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\keywords{Stirling numbers, Complex arguments, Hankel contours}
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% Entorno para escribir agradecimientos; similar al entorno abstract
% \begin{ackno} We thank Menganito Smith for ... \end{ackno}

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\begin{document}
\title[Stirling numbers and Hankel contours]
{On Stirling numbers for complex arguments\\ and Hankel contours}

\author[P.~Flajolet]
{Philippe Flajolet}
% \address[P.~Flajolet]{  
\address{\rm P.~Flajolet, Algorithms Project\\ INRIA \\ F-78150 Rocquencourt\\
France}
% \email{Philippe.Flajolet@@inria.fr\vskip0pt
\email{Philippe.Flajolet@inria.fr\vskip0pt
{\it WWW-address{\rm :\ 
}}\texttt{http://www-rocq.inria.fr/algo/flajolet/index.html}
}

\author[H.~Prodinger]
{Helmut Prodinger}
% \address[H.~Prodinger]{
\address{\rm H.~Prodinger,
Institut f\"ur Algebra und Diskrete Mathematik\\
Technical University of Vienna\\
Wiedner Hauptstrasse 8--10\\ A-1040 Vienna, Austria.}

% \email{Helmut.Prodinger@@tuwien.ac.at\vskip0pt
\email{Helmut.Prodinger@tuwien.ac.at\vskip0pt
{\it WWW-address{\rm :\ %%@
}}\texttt{http://info.tuwien.ac.at/theoinf/proding.htm}
}



%\date{December 29, 1997}

\date{January 9, 1998}


\begin{abstract}
Cauchy coefficient integrals and Hankel contours provide a natural
generalization of Stirling numbers for unrestricted complex values of their
arguments. Many classical identities survive such an extension.
\end{abstract}

\maketitle

\section{Introduction}  

Richmond and Merlini have introduced in~\cite{RiMe97}
an extension of Stirling's subset numbers $\stwo xy$ and
cycle numbers $\sone xy$  
when $x-y$ is an integer. They also propose a 
further generalization when
$x-y$ is not an integer, but most classical properties are no longer
preserved. As the authors  say about their
most general extension in~\cite[p.~76]{RiMe97}: ``{\em It
seems to us that 
the ideas used to derive identities and recurrences lead to
complicated formulas in general. There are significant terms resulting
from the fact that the integrands are not single valued and also from
the fact that the contours change.}''

In this note, we give an alternative and more natural extension of
Stirling numbers of complex arguments
for which most classical identities
are still satisfied. (We restrict ourselves to the most common properties
leaving it 
to the imagination of the reader to go further.)
Like in~\cite{RiMe97},
our approach starts with Cauchy coefficient integrals. However,
in contrast to~\cite{RiMe97}, we use a Hankel contour that has the merit of
{\em not\/} being dependent on particular index values. 
This intrinsic character of the contour precisely ensures the
permanence of identities.

Relevant references
for the classical theory are~\cite{Adamchik97, Comtet74, GrKnPa94, 
Knuth92}, the latter paper being an excellent 
historical account of Stirling numbers.

\section{Stirling numbers of complex index}

By definition, the Stirling subset numbers (`of the second kind')
$\stwo nk$ are for  $n,k\in\mathbb N$ given by
\begin{equation*}   
\stwo nk=\frac{n!}{k!}[z^n]\left(e^z-1\right)^k,
\end{equation*}
or, by Cauchy's coefficient formula,
\begin{equation}   \label{eq2}
\stwo nk=\frac{n!}{k!}\cauchy\int_\gamma\left(e^z-1\right)^k\frac{dz}{z^{n+1}},
\end{equation}
where the integration contour $\gamma$ is a small contour
encircling the origin. As $n$ is nonnegative in  (\ref{eq2}), 
the contour $\gamma$ can be deformed into a \emph{ Hankel contour}
${\mathcal{H}}$  (see~\cite{WhWa27})
that starts from $-\infty$ below the negative axis, surrounds the origin
counterclockwise and returns to $-\infty$  in the half plane $\Im z>0$.
Details of 
${\mathcal{H}}$ are of course immaterial
and  we need only assume that it is at distance
$\le1$ from the real axis.


This suggests the following definition
\begin{definition}
The Stirling numbers of complex arguments (``fractional order'')
are defined for $\Re(x)>0$ by
\begin{equation}   \label{eq3}
\stwo xy=\frac{x!}{y!}\cauchy\int_\H\left(e^z-1\right)^y\frac{dz}{z^{x+1}},
\end{equation}
where $s!=\Gamma(s+1)$.
The determination of $\left(e^z-1\right)^y$  is the principal determination  on the part
of the contour $\Re z>0$ extended by continuity to the whole of $\H$.
\end{definition}


This definition extends the Stirling subset numbers to 
{\em arbitrary} complex arguments $(x,y)$  satisfying $\Re x>0$.
When $\Re(x)\le 0$, the integral diverges. 
However,
through integration by parts, one finds, when $\Re(x)>1$,
\begin{equation}   \label{eq3bis}
\stwo
xy=\frac{(x-1)!}{(y-1)!}\cauchy\int_\H
e^z\left(e^z-1\right)^{y-1}\frac{dz}{z^{x}}.
\end{equation}
The integral in~(\ref{eq3bis}) now converges for all values of
$x$ and $y$. The variant form~(\ref{eq3bis})
shows that $\stwo xy$
can be continued for $\Re(x)\le 0$ into a meromorphic function
of $x$ (for any fixed $y$), with poles at the nonpositive integers.
As a function of $y$ (for any fixed~$x$ not a negative integer), it is
entire. 

Our definition of generalized Stirling numbers in~(\ref{eq3})
and~(\ref{eq3bis}) coincides with that of~\cite{RiMe97} {\em only\/}
when $x-y$ is an integer. It differs significantly in other cases,
since Richmond and Merlini propose to define the general form of $\stwo x y$ 
by means of a saddle point circle that, contrary to~$\H$, is 
dependent upon the particular values of $x,y$.


%%%%%% BULLSHIT!!!!
% The relation 
% (\ref{eq3}) defines $\stwo xy $ only when 
% $\Re x>0$ since otherwise the integral diverges. However from subtracting the
% behavior of the integrand at $-\infty$, we get analytic continuation.
% The principle is as follows. To extend the integral in (\ref{eq3}), 
% write, assuming initially 
% $\Re x>0$,
% \begin{equation*}   
% \cauchy\int_\H\left(e^z-1\right)^y\frac{dz}{z^{x+1}}=
% \cauchy\int_\H\Big(\left(e^z-1\right)^y-1\Big)\frac{dz}{z^{x+1}}
% +\cauchy\int_\H\frac{dz}{z^{x+1}}.
% \end{equation*}
% The first integral is an entire function of $x$. The second one is readily estimated by
% standard Hankel technology and has only a simple pole at $x=0$. Therefore: 
% \emph{The function $\stwo xy$  is meromorphic in the whole of ${\mathbb C}$ } with 
% (possibly) poles
% at  the nonpositive integers.
% 
% 
% 
% (Another way of removing the divergence of the integral would be
%  by \emph{integration by parts}.)



\section{Relations}

As announced, we show now that the most common properties 
are preserved for our generalized
Stirling numbers as defined  by~(\ref{eq3}) and~(\ref{eq3bis}).

\subsection*{Recurrence} In the integral representation for $\stwo xy$, perform integration
by parts. This gives for $\Re x >1$
\begin{equation*}   
\cauchy\int_\H\left(e^z-1\right)^y\frac{dz}{z^{x+1}}
=
\bigg[-\frac{1}{xz^x}\left(e^z-1\right)^y\bigg]_{\H}
+\frac{y}{x}\,\cauchy\int_\H\left(e^z-1\right)^{y-1}e^y\frac{dz}{z^{x}}
\end{equation*}  
and, upon writing $e^y=(e^y-1)+1$,
\begin{equation*}   
\cauchy\int_\H\left(e^z-1\right)^y\frac{dz}{z^{x+1}}
=\frac{y}{x}\,\cauchy\int_\H\left(e^z-1\right)^{y}e^y\frac{dz}{z^{x}}
+\frac{y}{x}\,\cauchy\int_\H\left(e^z-1\right)^{y-1}e^y\frac{dz}{z^{x}}
\end{equation*}  
or, in standard notation
\begin{equation*}   
\stwo{x}{y}=\stwo{x-1}{y-1}+y\stwo{x-1}{y}.
\end{equation*}
This relation originally established for $\Re x>1$ persists for all
complex $x$  by uniqueness of analytic continuation.

\subsection*{Binomial formula} A binomial expansion of
$(e^y-1)^k$ yields the  classical formula ($n,k\in\mathbb N$), 
\begin{equation*}   
\stwo nk=\frac{1}{k!}\sum_{j=1}^k\binom{k}{j}(-1)^{k-j} j^n.
\end{equation*}
This process naturally extends to complex $x$ giving for all $k\in \mathbb N$,
\begin{equation*}   
\stwo xk=\frac{1}{k!}\sum_{j=1}^k\binom{k}{j}(-1)^{k-j} j^x,
\end{equation*}
upon using the binomial expansion in~(\ref{eq3}) and appealing to
Hankel's original representation of the gamma function~\cite{WhWa27}.

\subsection*{Bell numbers} The Bell numbers of integral order are
defined by their exponential generating function
\begin{equation*}   
\B_n=n![z^n]e^{e^z-1},
\end{equation*}
and they satisfy the relation
\begin{equation*}   
\B_n=\sum_{k=0}^n\stwo nk .
\end{equation*}

This suggests to define the Bell numbers of any complex order $x$, $\Re x>0$,
as
\begin{equation}\label{eqbell}
\B_x=x!\,\cauchy\int_\H e^{e^z-1}\frac{dz}{z^{x+1}}.
\end{equation}  
When generalized in this way, the Bell numbers satisfy
\begin{equation*}   
\B_x=\sum_{k=0}^\infty\stwo xk,
\end{equation*}
as results from expanding the integrand of~(\ref{eqbell}),

\begin{equation*}   
e^{e^z-1}=\sum_{k=0}^\infty\frac{(e^z-1)^k}{k!}
\end{equation*}

\smallskip

\subsection*{Dobinski's formula}
This classical formula \cite{Comtet74}
also generalizes. If we expand
\begin{equation*}   
e^{e^z-1}=e^{-1}\sum_{k=0}^\infty\frac{e^{kz}}{k!}
\end{equation*}
we get
\begin{equation*}   
\B_x=\frac{x!}{e}\sum_{k=0}^\infty\frac{1}{k!}\,
\cauchy\int_\H e^{kz}\frac{dz}{z^{x+1}}.
\end{equation*}
The integral can now be evaluated by use of Hankel's formula for the gamma
function (substitute $kz=t$), and
\begin{equation*}   
\cauchy\int_\H e^{kz}\frac{dz}{z^{x+1}}=\frac{k^x}{x!};
\end{equation*} 
hence, the generalized Dobinski formula, 
\begin{equation}   
\B_x={e}^{-1}\sum_{k=0}^\infty\frac{k^x}{k!}.
\end{equation}



\subsection*{Bernoulli numbers} 
Given that their exponential generating
function is $z/(e^z-1)$, it is
natural to expect Bernoulli numbers to be related to Stirling numbers of type
$\stwo{x}{-1}$. Consider first the case of an integer index $n$. Then  
\begin{equation*} 
\stwo n{y-1}=\frac{n!}{(y-1)!}\cauchy\int_\H
\left(e^z-1\right)^{y-1}\frac{dz}{z^{n+1}}. 
\end{equation*}
As $y\to0$, we have by Cauchy's formula and the fact that
$(y-1)!\sim\frac{1}{y}$, 
\begin{equation*}   
\stwo n{y-1}\sim y\cdot n![z^{n+1}]\frac{z}{e^z-1}= y\,\frac{B_{n+1}}{n+1}.
\end{equation*}
% This holds because of the known expansion of the Gamma function at
% negative integers and the fact that we can deform the Hankel contour
% to a traditional contour used in Cauchy's formula.

This relation gives 
% us the first two terms in the Taylor expansion of $\stwo n{y-1}$: 
\begin{equation*}   
% \stwo n{-1}=0.\qquad 
\frac{d}{dy}\stwo n{y}\bigg|_{y=-1}=\frac{B_{n+1}}{n+1}=\zeta(-n),
\end{equation*}
and more generally, thanks to Hankel's representation of
the $\zeta$ function (see e.g. \cite{WhWa27}),
\begin{equation*}   
% \stwo x{-1}=0,\qquad 
\frac{d}{dy}\stwo x{y}\bigg|_{y=-1}=\zeta(-x).
\end{equation*}
In other words Bernoulli numbers of complex index that are naturally
defined by $B_{x+1}:=(x+1)\zeta(-x)$ are also obtained by
a simple limiting process applied to generalized Stirling numbers of
index $-1$:
\[
\frac{B_{x+1}}{x+1}=\lim_{\epsilon\to0}\frac{1}{\epsilon}{\stwo x
{-1+\epsilon}}=\frac{d}{dy}\stwo x{y}\bigg|_{y=-1}.
\]


\subsection*{Stirling cycle numbers.}
Formula~(\ref{eq3bis}) and the change of variables $z=\log(1+w)$ provide
a logarithmic form of Stirling
subset numbers,
\begin{equation}\label{s1}
\stwo xy=\frac{(x-1)!}{(y-1)!}\cauchy\int_{\H^\star}
\left(\log\frac{1}{1+w}\right)^{-x-1} w^{y-1}\, dw,
\end{equation}
where $\H^\star$ is a ``raindrop contour'' that is the image of $\H$ by
$z\mapsto w=e^z-1$.
(Thus, $\H^\star$ starts at $-1$ in the lower half plane, surrounds~0
anticlockwise and returns to $-1$ in the upper half plane.) 

On the other hand, Cauchy's coefficient formula applied to the
exponential generating function of Stirling cycle numbers gives
\begin{equation}\label{s2}
{\sone n k } =(-1)^{n-k} \frac{n!}{k!}
\cauchy\int_{\H^\star} \big(\log(1+w)\big)^k \, \frac{dw}{w^{n+1}}.
\end{equation}
A direct consequence of~(\ref{s1}) and~(\ref{s2}) is that the
Stirling
cycle numbers of integral arguments arise as limiting cases of
generalized Stirling subset numbers as defined in~(\ref{eq3}),
(\ref{eq3bis}),
\[
{\sone n k} =\lim_{\epsilon\to0} {\stwo {-k+\epsilon} {-n+\epsilon}}.
\]
One encounters once more an instance of the duality
relation ${\sone x y}={\stwo {-y} {-x}}$ that, together with~(\ref{s1}),
confirms that Stirling numbers eventually reduce to a single family
(see \cite{GrKnPa94,   Knuth92}).



% An analogous extension can be worked out via
% \begin{equation}\label{stir1}
% \sone xy=\frac{x!}{y!}\cauchy\int_{\H^\star}
% \left(\log\frac{1}{1-z}\right)^y\frac{dz}{z^{x+1}},
% \end{equation}
% with a suitably chosen Hankel contour $\H^\star$: in effect, 
% integrals of type~(\ref{stir1}) reduce to integrals of type~(\ref{eq3})
% through the change of variable $z\mapsto 1-e^t$.
% It is however simpler (and essentially equivalent) to use the connection
% \begin{equation*}   
% \sone xy=\stwo{-y}{-x}
% \end{equation*} 
% in order to define Stirling's cycle numbers
% in terms of subset numbers (see \cite{GrKnPa94,   Knuth92}).

\begin{small}
\smallskip
\noindent
{\bf Acknowledgements.}
This work was supported in part
by the Long Term Research Project {\em Alcom-IT\/} (\# 20244)
of the European Union, and by the French--Austrian 
Cooperation Project 
Amad\'ee 11j (\# 2868/1997)
\end{small}



   \bibliographystyle{plain}
%\bibliography{pro_bib}
 \begin{thebibliography}{1}

\bibitem{Adamchik97}
V.~Adamchik.
\newblock On Stirling numbers and Euler sums.
\newblock {\em Journal of Computational and Applied Mathematics}, 79:119--130,
  1997.

\bibitem{Comtet74}
L.~Comtet.
\newblock {\em Advanced Combinatorics}.
\newblock Reidel, Dordrecht, 1974.

\bibitem{GrKnPa94}
R.~L. Graham, D.~E. Knuth, and O.~Patashnik.
\newblock {\em Concrete Mathematics (Second Edition)}.
\newblock Addison Wesley, 1994.

\bibitem{Knuth92}
D.~E. Knuth.
\newblock Two notes on notation.
\newblock {\em American Mathematical Monthly}, 99:403--422, 1992.

\bibitem{RiMe97}
B.~Richmond and D.~Merlini.
\newblock Stirling numbers for complex arguments.
\newblock {\em SIAM Journal on Discrete Mathematics}, 10:73--82, 1997.

\bibitem{WhWa27}
E.~T. Whittaker and G.~N. Watson.
\newblock {\em A Course of Modern Analysis}.
\newblock Cambridge University Press, fourth edition, 1927.
\newblock Reprinted 1973.

\end{thebibliography}  
\end{document}




