
this first-order  
approximation slightly by showing that $\,|G_n|\sim\Gamma(1+\xi) \big(n\ln^2 n + n\pi^2\big)^{-1}\,$ at $\,n\to\infty\,$ 
for some $\,\xi\in[0,1]\,$, without noticing that   
7 years earlier S.\ C.\ Van Veen had already obtained the complete asymptotics for them \cite[p.\ 336]{van_veen_01}, \cite[p.\ 29]{norlund_01}. 
The identical complete asymptotics were recently rediscovered in a slightly different form by Charles Knessl \cite[p.\ 473]{coffey_02},  
and later by Gerg\H{o} Nemes  \cite{nemes_01}. 
An alternative demonstration of the same result was also presented by the author \cite[p.\ 414]{iaroslav_08}. 
\begin{figure}[!t]    
\centering 
\includegraphics[width=0.5\textwidth]{Schroder-asymptotic-formula.eps} 
\caption{A fragment of p.\ 115 from Schr\"oder's paper \cite{schroder_01}.} 
\label{hg28g3dc} 
\vskip 0.25in  
\end{figure} 
 
 
 
 
 
\section{Acknowledgments}  
The author is grateful to Jacqueline Lorfanfant from the University of Strasbourg  
for sending a scanned version of \cite{schroder_01}.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
\vskip 0.25in  
 
 
\begin{thebibliography}{10} 
 
\bibitem{binet_01} 
J.\ Binet,  
M\'emoire sur les int\'egrales d\'efinies eul\'eriennes et sur leur application \`a la th\'eorie  
des suites, ainsi qu'\`a l'\'evaluation des fonctions des grands nombres,  
{\it J.\ \'Ec.\ Roy.\ Polytech.}, {\bf 16} (27) (1839), 123--343. 
 
\bibitem{iaroslav_09} 
Ia.\ V.\ Blagouchine,  
Expansions of generalized \text{Euler's} constants into the series of polynomials 
in $\pi^{-2}$ and into the formal enveloping series with rational coefficients only,  
{\it J.\ Number Theory}, {\bf 158}  (2016), 365--396. Corrigendum: {\bf 173} (2017), 631--632.  
 
\bibitem{iaroslav_08} 
Ia.\ V.\  Blagouchine,  
Two series expansions for the logarithm of the gamma function involving \text{Stirling} numbers and  
containing only rational coefficients for certain arguments related to $\pi^{-1}$,  
{\it J.\ Math.\ Anal.\ Appl.}, {\bf 442} (2016), 404--434. 
 
\bibitem{chikhi_01} 
J.\ Chikhi, 
Integral representation for some generalized \text{Poly-Cauchy} numbers, preprint, 2016,  
\url{https://hal.archives-ouvertes.fr/hal-01370757v1}. 
 
\bibitem{coffey_02} 
M.\ W.\ Coffey,  
Series representations for the \text{Stieltjes} constants,  
{\it Rocky Mountain J.\ Math.}, {\bf 44} (2014), 443--477. 
 
\bibitem{comtet_01} 
L.\ Comtet,  
{\it Advanced Combinatorics. The Art of Finite and Infinite Expansions (revised and enlarged edition)},  
D.\ Reidel Publishing Company, 1974. 
 
\bibitem{davis_02} 
H.\ T.\ Davis,  
The approximation of logarithmic numbers,  
{\it Amer.\ Math.\ Monthly}, {\bf 64} (1957), 11--18. 
 
\bibitem{nemes_01} 
G.\ Nemes,  
An asymptotic expansion for the Bernoulli numbers of the second kind,  
{\it J.\ Integer Seq.}, {\bf 14} (2011),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL14/Nemes/nemes4.pdf}{Article 11.4.8}.   
 
\bibitem{norlund_01} 
N.\ E.\ N\"orlund, Sur les valeurs asymptotiques des nombres et des polyn\^omes de \text{Bernoulli},  
{\it Rend.\ Circ.\ Mat.\ Palermo}, {\bf 10} (1) (1961), 27--44. 
 
\bibitem{qi_02} 
F.\ Qi and X.-J.\ Zhang,  
An integral representation, some inequalities, and complete monotonicity of \text{Bernoulli} numbers of the second kind,  
{\it Bull. Korean Math. Soc.}, {\bf 52} (3) (2015), 987--998. 
 
\bibitem{schroder_01} 
E.\ Schr\"oder,  
Bestimmung des infinit\"aren \text{Werthes} des \text{Integrals} $\int\limits_0^1 (u)_n\, du$, 
{\it Z.\ Math.\ Phys.}, {\bf 25} (1880), 106--117. 
 
\bibitem{steffensen_01} 
J.\ F.\ Steffensen,  
On \text{Laplace's} and \text{Gauss'} summation--formulas,  
{\it Skandinavisk Aktuarietidskrift (Scand.\ Actuar.\ J.)}, {\bf 1} (1924), 1--15. 
 
\bibitem{steffensen_02} 
J.\ F.\ Steffensen,  
{\it Interpolation}, 2nd edition,  
Chelsea Publishing, 1950. 
 
\bibitem{van_veen_01} 
S.\ C.\ Van Veen,  
Asymptotic expansion of the generalized \text{Bernoulli} numbers \text{$B_n^{(n-1)}$} for large values of $n$ ($n$ integer),  
{\it Indag.\ Math.}, {\bf 13} (1951), 335--341. 
 
\end{thebibliography} 
 
\bigskip
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\noindent 2010 {\it Mathematics Subject Classification:}
Primary 11B68; Secondary 11B83, 30E20, 11-03, 05A10, 01A55. \\

\noindent {\it Keywords:}
Bernoulli number of the second kind, 
Gregory coefficient, 
Cauchy number, 
logarithmic number, 
Schr\"oder, 
rediscovery, 
state of art, 
complex analysis, 
theory of complex variable, 
contour integration, 
residue theorem. 

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A002206},  
\seqnum{A002207}, \seqnum{A002657}, \seqnum{A002790},  
\seqnum{A195189}, \seqnum{A075266}, and \seqnum{A262235}.)

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received December 20 2016;
revised versions received January 24 2017; January 25 2017; January 26 2017.
Published in {\it Journal of Integer Sequences}, January 27 2017.

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