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\begin{center}
\vskip 1cm{\LARGE\bf 
An Asymptotic Expansion for the Bernoulli \\
\vskip .05in
Numbers of the Second Kind
}
\vskip 1cm
\large
Gerg\H{o} Nemes\\
Lor\'and E\"otv\"os University\\ 
P\'azm\'any P\'eter s\'et\'any 1/C\\ 
H-1117 Budapest\\
Hungary\\
\href{mailto:nemesgery@gmail.com}{\tt nemesgery@gmail.com} \\
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\begin{abstract}
In this paper we derive a complete asymptotic series for the Bernoulli
numbers of the second kind, and provide a recurrence relation for the
coefficients.
\end{abstract} 

\section{Introduction}

The Bernoulli numbers of the second kind $b_n$ (also known as the Cauchy numbers, Gregory coefficients or logarithmic numbers)
are defined by the generating function
\[
\frac{x}{{\log \left( {1 + x} \right)}} = \sum\limits_{n \ge 0} {b_n x^n } .
\]
The first few are $b_0 = 1$, $b_1 = 1/2$, $b_2 = -1/12$, $b_3 = 1/24$, $b_4 = -19/720$.  The numerators and denominators are given by
Sloane's sequences \seqnum{A002206} and \seqnum{A002207}. 
They are related to the generalized Bernoulli numbers \cite[p.\ 596]{NIST} by
\[
b_n  =  - \frac{{B_n^{\left( {n - 1} \right)} }}{{\left( {n - 1} \right)n!}}.
\]
The ${B_n^{\left( {t} \right)} }$'s are the coefficients of the exponential generating function $x^t /\left( {e^x  - 1} \right)^t$. It is known \cite{HTD} that
\[
b_n  = \int_0^1 {\frac{{t\left( {t - 1} \right) \cdots \left( {t - n + 1} \right)}}{{n!}}dt},
\]
which can be shown as follows:
\begin{align*}
\frac{x}{{\log \left( {1 + x} \right)}} & = \int_0^1 {\exp \left( {t\log \left( {1 + x} \right)} \right)dt}  = \int_0^1 {\left( {1 + x} \right)^t dt}  = \int_0^1 {\sum\limits_{n \ge 0} {\binom{t}{n}x^n } dt}\\
& = \sum\limits_{n \ge 0} {\left( {\int_0^1 {\binom{t}{n}dt} } \right)x^n }  = \sum\limits_{n \ge 0} {\left( {\int_0^1 {\frac{{t\left( {t - 1} \right) \cdots \left( {t - n + 1} \right)}}{{n!}}dt} } \right)x^n } .
\end{align*}
Using the $s\left( {n,k} \right)$ Stirling numbers of the first kind (Sloane's
\seqnum{A048994}) defined by
$$t\left( {t - 1} \right) \cdots \left( {t - n + 1} \right) = \sum\nolimits_{k = 0}^n {s\left( {n,k} \right)t^k },$$ 
we immediately obtain the representation
\[
b_n  = \frac{1}{n!}\sum\limits_{k = 0}^n {\frac{{s\left( {n,k} \right)}}{{k + 1}}} .
\]

The investigation of the asymptotic behavior of these numbers was begun by 
Steffensen \cite{JFS}, who proved that
\[
b_n  \sim \frac{{\left( { - 1} \right)^{n + 1} }}{{n\log ^2 n}} =: a_n
\]
as $n\rightarrow +\infty$. However, the ratio $b_n/a_n$ converges very slowly toward $1$, as was pointed out by Davis \cite{HTD},
who derived better approximations including the following:
\[
b_n  \approx \frac{{\left( { - 1} \right)^{n + 1} \varGamma \left( {\xi_n  + 1} \right)}}{{n\left( {\log ^2 n + \pi ^2 } \right)}}, 
\]
where $0 < \xi_n < 1$ and $\varGamma$ is the gamma function. The aim of this paper is to extend Steffensen's asymptotic approximation into a complete asymptotic expansion in terms of $1/\log n$.

\section{The asymptotic expansion}

\begin{theorem}\label{theorem} The Bernoulli numbers of the second kind
$b_n$ have an asymptotic expansion of the form
\begin{equation}\label{eq1}
b_n  \sim \frac{{\left( { - 1} \right)^{n + 1} }}{{n\log ^2 n}}\sum\limits_{k \ge 0} {\frac{{\beta _k }}{{\log ^k n}}} 
\end{equation}
as $n\rightarrow +\infty$, where
\begin{equation}\label{eq2}
\beta _k = \left( { - 1} \right)^k \left[ {\frac{{d^{k + 1} }}{{ds^{k + 1} }}\left( {\frac{1}{{\varGamma \left( s \right)}}} \right)} \right]_{s = 0} .
\end{equation}
\end{theorem}
Note that the main term of this asymptotic series is just Steffensen's approximation. Computing the first few coefficients $\beta _k$, our expansion takes the form
\[
b_n  \sim \frac{{\left( { - 1} \right)^{n + 1} }}{{n\log ^2 n}}\left( {1 - \frac{{2\gamma }}{{\log n}} - \frac{{\pi ^2  - 6\gamma ^2 }}{{2\log ^2 n}} + \frac{{2\pi ^2 \gamma  - 4\gamma ^3  - 8\zeta \left( 3 \right)}}{{\log ^3 n}} +  \cdots } \right),
\]
where $\gamma$ is the Euler-Mascheroni constant and $\zeta$ is the Riemann zeta function.

In our proof we will use the following special case of Watson's lemma.

\begin{lemma}Let $g\left(s\right)$ be a function of the positive real variable $s$, such that
\[
g\left( s \right) = \sum\limits_{k \ge 1} {g_k s^k } 
\]
as $s \rightarrow 0+$. Then for each nonnegative integer $N$
\[
\int_0^{ + \infty } {g\left( s \right)e^{ - ms} ds}  = \sum\limits_{k = 0}^{N - 1} {\frac{{\left( {k + 1} \right)!g_{k+1} }}{{m^{k + 2} }}}  + \mathcal{O}\left( {\frac{1}{{m^{N + 2} }}} \right)
\]
as $m\rightarrow +\infty$, provided that this integral converges throughout its range for all sufficiently large $m$.
\end{lemma}
For a more general version and proof see, e.g., Olver \cite[p.\ 71]{FWJO} or Wong \cite[p.\ 20]{RW}. We will also need sharp bounds for the ratio of two gamma functions.

\begin{lemma}\label{lemma2} For $n>2$ and $0 \leq s \leq 1$ we have
\[
\frac{1}{n}\frac{1}{{n^s }} \le \frac{{\varGamma \left( {n - s} \right)}}{{\varGamma \left( {n + 1} \right)}} \le \frac{1}{n}\frac{1}{{n^s }} + \frac{2}{{n^2 }}\frac{s}{{n^s }}.
\]
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lemma2}] Fix $n>1$ and let
\begin{align*}
f_1 \left( s \right) & = -\left( {s + 1} \right)\log n,\\
f_2 \left( s \right) & = \log \varGamma \left( {n - s} \right) - \log \varGamma \left( {n + 1} \right)
\end{align*}
for $0 \leq s \leq 1$. The function $f_1$ is affine while the function $f_2$ is convex (since $\log \varGamma$ is convex). Furthermore, $f'_1 \left( 0 \right) =  - \log n$, $f'_2 \left( 0 \right) =  - \psi \left( n \right)$, where $\psi := \varGamma' / \varGamma$ is the Digamma function. From the simple inequality $\psi \left( n \right) < \log n$ we see that $f'_1 \left( 0 \right) < f'_2 \left( 0 \right)$, hence
\[
\frac{1}{n}\frac{1}{{n^s }} \le \frac{{\varGamma \left( {n - s} \right)}}{{\varGamma \left( {n + 1} \right)}}
\]
holds for $n > 1$ and $0 \leq s \leq 1$. To prove the upper bound, we first show that
\begin{equation}\label{gammainequality}
\frac{{\varGamma \left( {n + a} \right)}}{{\varGamma \left( {n + 1} \right)}} \le \frac{1}{{n^{1 - a} }}
\end{equation}
for $n\geq1$ and $0 \leq a \leq 1$. Fix $n\geq1$ and let
\begin{align*}
g_1 \left( a \right) & = \log \varGamma \left( {n + a} \right) - \log \varGamma \left( {n + 1} \right),\\
g_2 \left( a \right) & = \left( {a-1} \right)\log n
\end{align*}
for $0 \leq a \leq 1$. The function $g_1$ is convex while the function $g_2$ is affine. Since $g_1 \left( 0 \right) = g_2 \left( 0 \right) =  - \log n$ and $g_1 \left( 1 \right) = g_2 \left( 1 \right) = 0$, the inequality \eqref{gammainequality} holds. From this it follows that for $n>2$ and $0 \leq s \leq 1$
\[
\frac{{\varGamma \left( {n - s} \right)}}{{\varGamma \left( {n + 1} \right)}} = \frac{{\varGamma \left( {n + \left( {1 - s} \right)} \right)}}{{\left( {n - s} \right)\varGamma \left( {n + 1} \right)}} \leq \frac{1}{n}\frac{1}{{n^s \left( {1 - \frac{s}{n}} \right)}} \le \frac{1}{n}\frac{1}{{n^s }} + \frac{2}{{n^2 }}\frac{s}{{n^s }}.
\]
\end{proof}

\begin{proof}[Proof of Theorem \ref{theorem}] As shown by Steffensen,
\[
b_n  = \frac{{\left( { - 1} \right)^{n + 1} }}{\pi }\int_0^1 {\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right)\frac{{\varGamma \left( {n - s} \right)}}{{\varGamma \left( {n + 1} \right)}}ds} .
\]
By Lemma \ref{lemma2} we find that
\begin{align*}
0 \leq \int_0^1 {\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right)\frac{{\varGamma \left( {n - s} \right)}}{{\varGamma \left( {n + 1} \right)}}ds} & - \frac{1}{n}\int_0^1 {\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right)e^{ - ms} ds}\\
& \le \frac{2}{{n^2 }}\int_0^1 {s\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right)e^{ - ms} ds} 
\end{align*}
where $m := \log n$. Hence, we conclude that
\begin{align*}
\left| {b_n   - \frac{{\left( { - 1} \right)^{n + 1} }}{{\pi n}}\int_0^1 {\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right)e^{ - ms} ds} } \right| & \le \frac{2}{{\pi n^2 }}\int_0^1 {s \varGamma \left( {s + 1} \right) \sin \left( {\pi s} \right) e^{ - ms} ds} \\
& < \frac{2}{{\pi n^2 }}\int_0^{ + \infty } {s e^{ - ms} ds}  = \frac{2}{{\pi n^2 \log^2 n}}.
\end{align*}
Thus, we derived the asymptotic formula
\begin{align*}
b_n & = \frac{{\left( { - 1} \right)^{n + 1} }}{{\pi n}}\int_0^1 {\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right)e^{ - ms} ds}  + \mathcal{O}\left( {\frac{1}{{n^2 \log^2 n}}} \right)\\
& = \frac{{\left( { - 1} \right)^{n + 1} }}{n}\int_0^1 {\frac{s}{{\varGamma \left( {1 - s} \right)}}e^{ - ms} ds}  + \mathcal{O}\left( {\frac{1}{{n^2 \log ^2 n}}} \right)
\end{align*}
as $n\rightarrow +\infty$. Here we used the reflection formula $\varGamma \left( {s + 1} \right)\sin \left( {\pi s} \right) = \pi s/\varGamma \left( {1 - s} \right)$. The function $s/\varGamma \left( {1 - s} \right)$ is analytic in the range $0 < s < 1$ (in fact, it is an entire function), let
\[
\frac{s}{\varGamma \left( {1 - s} \right)} = \sum\limits_{k \ge 1} {\gamma _k s^k } .
\]
Define the function $\Delta\left(s\right)$ in the positive real variable $s$ by
\[
\Delta \left( s \right) := \begin{cases} s/\varGamma \left( {1 - s} \right), & \text{if } 0 < s < 1;\\
0, & \text{if } s \geq 1. \end{cases}
\]
Then our asymptotic formula becomes
\[
b_n  = \frac{{\left( { - 1} \right)^{n + 1} }}{{n}}\int_0^{ + \infty } {\Delta \left( s \right)e^{ - ms} ds}  + \mathcal{O}\left( {\frac{1}{{n^2 \log^2 n}}} \right).
\]
The integral satisfies the conditions of Watson's Lemma and we obtain that for each nonnegative integer $N$
\begin{align*}
b_n & = \frac{{\left( { - 1} \right)^{n + 1} }}{{n}}\left(\sum\limits_{k = 0}^{N-1} {\frac{{\left( {k + 1} \right)! \gamma _{k+1}}}{{\log ^{k + 2} n}}}  + \mathcal{O}\left( {\frac{1}{{\log ^{N + 2} n}}} \right)\right) + \mathcal{O}\left( {\frac{1}{{n^2 \log^2 n}}} \right)\\
& = \frac{{\left( { - 1} \right)^{n + 1} }}{{n\log ^2 n}}\left( {\sum\limits_{k = 0}^{N-1} {\frac{{\beta _k }}{{\log ^k n}}}  + \mathcal{O}\left( {\frac{1}{{\log ^N n}}} \right) + \mathcal{O}\left( {\frac{{1}}{n}} \right)} \right)
\end{align*}
as $n\rightarrow +\infty$, where
\begin{align*}
\beta _k : = \left( {k + 1} \right)!\gamma _{k + 1} & = \left[ {\frac{{d^{k + 1} }}{{ds^{k + 1} }}\left( {\frac{s}{\varGamma \left( {1 - s} \right)}} \right)} \right]_{s = 0}\\
& = \left[ {\frac{{d^{k + 1} }}{{ds^{k + 1} }}\left( { - \frac{1}{{\varGamma \left( { - s} \right)}}} \right)} \right]_{s = 0} \\
& = \left( { - 1} \right)^k \left[ {\frac{{d^{k + 1} }}{{ds^{k + 1} }}\left( {\frac{1}{{\varGamma \left( s \right)}}} \right)} \right]_{s = 0} .
\end{align*}
Since for every $N\geq0$
\[
\frac{1}{n} = o\left( {\frac{1}{{\log ^N n}}} \right)
\]
as $n\rightarrow +\infty$, we have proved the theorem.
\end{proof}

\section{Recurrence for the coefficients $\beta_k$}

Here we derive a recurrence formula for the coefficients $\beta_k$ in the asymptotic expansion \eqref{eq1}. Since the reciprocal of the Gamma function is an entire function, we can write it as a power series around $0$, say
\[
\frac{1}{{\varGamma \left( s \right)}} = \sum\limits_{k \ge 1} {\alpha _k s^k } .
\]
According to the formula for the Taylor coefficients and equation \eqref{eq2}, we have
\begin{equation}\label{eq3}
\alpha _k  = \frac{1}{{k!}}\left[ {\frac{{d^k }}{{ds^k }}\left( {\frac{1}{{\varGamma \left( s \right)}}} \right)} \right]_{s = 0}  = \frac{{\left( { - 1} \right)^{k - 1} \beta _{k - 1} }}{{k!}}.
\end{equation}
It is known that $\alpha_1 = 1$, $\alpha_2 = \gamma$ and
\[
k\alpha _{k + 1}  = \gamma \alpha _k  - \sum\limits_{j = 2}^k {\left( { - 1} \right)^j \zeta \left( j \right)\alpha _{k - j + 1} } 
\]
for $k\geq2$ (cf.~\cite[p.\ 139]{NIST}). This can be seen as follows. The Digamma function has the power series
\[
\psi \left( {s + 1} \right) =  - \gamma  + \sum\limits_{k \ge 2} {\left( { - 1} \right)^k \zeta \left( k \right)s^{k - 1} } 
\]
(see, e.g., \cite[p.\ 139]{NIST}) and differentiating
\[
\frac{1}{{\varGamma \left( {s + 1} \right)}} = \frac{1}{{s\varGamma \left( s \right)}} = \sum\limits_{k \ge 1} {\alpha _k s^{k - 1} } 
\]
we find the power series for
\[
 - \frac{{\varGamma '\left( {s + 1} \right)}}{{\varGamma ^2 \left( {s + 1} \right)}} =  - \frac{{\psi \left( {s + 1} \right)}}{{\varGamma \left( {s + 1} \right)}},
\]
but this power series can be obtained by Cauchy multiplication of the two previous ones. In this way we get the recursion formula for the $\alpha _k$'s. From this recursion formula and \eqref{eq3} it follows that $\beta _0  = 1$, $\beta _1  =  - 2\gamma$ and
\[
k\beta _k  =  - \gamma \left( {k + 1} \right)\beta _{k - 1}  - \sum\limits_{j = 2}^k {\binom{k + 1}{j}j!\zeta \left( j \right)\beta _{k - j} } 
\]
for $k\geq2$.

\section{Acknowledgement}

I would like to thank the anonymous referee for his/her thorough, constructive and helpful comments and suggestions on the manuscript.

\begin{thebibliography}{1}
\setlength{\itemsep}{2pt}

\bibitem{HTD}
H.~T.~Davis, The approximation of logarithmic numbers,
{\em Amer. Math. Monthly\/} {\bf 64} (1957), 11--18.

\bibitem{FWJO}
F.~W.~J.~Olver,
\emph{Asymptotics and Special Functions.}
A. K. Peters. Reprint, with corrections,
of original Academic Press edition, 1974.

\bibitem{NIST}
F.~W.~J.~Olver, D.~W.~Lozier, R.~F.~Boisvert and C.~W.~Clark (eds.), 
\emph{NIST Handbook of Mathematical Functions.}
Cambridge University Press, New York, 2010.

\bibitem{JFS}
J.~F.~Steffensen,
On Laplace's and Gauss' summation-formulas,
{\it Skandinavisk Aktuariettidskrift} (1924), 2--4.

\bibitem{RW}
R.~Wong, 
\emph{Asymptotic Approximations of Integrals.}
Academic Press.  Reprinted with
corrections by SIAM, Philadelphia, PA, 2001.

\end{thebibliography}


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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B83; Secondary 41A60.

\noindent \emph{Keywords: }
Bernoulli numbers of the second kind, asymptotic expansions, gamma function.

\bigskip
\hrule
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\noindent (Concerned with sequences
\seqnum{A002206},
\seqnum{A002207}, and
\seqnum{A048994}.)

\bigskip
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\vspace*{+.1in}
\noindent
Received  January 7 2011;
revised version received March 27 2011. 
Published in {\it Journal of Integer Sequences}, April 15 2011.

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