%On Generalizations of the Stirling Number Triangles,
%Wolfdieter Lang February, 11  2000, update May 31 2000
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\begin{document}
\begin{center}
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{\LARGE\bf On Generalizations of the Stirling Number Triangles\footnote 
{In memory of my mother Else Gertrud Lang.} }\\
\vskip 1.5cm
\large Wolfdieter Lang\\ \medskip
Institut f\"ur Theoretische Physik \\ Universit\"at Karlsruhe \\
Kaiserstra\ss e 12, D-76128 Karlsruhe, Germany\\ \medskip
Email address:
\href{mailto:wolfdieter.lang@physik.uni-karlsruhe.de}{wolfdieter.lang@physik.uni-karlsruhe.de} \\
Home page:
\htmladdnormallink{http://www-itp.physik.uni-karlsruhe.de/${\ \tilde{}}\ $wl}
{http://www-itp.physik.uni-karlsruhe.de/~wl} \medskip
\vskip2.5cm
\bf {Abstract}
\end{center}

{\em
Sequences of generalized Stirling numbers of both kinds are introduced.
These sequences of triangles (i.e. infinite-dimensional lower 
triangular matrices)
of numbers will be denoted by $S2(k;n,m)$ and $S1(k;n,m)$
with $k\in \bf Z $. 
The original Stirling number triangles of the second and first kind 
arise when $k=1$. $S2(2;n,m)$ is identical with
the unsigned $S1(2;n,m)$ triangle, called $S1p(2;n,m)$, which also represents the 
triangle of signless Lah numbers. 
Certain associated number triangles, denoted by $s2(k;n,m)$ and $s1(k;n,m)$, are also 
defined.
Both $s2(2;n,m)$ and $s1(2;n+1,m+1)$ form Pascal's triangle, 
and $s2(-1,n,m)$ turns out to be Catalan's triangle.

Generating functions
are given for the columns of these triangles.
Each ${\bf S2}(k)$ and ${\bf S1}(k)$ matrix is an example of a
Jabotinsky matrix.
The generating functions for the rows 
of these triangular arrays therefore constitute exponential convolution polynomials.
The sequences of the row sums of these triangles are also considered.

These triangles are related
to the problem of obtaining finite transformations
from infinitesimal ones generated by $x^k\,\Dx{}$, for $k\in \bf Z$.
}

\vspace*{+.1in}
\noindent AMS MSC numbers: 11B37, 11B68, 11B83, 11C08, 15A36 
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\section{Overview}
{\it Stirling's numbers of the second kind} (also called {\it subset numbers}),
and denoted by $S2(n,m)$ (or $\left\{\begin{array}{c} n\\m \end{array}\right\}$ 
in the notation of \cite{GKP}, or ${\cal S}_{n}^{(m)}$ in \cite {AS}, or
sequence \seqnum{A008277}
in the database \cite {Sloane}) can be 
defined by
\begin{equation}
E_{x}^{\ n}\equiv (x\,d_{x})^n\speq \sum_{m=1}^n\, S2(n,m)\,x^m\,d_{x}^{\ m}\ \ , \ \ 
n\in {\bf N},\label{(1.1)} 
\end{equation}
where the derivative operator $d_{x}\equiv \Dx{}{}$, and $E_{x}$ is the {\sl Euler}
operator satisfying $E_{x}\,x^k\speq k\,x^k$. A recursion relation can be derived from 
eq.~\ref{(1.1)} by considering 
$x\,d_{x}(x\,d_{x})^{n-1}$, using the convention $S2(n,m)=0$ if $n<m$
to interpret $S2(n,m)$ as a lower triangular, infinite-dimensional matrix 
$\bf {S2}$:
\begin{equation}
S2(n,m)\speq m\,S2(n-1,m) \spp S2(n-1,m-1), \label{(1.2)}
\end{equation}
with initial values $S2(n,0)\equiv 0$ and $S2(1,1)=1$. Because of eq.~\ref{(1.1)} 
these numbers arise when one asks how finite scale transformations (dilations)
look, given infinitesimal ones. This is a special case of the 
exponentiation operation for Lie groups. The generator of 
the abelian  Lie group of scale transformations 
$x^{\prime}=\lambda\, x$, $\lambda \in \bf R_{+}$, is $E_{x}$. In order to 
exhibit these numbers within this framework consider first
\begin{eqnarray}
 e^{c\, x\, d_{x}}&\; =\; & \sum_{n=0}^{\infty}\,\frac{c^n}{n!}\ E_{x}^{\ n} 
\; = \; 1+\sum_{n=1}^{\infty}\,\frac{c^n}{n!}\sum_{m=1}^{n}\, S2(n,m)\, 
x^m\,d_{x}^{\ m}  \label{(1.3)}\\
 &\; = \;& 1+\sum_{m=1}^{\infty}\left ( \sum_{n=m}^{\infty}\frac{c^n}{n!}\, 
S2(n,m)\right )\, x^m\,d_{x}^{\ m} \; =\; 1+\sum_{m=1}^{\infty}\, G2_{m}(c)\, x^m\, d_{x}^{\ m}\ . \ \nonumber  
\end{eqnarray} 
In the third step an interchange of summation has been performed (we
ignore questions of convergence here), and in the last step an exponential generating 
function (e.g.f.) has been introduced
for the $m-$th column of the number triangle, or lower 
triangular matrix, ${\bf S2}$.
The recursion relation 
implies 
$G2_{m}(c)=\frac{1}{m!}\,(G2(c))^m$, with $G2(c)=exp(c)-1$; therefore 
we obtain
\begin{equation}
e^{c\,x\,d_{x}}\speq \sum_{m=0}^{\infty}\frac{1}{m!}\,(G2(c)\,x)^m\,d_{x}^{\ m}
\speq :e^{(exp(c)-1)x\,d_{x}}:\ , \ \label{(1.4)}
\end{equation}
where we have used the linear normal order symbol $:A:$ from quantum
physics.
($:A:$ means expand $A$ in powers of $x$ and $d_x$, and move all operators $d_x$ to the right-hand side, ignoring the usual
commutation rule $[d_{x},x]\equiv d_{x}\,x-x\,d_{x} = 1$. 
For example,
$:(x\,d_{x})^m:\speq x^m\,d_{x}^m.$)
This normal order prescription is applied to each term of the expanded exponential in eq. ~\ref{(1.4)}. 
From Taylor's theorem, we see that for suitable functions $f$ we have
\begin{equation}
e^{c\,x\,d_{x}}\, f(x)\speq :e^{(exp(c)-1)x\,d_{x}}:\,f(x)\speq f(x+(e^c-1)x)\ 
=\ f(x^{\prime}) \ \ , \  \label{(1.5)}
\end{equation}
with $x^{\prime}=e^c\,x$. Therefore the parameter $\lambda$ for finite scale
transformations is $\lambda=e^c$ if $c$ is the parameter for infinitesimal 
transformations.
In the context of
Lie groups this fact is found by
integrating the ordinary differential equation
(see for example \cite{Dr} and the references given there):
\begin{equation}
\D{x(\alpha)}{\alpha\phantom{x()}}\speq c\,x(\alpha), \label{(1.6)}
\end{equation}
for curves starting at a fixed $x:=x(\alpha=0)$. The finite transformation 
maps $x$ to $x^{\prime}:= x(\alpha=1)$. $x^{\prime}$ should not be 
confused with a derivative.   
In this work we will generalize this to the case $E_{k;x}\equiv x^k\,d_{x}$ with
$k\in\bf Z$. It is clear from the solution of the differential equation
\begin{equation}
\D{x(\alpha)}{\alpha\phantom{x()}}\speq c\,x^{k}(\alpha),\ \ \mbox{with initial condition}\ \ x(\alpha=0)\ =: x\ \ \mbox{and its transform}\ 
\ x^{\prime}:= x(\alpha=1), \label{(1.7)}
\end{equation}
that the scale transformation case $k=1$ which has been treated above is special. 
For $k \neq 1$, after separation of variables, we obtain the equation
$(x^{\prime})^{1-k}\spm x^{1-k}\speq (1-k)\,c$. 
Setting
\begin{equation}
x^{\prime}\speq (1+g(k;c;x))\,x \label{(1.8)}
\end{equation}	
we have
\begin{equation}
1+g(k;c;x)\speq \bigl{(}1-(k-1)\,c\,x^{k-1}\bigr{)}^{-\frac{1}{k-1}}\ . 
\label{(1.9)}
\end{equation}
Therefore
\begin{equation}
e^{c\,x^k\,d_{x}}\,f(x)\speq f(x^{\prime})\speq f
\left( (1-(k-1)\,c\,x^{k-1})^{-\frac{1}{k-1}}\,x\right) \label{(1.10)}
\end{equation}
for $k\in {\bf Z}\setminus \{1\}$. The case $k=1$ has been dealt with in 
eq.~\ref{(1.5)}. It can be recovered from
eq.~\ref{(1.10)} by taking the limit $k-1\to 0$. 

$k-${\it Stirling numbers of the second kind}, which
we will denote by $S2(k;n,m)$,
with $S2(1;n,m)=S2(n,m)$ the ordinary Stirling subset numbers, emerge 
in a proof, independent of the one implied by eq.~\ref{(1.10)}, of the following operator identity, valid for 
$k\in \bf Z$,
\begin{equation}
e^{c\,x^k\,d_{x}}\speq :e^{g(k;c;x)\,x\,d_{x}}: \ \  , \label{(1.11)}
\end{equation}
where $g(k;c;x)$ is defined by eq.~\ref{(1.9)} for $k\neq 1$ and 
$g(1;c;x)=G2(c)$ (see eq.~\ref{(1.4)}).
By analogy with eq.~\ref{(1.1)}
the $S2(k;n,m)$ number triangle is defined by
\begin{equation}
E_{k;x}^{\ \ \ n}\equiv (x^k\,d_{x})^n\speq \sum_{m=1}^n\, S2(k;n,m)\,
x^{m+(k-1)\,n}\,d_{x}^{\ m}\ \ , \ \ 
n\in {\bf N}, \ \ k\in {\bf Z} \ ,\label{(1.12)} 
\end{equation}
with the further convention that $S2(k;n,m)=0$ for $n<m$ and $S2(k;n,0)=0$.
These numbers will be shown to satisfy the recursion relation
\begin{equation}
S2(k;n,m)\speq ((k-1)(n-1)+m)\,S2(k;n-1,m) \spp S2(k;n-1,m-1), \label{(1.13)}
\end{equation}
with $S2(k;1,1)=1$ from eq.~\ref{(1.12)}.
The e.g.f. for the $m-$th 
column of the ${\bf S2} (k)$ triangle,
\begin{equation}
G2(k;m;x)\spdef \sum_{n=m}^{\infty}\, S2(k;n,m)\,\frac{x^n}{n!} \ , 
\label{(1.14)}
\end{equation} 
satisfies
\begin{equation}
G2(k;m;x)\speq \frac{1}{m!}\,(G2(k;x))^m  \label{(1.15)}
\end{equation}  
for $k \neq 1$, with
\begin{equation}
G2(k;x)\speq (k-1)\, g2(k;\frac{x}{k-1})\ , \label{(1.16)}\, 
\end{equation}
where 
\begin{equation}
g2(k;y)\spdef \sum_{n=1}^{\infty}\, s2(k;n,1)\,y^n \label{(1.17)}
\end{equation}  
is the ordinary generating function (o.g.f.) for the first column of
the triangle of numbers $s2(k;n,m)$ which is associated to triangle 
$S2(k;n,m)$ by\footnote{These {\it associated Stirling numbers of
the second kind} are not the ones of \cite {Riordan ICA}, p.~76, Table 2.} 
\begin{equation}
s2(k;n,m)\spdef (k-1)^{n-m}\, \frac{m!}{n!}\, S2(k;n,m)\ . \label{(1.18)}
\end{equation}   
These number triangles, or lower triangular infinite-dimensional matrices,
${\bf s2}(k)$, which are here only defined for $k\in {\bf Z}\setminus \{1\}$,
obey the recursion relation
\begin{equation}
 s2(k;n,m)\speq \frac{k-1}{n}\,\left[ (k-1)(n-1)+m\right]\, s2(k;n-1,m)\spp
\frac {m}{n}\, s2(k;n-1,m-1)\ \ , \label{(1.19)}
\end{equation}  
with 
\begin{equation}
s2(k;n,m)\speq 0,\ n<m\ \ ,\ \ s2(k;n,0)\speq 0,\ \  s2(k;1,1)=1\ . \label{(1.20)}
\end{equation}   
It follows that these numbers are nonnegative. At this stage it is not 
obvious that they are integers for every $k\in {\bf Z}\setminus \{1\}$.

\noindent
The o.g.f. of the $m$-th column of the ${\bf s2}(k)$ matrix is
\begin{equation}
g2(k;m;y)\spdef \sum_{n=m}^{\infty}\, s2(k;n,m)\, y^n\speq (g2(k;y))^m\ , 
\label{(1.21)}
\end{equation}
with
\begin{equation}
g2(k;y)\speq y\,c2(1-k;y), \label{(1.22)}
\end{equation} 
and, for $l\in {\bf Z}\setminus \{0\}$,  
\begin{equation}
c2(l;y)\speq \frac{1-(1-l^2\,y)^{\frac{1}{l}}}{l\,y}\ .
\label{(1.23)}
\end{equation}  
It is  clear from eq.~\ref{(1.21)} that ${\bf s2}(k)$ is a convolution
triangle generated from its first column (cf.
\cite{Knuth},\cite {Rogers},\cite {SGWW}). Such ordinary convolution triangles
will be called {\it Bell} matrices \cite{SGWW} (see Note 7). 
For $k\in{\bf Z}\setminus \{1\}$, eq.~\ref{(1.16)} now yields
\begin{equation}
G2(k;x)\speq -1\spp (1+(1-k)\,x)^{\frac{1}{1-k}}~,\ \label{(1.24)}
\end{equation}  
and letting $k-1 \to 0$ we obtain
$G2(1;x)\speq e^x -1\speq G2(x)$.
The infinite-dimensional lower triangular matrices ${\bf S2}(k)$ with integer 
entries are examples of  {\it Jabotinsky} matrices 
({\it cf.} \cite{Knuth}, which also contains earlier
references).
Therefore the o.g.f. of the rows of the triangle ${\bf S2}(k)$ are exponential 
(or binomial) convolution polynomials.
In other words,
the polynomials 
\begin{equation}
S2_{n}(k;x)\spdef \sum_{m=1}^{n}\, S2(k;n,m)\, x^m\ \ , \ \ S2_{0}(k;x):=1\ ,
\label{(1.25)} 
\end{equation}
satisfy
\begin{equation}
S2_{n}(k;x+y)\speq \sum_{p=0}^{n}\, \binom n p \, S2_{p}(k;x)\, S2_{n-p}(k;y)
\speq \sum_{p=0}^{n}\, \binom n p \, S2_{p}(k;y)\, S2_{n-p}(k;x)
\label{(1.26)}
\end{equation}
for $k \in {\bf Z}$.
In the notation of the umbral calculus ({\it cf.} \cite{Roman}) the polynomials 
$S2_{n}(k;x)$ are a special type of {\it Sheffer} polynomials called
{\it associated polynomial sequences}.
An equivalent notation used there for
the general case is ``Sheffer for $(1,f(t))$''. In our case
$f(t)\speq \overline{G2(k;t)})$, where  $\overline{G2(k;t)}\speq (-1+(1+t)^{1-k})/(1-k)$ if $k\neq 1$. This is the compositional inverse of $G2(k;t)$ from eq.~\ref{(1.24)}.
Also $\overline{G2(1;t)}= ln(1+t)$ can be obtained in the limit as $1-k\to 0$. 

\noindent
For negative $k$ the ${\bf S2}(k)$ matrices also contain negative entries. 
The recursion of eq.~\ref{(1.13)} shows that it is possible to define 
nonnegative matrices by
\begin{equation}
  S2p(-k;n,m)\spdef (-1)^{n-m}\, S2(-k;n,m)\ \ ,\ \  k\in \N0.
\label{(1.27)}
\end{equation}
The e.g.f. for column $m$ of the triangle ${\bf S2}p(-|k|)$ is 
\begin{equation}
G2p(-|k|;m;x)\speq \frac{1}{m!}\,(G2p(-|k|;x))^m  \label{(1.28)}
\end{equation}  
with
\begin{equation}
G2p(-|k|;x)\speq -\, G2(-|k|;-x)
\speq 1\spm (1-(|k|+1)\,x)^{\frac{1}{|k|+1}} . \label{(1.29)}\, 
\end{equation}
Eqs.~\ref{(1.19)} and ~\ref {(1.20)} will be seen to imply that the 
${\bf s2}(-|k|)$  matrices have always nonnegative entries.
In Tables 1 and 2
we have listed for some of these 
${\bf s2}(k)$ and
${\bf S2}(k)$ triangles the $A$-numbers under which they can be
viewed in the on-line data-base 
\cite{Sloane2}
(see also \cite{Sloane}). This data-base will henceforth 
be quoted as  {\it EIS} (Encyclopedia of Integer Sequences).
These tables also give
the $A$-numbers of the sequences formed by the 
first columns of the lower triangular matrices, and
of the sequences of the row sums of these matrices.

\noindent
For $l=2$ the
function $c2(l;y)$ defined in eq.~\ref{(1.23)} generates the well-known
{\it Catalan} numbers. For $l\in {\bf Z}\setminus \{0\}$ it defines what we 
call $l-${\it Catalan} numbers. For positive $l$ these sequences were
introduced by O. Gerard in {\it EIS}, who called them {\it Patalan} numbers.
It will be proved later (see Note 11) that $c2(l;y)$ does indeed generate 
integers. Their explicit form can be found in eq.~\ref{E77}.

The {\it Stirling numbers of the first kind}, $S1(n,m)$ ($S_{n}^{(m)}$ in 
\cite {AS}, {\it EIS}: 
\seqnum{A008275}
can be defined from the inversion of Eq.~\ref{(1.1)} by
\begin{equation}
 x^n\,d_{x}^{\ n} \speq \sum_{m=1}^{n} S1(n,m)\, (x\,d_{x})^m \ \ \ .
\label{(1.30)}
\end{equation}  
We also set $S1(n,0)\equiv 0$ and $S1(n,m):= 0$ for $n<m$. In 
(infinite-dimensional) matrix notation we can write \cite{Knuth}
\begin{equation}
{\bf S1} \cdot {\bf S2} \spdef {\bf 1}\speq {\bf S2} \cdot {\bf S1}\ \ . \label{(1.31)}
\end{equation}
The {\it signless Stirling numbers of the first kind}, also known as {\it cycle numbers}, $S1p(n,m)$
% $\left[{n \atop m}\right]$
(or $\left[\begin{array}{c} n\\m \end{array}\right]$ in the notation of 
\cite {GKP}), are 
\begin{equation}
S1p(n,m)\spdef (-1)^{n-m}\, S1(n,m) \ \ . \label{(1.32)}
\end{equation}   
Their recurrence formula is
\begin{equation}
S1p(n+1,m)\speq n\,S1p(n,m)\spp S1p(n,m-1)\ \ , \label{(1.33)}
\end{equation} 
with $S1p(1,1)=1$, $S1p(n,0)=0$ and $S1p(n,m)=0$ for $n<m$.

The {\it generalized} $k-${\it Stirling numbers of the first kind} $S1(k;n,m)$
are defined analogously by inverting eq.~\ref{(1.12)}, {\it i.e.}
\begin{equation}
x^{kn}\,d_{x}^{\ n}\speq \sum_{m=1}^{n} S1(k;n,m)\, x^{(k-1)(n-m)}\,(x^k\,d_{x})^m \,\ \   {\rm for}\ \  k\in {\bf Z}\ ,\ n\in \Na\ .  
\label{(1.34)}
\end{equation} 
In matrix notation:
\begin{equation}
{\bf S1}(k) \cdot {\bf S2}(k) \speq {\bf 1}\speq {\bf S2}(k) \cdot {\bf S1}(k)\ \ \ {\rm for}\ \ k \in {\bf Z}.   \label{(1.35)}
\end{equation} 
For $k\in {\bf N}$ we define the {\it nonnegative} $k-${\it Stirling numbers of the 
first kind} by 
\begin{equation}
S1p(k;n,m)\spdef (-1)^{n-m}\, S1(k;n,m) \ .\label{(1.36)}
\end{equation} 
For $-k \in \N0$ the numbers $S1(k;n,m)$ are nonnegative.

\noindent
The recurrence relation for $k-$Stirling numbers of the 
first kind is  
\begin{equation}
S1(k;n,m)\speq -[(k-1)\,m+n-1]\,S1(k;n-1,m)\spp S1(k;n-1,m-1)\ \ , 
\label{(1.37)}
\end{equation} 
with $S1(k;1,1)=1$, $S1(k;n,0)=0$ and $S1(k;n,m)=0$ for $n<m$.
For $k\neq 1$ we also introduce the {\it associated $k-${Stirling} numbers of the first 
kind} \footnote{These associated Stirling numbers of the first kind are 
not the ones appearing in \cite {Riordan ICA}, p.~75, table 2.}
\begin{equation}
s1(k;n,m)\spdef (1-k)^{n-m}\,\frac{m!}{n!}\, S1(k;n,m)\ ,\label{(1.38)}
\end{equation}  
which turn out to be always nonnegative.
They satisfy the recursion 
\begin{equation}
s1(k;n,m)\speq \frac {k-1}{n}\bigl{(}(k-1)\,m+n-1\bigr{)}\, s1(k;n-1,m) +\frac{m}{n}\, s1(k;n-1,m-1)\,
\label{(1.39)}
\end{equation} 
with $s1(k;n,m)= 0$ for $n<m$, $s1(k;1,1) = 1$ and $s1(k;n,0):= 0$. At this 
stage it is not obvious that the $s1(k;n,m)$ are in fact integers.

\noindent
The o.g.f. for the $m-$th column of the number triangle 
$s1(k;n,m)$ 
will be shown to be
\begin{equation}
g1(k;m;y) \speq \sum_{n=m}^{\infty}\, s1(k;n,m)\, y^n \speq (g1(k;y))^m\ ,
\label{(1.40)}
\end{equation} 
for $k\in {\bf Z}\setminus \{1\}$,
with
\begin{equation}
g1(k;y)\speq y\, c1(k-1;y),\label{(1.41)}
\end{equation} 
and, for $l \in {\bf Z}\setminus \{0\}$, 
\begin{equation}
c1(l;y)\speq \frac{-1\spp (1-l\, y)^{-l}}{l^2\,y}\ .\label{(1.42)}
\end{equation} 
Hence ${\bf s1}(k)$ is, like ${\bf s2}(k)$, a convolution triangle generated
from its $m=1$ column, {\it i.e.} both are Bell matrices.

\noindent
For $l \in {\bf N}$ the function $c1(l;y)$ generates the numbers	
\begin{equation}
c1(l;y)\speq \sum_{n=0}^{\infty}\, c1_{n}^{(l)}\,y^n\ ,\ \ 
c1_{n}^{(l)}\speq \binom {n+l} {l-1} \, l^{n-1}\ . \label{(1.43)}
\end{equation}
For $l=2$ this is the {\it EIS} sequence 
\seqnum{A001792} $\{1,3,8,20,48,...\}$.
For negative $l$,
$c1(l;y)$ becomes a polynomial in $y$; {\it e.g.} $c1(-2;y)=1+y$,
or $g1(-1;y) = y+ y^2.$ The coefficients of these polynomials define a 
triangle of numbers found under the {\it EIS} number 
\seqnum{A049323}.  Their explicit
form is, for $l\in \Na$, 
\begin{equation}
c1_{n}^{(-l)}\speq {\binomial{l}{n+1}}\,l^{n-1}\ \ 
{\rm for}\  n=0,1,...,l-1, \ {\rm and}\  0 \ {\rm otherwise}. \label{(1.44)}
\end{equation}
Eq. ~\ref{(1.43)} now implies, using eqs. ~\ref{(1.41)} and ~\ref{(1.40)},
that $s1(k;n,m)$ is indeed an integer for every $k\in {\bf Z}\setminus \{1\}$.
An explicit form for the entries in the first column is
\begin{equation}
s1(k;n,1)\speq \left\{ \begin{array}{ll}
{\binomial{k-2-n}{k-2}}\,(k-1)^{n-2}& \mbox{for $k=2,3,..., {\rm and} 
\ n\in \Na$}\\
 \\
{\binomial{|k|+1}{n}}\,(|k|+1)^{n-2}& \mbox{for $-k\in \N0\, ,\,n= 1,2,... 
|k|+1\ . $}
\end{array}\right. \label{(1.45)}
\end{equation}
The e.g.f.s for the $m-$th column of the signless $k-$Stirling
numbers of the first kind are then, for $k=2,3,...$,
\begin{eqnarray}
G1p(k;m;x) &\spdef& \sum_{n=m}^{\infty}\, S1p(k;n,m)\, \frac{x^n}{n!}\ ,
\label{(1.46)}\\
&\speq & \frac{1}{m!}\, \Bigl{[}(k-1)\,g1(k;\frac{x}{k-1})\Bigr{]}^m\ . 
\label{(1.47)}
\end{eqnarray} 
The case $k=1$ corresponds to the ordinary unsigned Stirling numbers
$S1p(n,m)$ with e.g.f. for column $m$ given by 
$G1p(1;m;x)\speq \frac{1}{m!}\,\bigl{(}-ln(1-x)\bigr{)}^m$.
From 
eqs.~\ref{(1.47)}, ~\ref{(1.41)} and ~\ref{(1.42)},
\begin{equation}
G1p(k;1;x)\speq (k-1)\, g1(k;\frac{x}{1-x})\speq \frac{1}{k-1}\, 
(-1\spp \frac{1}{(1-x)^{k-1}})\ ,\label{(1.48)}
\end{equation}
and we recover the result for $k=1$ from l'H\^{o}pital's rule in the limit 
$k-1\to 0 $.
Note that  $G1(k;1;x)\sspeq -G1p(k;1;-x)\sspeq \overline{G2(k;1;x)}$ for $k \in
\bf{Z}$.

\noindent
For $-k\in \N0$ the e.g.f. for the $m-$th column of the nonnegative 
triangular matrix ${\bf S1}(-|k|)$ is, from eqs.~\ref{(1.36)} and 
~\ref{(1.46)}, $G1(-|k|;m;x)\speq (-1)^m\,G1p(-|k|;m;-x)$, hence
\begin{equation}
G1(k;1;x)\speq \frac{1}{1+|k|}\,(-1\spp (1+x)^{1+|k|})\ \ \ \text{for} 
\ \ -k\in \N0\ .       \label{(1.49)}
\end{equation}
For the signed matrix ${\bf S1}(-|k|)$  with elements 
defined by $S1s(-|k|;n,m):= (-1)^{n-m}\,S1(-|k|,n,m)$ the e.g.f. of the 
$m-$th column is $G1s(-|k|;m;x)\speq (-1)^m\, G1(-|k|;m;-x)$, {\it i.e.}
$G1s(-|k|;x)\, \equiv \, G1s(-|k|;1;x)\speq (1-(1-x)^{1+|k|})/(1+|k|)$ for 
$k\in \N0$.

Tables 3 and 4 give the {\it EIS} A-numbers of some of the number
triangles ${\bf s1}(k)$, ${\bf S1}(k)$ and ${\bf S1p}(k)$.
The A-numbers
of the $m=1$ column and of the sequence of row sums for each triangle are 
also given there. 

The o.g.f. of the row sequences of triangle ${\bf S1}(k)$ are also 
exponential (or binomial) convolution polynomials.
In other words the polynomials
\begin{equation}
S1_{n}(k;x)\spdef \sum_{m=1}^{n}\, S1(k;n,m)\, x^m\ \ , \ \ S1_{0}(k;x):=1\ , \ \ k \in {\bf Z} \ ,
\label{(1.50)} 
\end{equation}
satisfy eq.~\ref{(1.26)} with $S2$ replaced by $S1$. 
In the notation of the umbral calculus ({\it cf.} \cite{Roman}) the polynomials 
$S1_{n}(k;x)$ are a special type of Sheffer polynomials called
associated polynomial sequences 
or ``Sheffer for
$(1,f(t))$.''
Here $f(t)\speq \overline{G1(k;t)})$, where
$\overline{G1(k;t)}\speq G2(k;t)$ is given, for $k\neq 1$, in  eq.~\ref{(1.24)}.
Also $\overline{G1(1;t)}= G2(1;t)= exp(t)-1$ is obtained in the limit as $1-k\to 0$.

Each sequence of row sums of a triangle of the type considered in this
work is generated by a function which depends on the generating function of 
the triangle's first $(m=1$) column.
For the ${\bf s2}(k)$ and
${\bf s1}(k)$ triangles, which can be considered as Bell matrices,
these o.g.f.s are, for $k \in {\bf Z}\setminus \{1\}$,
\begin{eqnarray}
r2(k;x)& \speq&  \frac{g2(k;x)}{1-g2(k;x)}\speq 
\frac{-1\spp [1-(1-k)^2\, x]^{\frac{1}{1-k}}}
{k\spm [1-(1-k)^2\, x]^{\frac{1}{1-k}}} , \label{(1.51)}\\
&& \nonumber\\
r1(k;x)& \speq&  \frac{g1(k;x)}{1-g1(k;x)}\speq 
\frac{1\spm [1-(k-1)\, x]^{k-1}}{(1+(1-k)^2)\,[1-(k-1)\, x]^{k-1}\spm 1}
\label{(1.52)}
\end{eqnarray}
For the ${\bf S2}(k)$ ($k\in \N0$) and ${\bf S2p}(k)$ ($-k\in \N0$) triangles,
which can be interpreted as Jabotinsky matrices,
the e.g.f.s for the sequences of row sums are 
\begin{eqnarray}
R2(k;x)& \speq&  e^{G2(k;x)}\spm 1\speq 
exp\,[-1+(1-(k-1)\,x)^{\frac{1}{1-k}}]\spm 1\ \  , \label{(1.53)}\\
&  &  \nonumber \\
R2p(-|k|;x)& \speq&  e^{G2p(-|k|;x)}\spm 1\speq 
exp\, [1-(1-(1+|k|)\, x)^{\frac{1}{1+|k|}}]\spm 1 \ \ . \label{(1.54)}
\end{eqnarray}
For the ${\bf S1p}(k)$ ($k\in \N0$) and ${\bf S1}(k)$ ($-k\in \N0$) triangles,
which can also be interpreted as Jabotinsky matrices,
the e.g.f.s for the sequences of row sums are  
\begin{eqnarray}
R1p(k;x)& \speq&  e^{G1p(k;x)}\spm 1\speq 
exp \Bigl{(}\frac{1}{k-1}\,[-1+(1-x)^{\frac{1}{k-1}}]\Bigr{)} \spm 1\ \  , 
\label{(1.55)}\\
& &  \nonumber \\
R1(-|k|;x)& \speq&  e^{G1(-|k|;x)}\spm 1\speq 
exp \Bigl{(}\frac{1}{1+|k|} [-1+(1+x)^{1+|k|}]\Bigr{)}\spm 1 \ \ . 
\label{(1.56)}
\end{eqnarray}
The special case $k=1$ can be obtained for $R2(k;x)$ and $R1p(k;x)$ 
by taking the limit as $k-1\to 0$.

In Sections 2 and 3 
we will give proofs of the results stated above.

\section{${\bf k-}$Stirling numbers of the second kind}
{\bf Definition 1:} $S2(k;n,m)$.
The $k$-{\it Stirling numbers of the second kind}, $S2(k;n,m)$, are defined 
for $k \in {\bf Z}$ by eq.~\ref{(1.12)}.

\vspace*{+.1in}
\noindent
{\bf Lemma 1:} The numbers $S2(k;n,m)$ satisfy the
recursion relation eq.~\ref{(1.13)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Consider $(x^k\,d_{x})^n\speq x^k\,d_{x}\,(x^k\,d_{x})^{n-1}$ 
and use eq.~\ref{(1.12)}
with $n\to n-1$ together with the lower triangular matrix conditions given 
after this eq.
Then compare coefficients of $\{x^m\,d_{x}^{\ m}\}_{1}^{n}$.\ \  \eop 

\vspace*{+.1in}
\noindent
{\bf Note 1:} It follows from eq.~\ref{(1.13)} and the initial conditions
that the $S2(k;n,m)$ are always integers.

\vspace*{+.1in}
\noindent
{\bf Definition 2:} $s2(k;n,m)$.
The {\it associated} $k$-{\it Stirling numbers of the second kind},
$s2(k;n,m)$, are defined for $k \in {\bf Z}\setminus \{1\}$ by 
eq.~\ref{(1.18)}.

\vspace*{+.1in}
\noindent
{\bf Lemma 2:} The numbers $s2(k;n,m)$ satisfy the
recursion relation 
given by eqs.~\ref{(1.19)} and ~\ref{(1.20)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Rewrite eq.~\ref{(1.13)} for $s2(k;n,m)$. \eop

\vspace*{+.1in}
\noindent
{\bf Note 2:} That the $s2(k;n,m)$ are indeed integers will be proved 
much later in Lemma 19.

\vspace*{+.1in}
\noindent
{\bf Note 3:} For $k=1$ eqs. 19 and 20 give
the (infinite-dimensional) unit matrix 
${\bf s2}(1)= {\bf 1}$. This will be used
as the definition of {\bf s2}(1).

\vspace*{+.1in}
\noindent
{\bf Lemma 3:} Nonnegativity of ${\bf s2}(k)$.
The entries of the
lower triangular matrix ${\bf s2}(k)$ are nonnegative 
for each $k\in {\bf Z}$.

\vspace*{+.1in}
\noindent
{\it Proof}: If $k-1\geq 0$ this follows from 
eq.~\ref{(1.19)}. For $1-k\in \Na$ the first term in
eq.~\ref{(1.19)} becomes negative if and only if  $(1-k)\,(n-1)<m$ and $n-1\geq m$ 
(otherwise $s2(k;n-1,m)$ vanishes). But the first condition contradicts the
second.\ \ \eop

\vspace*{+.1in}
\noindent
{\bf Lemma 4:} The o.g.f. $g2(k;m,y)$
defined in the first of eqs.~\ref{(1.21)} for
the $m-$th column sequence 
of the lower triangular matrix ${\bf s2}(k)$ with $k\in {\bf Z}\setminus 
\{1\}$ satisfies the first 
order linear differential-difference equation
\begin{eqnarray}
[1-(k-1)^2\,y]\,g2^{\prime}(k;m,y)\spm m\,(k-1)\,g2(k;m,y)\spm m \,
g2(k;m-1,y)
\speq 0\ \ , \  \ && \label{(2.1)}\\ 
g2(k;m,0)=0\ ,\ m\in \Na; \  \ g2^{\prime}(k;m,y)|_{y=0}= 0\ ,\ 
m\in \{2,3,...\}\ ;\  g2^{\prime}(k;1,y)|_{y=0}= s2(k;1,1)= 1 . 
 &&\label{(2.2)}
\end{eqnarray} 
The prime denotes differentiation with respect to the variable $y$.

\vspace*{+.1in}
\noindent
{\it Proof}: Compute $y\,{\frac{d\ }{dy}}\,\sum_{n=m}^{\infty}\,n\,s2(k;n,m)\, y^n$
with the help of the recurrence relation in eqs.~\ref{(1.19)} and ~\ref{(1.20)} 
for $y\neq 0$.
For $y=0$ the conditions given in eq.~\ref{(2.2)} follow from the definition
of $g2(k;m,y)$.\ \ \  \eop 

\vspace*{+.1in}
\noindent
{\bf Lemma 5:} Using
$g2(k;m,y)\speq(g2(k;1,y))^m$,
$g2(k;y):= g2(k;1,y)$
satisfies
the first order differential equation
\begin{equation}
[1-(k-1)^2\,y]\,g2^{\prime}(k;y)\spm (k-1)\,g2(k;y)\spm 1\speq 0 \ . 
 \label{(2.3)}
\end{equation} 
for $k\in {\bf Z}\setminus \{1\}$.

\vspace*{+.1in}
\noindent
{\it Proof}: Immediate from Lemma 4. \eop

\vspace*{+.1in}
\noindent
{\bf Lemma 6:} The solution to the differential  eq.~\ref{(2.3)} with initial
condition $g2(k;0)=0$ is, for $k\in {\bf Z}\setminus \{1\}$,
\begin{equation}
g2(k;y)\speq {\frac{1}{[1-(k-1)^2\,y]^{\frac{1}{k-1}}}}\, 
{\frac{1-[1-(k-1)^2\,y]^{\frac{1}{k-1}}}{k-1}}\ =:\  
{\frac{y}{[1-(k-1)^2\,y]^{\frac{1}{k-1}}}}\, c2(k-1;y)\ .
\label{(2.4)}
\end{equation}
{\it Proof}: Standard integration of a first order inhomogeneous differential equation
of the form $g^{\prime}(y)+ f(y)\, g(y) = k(y)$. \eop 

\vspace*{+.1in}
\noindent
{\bf Note 4:} {\it Generalized Catalan numbers}.
The $l$-Catalan numbers (for $l \in {\bf Z} \setminus \{0\}$) have
\begin{equation}
c2(l;x)\spdef {\frac{1-[1-l^2\,x]^{\frac{1}{l}}}{l\,x}}\
\label{(2.5)}
\end{equation}
as o.g.f.
The case $l=2$ corresponds to the ordinary Catalan numbers ({\it EIS} 
\seqnum{A000108}.
For positive $l$ these numbers have been called Patalan numbers by 
Gerard in {\it EIS} ({\it cf.} 
\seqnum{A025748}--\seqnum{A025755}
for $l=3..10$).

That $c2(l;y)$ generates integers will follow later from the fact that 
$s2(k;n,m)$ is always an integer (see Notes 2 and 11).
Because $c2(-l;x)\speq c2(l;x)/(1-l^2\,x)^{\frac{1}{l}}$, one
can write $g2(k;y)\speq y\,c2(1-k;y)$, as stated in  eq.~\ref{(1.22)}.

\vspace*{+.1in}
\noindent
Consider the expansion $1/(1-l^2\,x)^{1/l}\sspeq \sum_{n=0}^{\infty}\, 
b_{n}^{(l)} \, x^n\ $, where $b_{n}^{(l)}\sspeq l^n\,(\prod_{j=1}^{n}\,(j\,l\sspp 1 \sspm l))/n!$ and 
$b_{0}^{(l)}\sspeq 1$, $n\geq 1$.
Therefore the sequence $\{c2^{(-l)}_{n}\}_{n=0}^{\infty}$ generated by 
$c2(-l;x)$ for $l\in \Na$ is
the (ordinary) convolution of the sequence $\{b^{(l)}_{n}\}_{n=0}^{\infty}$ 
with the sequence $\{c2^{(l)}_{n}\}_{n=0}^{\infty}$. See {\it e.g.} 
{\it EIS} 
\seqnum{A035323}
for $l=-10$.

\vspace*{+.1in}
\noindent
Since we have put ${\bf s2}(1)\,=\,{\bf 1}$ we take $g2(1;y)\,=\,y$.

\vspace*{+.1in}
\noindent
{\bf Lemma 7:} The e.g.f. for the $m$-th column sequence of the 
$k-$Stirling triangle of the
second kind ${\bf S2}(k)$, defined in eq.~\ref{(1.14)}, is 
$G2(k;m;x)\speq \frac{1}{m!}\,(G2(k;1;x))^m$, $m\in \Na$, with 
$G2(k;1;x)\spequiv G2(k;x)\speq (k-1)\, g2(k;{\frac{x}{k-1}})$ for $k\neq 1$
and $G2(1;1;x)\spequiv G2(1;x)\speq exp(x)-1$.

\vspace*{+.1in}
\noindent
{\it Proof}: For $k\neq 1$ substitute $S2(k;n,m)$ from eq.~\ref{(1.18)} into the definition of
$G2(k;m;x)$,
and then use eq.~\ref{(1.21)}.
For the ordinary Stirling numbers, {\it i.e.} for $k=1$, 
the stated result is well-known \cite{AS}. \eop

\vspace*{+.1in}
\noindent
{\bf Lemma 8:} For  $k\in {\bf Z}\setminus \{1\}$,
\begin{equation}
e^{c\,x^k\,d_{x}}\speq \sum_{0}^{\infty}{\frac{1}{m!}}\,\Bigl{[}
g2(k;{\frac{c}{k-1}}\,x^{k-1})\,(k-1)\,x\Bigr{]}^m\,d_{x}^{\ m}\speq 
:e^{g(k;c;x)\,x\,d_{x}}:\ , \label{(2.6)}
\end{equation} 
where $g(k;c;x):= g2(k;{\frac{c}{k-1}}\,x^{k-1})\,(k-1)$, and
the normal order $:A:$ notation has been explained in the 
paragraph following
eq.~\ref{(1.4)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Similar to that for ordinary Stirling
numbers of the second kind, as explained in Section 1,
eqs.~\ref{(1.3)} and ~\ref{(1.4)}. 
Expand the exponential and insert the definition of $S2(k;n,m)$ from 
eq.~\ref{(1.12)} using the triangle convention stated there. Then 
exchange the row summation with the column summation (ignoring questions of
convergence).
After replacing $S2(k;n,m)$ by 
$s2(k;n,m)$, using
eq.~\ref{(1.18)} (and remembering that $k\neq 1$) we find the o.g.f. 
$g2(k;m;{\frac{c}{k-1}}\,x^{k-1})$ inside the column summation.
The convolution property eq.~\ref{(1.21)} (Lemmas 4,5 and 6) then yields 
the first eq. of the lemma.
The second follows from the definition 
of normal order, which is applied to each term in the expanded 
exponential. \eop

\vspace*{+.1in}
\noindent
{\bf Note 5:} For $k \neq 1$, if we insert the o.g.f. $g2(k;y)$ given in 
Lemma 6, or eqs.~\ref{(1.22)} and ~\ref{(1.23)}, we obtain the formula
for $g(k;c;x)$ 
given in eq.~\ref{(1.9)}.
For $k=1$ we obtain
$g(1;c;x)\speq exp(c)-1$
from eq.~\ref{(1.5)}.

\vspace*{+.1in}
\noindent
{\bf Corollary 1:} The operator identity in eq.~\ref{(1.11)}, proved in Lemma 8, implies
the shift property shown in eq.~\ref{(1.10)}.

\vspace*{+.1in}
\noindent
{\it Proof}: An applicaton of Taylor's theorem.\ \ \ \eop

\vspace*{+.1in}
\noindent
{\bf Note 6:} A third proof of the shift property in eq.~\ref{(1.10)} can be 
given by using the well-known multiple commutator formula for 
$exp\,({\bf B})\, x^l\,exp(-{\bf B})$ for $l\in \N0$, setting the operator 
${\bf B}=\,c\,E_{k;x}=c\,x^k\,d_{x}$ for $k\in {\bf Z}$ and the commutator
$[E_{k;x},x^l]\,=\, l\,x^{l+k-1}$.
For $k=1$ we find
$exp(c\,x\,d_{x})\,x^l\,1 \sspeq (exp(c)\,x)^l\,exp(c\,x\,d_{x})\,1\sspeq 
(exp(c)\,x)^l$. For $k\neq 1$ we first obtain
$exp(c\,E_{k;x})\,x^l\,exp(-c\,E_{k;x})
\sspeq \sum_{n=0}^{\infty}{\frac{1}{n!}}(l/(k-1))_{n}\,(c\,(k-1)\,x^{k-1})^n$
using the rising factorial (or {\it Pochhammer}) symbol $(\nu)_{n}:=
\nu\,(\nu +1)\cdots (\nu +n-1)$. This implies $exp(c\,x^k\,d_{x})\,x^l\,1\sspeq
[(1+g(k;c;x))\,x]^l\,1$ with $1+g(k;c;x)$ given in eq.~\ref{(1.9)}.
The $1$ on the right-hand side stands for any $x-$independent operator or function.
Hence the shift property eq.~\ref{(1.10)} holds for polynomials 
and (formally) for power series $f(x)$.

\vspace*{+.1in}
\noindent
{\bf Lemma 9:} For $k\in {\bf Z}$ the e.g.f. of the row polynomials 
$S2_{n}(k;x)$ defined in  eq.~\ref{(1.25)},
${\cal G}2(k;z,x):=\sum_{n=0}^{\infty}S2_{n}(k;x)\,z^n/n!$, is given by 
\begin{equation}
{\cal G}2(k;z,x)\speq e^{x\,G2(k;z)} \ , 
\label{(2.7)}
\end{equation} 
where $G2(k;z)$ is the e.g.f. for the first $(m=1)$ column 
sequence of the triangular matrix ${\bf S2}(k)$ given in eq.~\ref{(1.24)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Separate the $n=0$ term in the definition of ${\cal G}2(k;z,x)$ and 
insert in the remaining expression the definition of the row polynomials 
eq.~\ref{(1.25)}.
Then interchange the row and column summation indices 
and use the definition of the 
e.g.f. $G2(k;m;z)$ given in eq.~\ref{(1.14)}. The convolution property 
Lemma 7, or eq.~\ref{(1.15)}, then leads to the desired result.\ \eop

\vspace*{+.1in}
\noindent
{\bf Note 7:} Another way to state Lemma 9 is to write
\begin{equation}
S2(k;n,m)\speq \left[ {\frac{z^n}{n!}} \right ]\,[x^m]\, e^{x\,G2(k;z)}\ ,
\label{(2.8)}
\end{equation}
where $[y^k]\,f(y)$ denotes the coefficient of $y^k$ in the expansion of 
$f(y)$. For each $k\in {\bf Z}$ a matrix constructed in this way from
the entries of its first ($m=1$) column (collected in the 
e.g.f. $G2(k;z)$) is called a Jabotinsky matrix. 
(See \cite{Knuth} for references to the original works. Note that we 
use Knuth's $n!\,F_{n}(x)$ as row (or Jabotinsky) polynomials.
Knuth's $f(z)$ corresponds to our e.g.f. 
for the $m=1$ column sequence.)

\vspace*{+.1in}
\noindent
Another notation is used in the umbral calculus ({\it cf.} \cite{Roman}). 
The row polynomials $E_{n}(x) \sspeq \sum_{n=1}^{m}J(n,m)\,x^n$ built from 
a lower triangular Jabotinsky matrix $J(n,m)$ are there called
associated polynomial sequences.
Their defining function is
the compositional inverse of  the e.g.f. $f(t)$ used by Knuth and 
in the present work ({\it cf.} \cite{Roman}, p.~53). $\{E_{n}(x)\}$ are special 
Sheffer polynomials for $(1,{\bar f}(t))$ in the umbral notation 
({\it cf.} \cite{Roman}, p.~107).

\vspace*{+.1in}
\noindent
Yet another description of such convolution polynomials can be found in \cite
{SGWW}, where Jabotinsky matrices appear as a special case of so-called
{\it Riordan} matrices (if one uses exponential generating functions). The 
corresponding matrix product furnishes a so-called {\it Bell} subgroup of the
Riordan group ({\it cf.} \cite{SGWW}, p.~238). In the sequel we shall 
reserve the names Riordan and Bell matrices for the case of 
ordinary convolutions.

\vspace*{+.1in}
\noindent
{\bf Lemma 10:} The exponential (or binomial) convolution property
given in eq.~\ref{(1.26)} for polynomials $S2_{n}(k;x), n\in \N0$ and 
fixed $k$, is equivalent to the functional equation
\begin{equation}
{\cal G}2(k;z,x+y)\speq {\cal G}2(k;z,x)\,{\cal G}2(k;z,y)\ ,
\label{(2.9)}
\end{equation} 
which follows from eq.~\ref{(2.7)} for the e.g.f. ${\cal G}2(k;z,x)$ 
defined in Lemma 9.

\vspace*{+.1in}
\noindent
{\it Proof}: Fix $k$ and compare the coefficients of
$z^n/n!$ on both 
sides of this equation. \ \eop

\vspace*{+.1in}
\noindent
{\bf Proposition 1:} Exponential convolution property of the $S2_{n}(k;x)$
polynomials.
The row polynomials $S2_{n}(k;x)$ defined in eq.~\ref{(1.25)} for $n\in \N0$
satisfy for each $k\in {\bf Z}$ the exponential
convolution  property shown in eq.~\ref{(1.26)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Lemma 10 with Lemma 9.\ \ \eop\pbn
{\bf Lemma 11:} Row sums of ordinary convolution matrices\ \cite{Rogers}.
The o.g.f. $r(x)\sspdef \sum_{n=1}^{\infty}\, r_{n}\, x^n$ of the 
row sums $r_{n}\sspdef \sum_{m=1}^{n}\,s(n,m)$ of a lower triangular ordinary
convolution matrix $\{s(n,m)\}_{n\geq m\geq 1}$ is given by
\begin{equation}
r(x)\spdef {\frac{g(x)}{1-g(x)}}\ ,\ 
\label{(2.10)}
\end{equation}   
where $g(x)$ is the o.g.f. of the first ($m=1$) column of the matrix 
$s(n,m)$.

\vspace*{+.1in}
\noindent
{\it Proof}: Consider a lower triangular convolution matrix.
By definition,
the o.g.f. $g(m;x)$ for its $m-$th column sequence is given by 
$g(m;x)\sspeq (g(1;x))^m = g(x)^m$ for $m\in \Na$.
The result follows by inserting into $r(x)$ the definition 
of the row sums $r_{n}$, interchanging row and column summation indices and
using the definition and convolution property of $g(m;x)$.\ \ \eop

\vspace*{+.1in}
\noindent
{\bf Lemma 12:} Row sums of exponential convolution matrices.
The e.g.f. $R(x)\sspdef \sum_{n=1}^{\infty}\, R_{n}\, x^n/n!$ of the 
row sums $R_{n}\sspdef \sum_{m=1}^{n}\,S(n,m)$ of a lower triangular 
exponential convolution matrix $\{S(n,m)\}_{n \geq m \geq 1}$ is given by
\begin{equation}
R(x)\spdef  e^{G(x)}\spm 1\ , \label{(2.11)}
\end{equation}   
where $G(x)$ is the e.g.f. of the first ($m=1$) column sequence of the matrix 
$\{S(n,m)\}_{n\geq m\geq 1}$.

\vspace*{+.1in}
\noindent
{\it Proof}: Analogous to the proof of Lemma 11.\ \  \eop

\vspace*{+.1in}
\noindent
{\bf Proposition 2:} O.g.f. for row sums of the ${\bf s2}(k)$ triangles.
For $k\in {\bf Z}\setminus \{1\}$ the o.g.f. of the sequence of 
row sums of the lower triangular matrix ${\bf s2}(k)$ is given by 
eq.~\ref{(1.51)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Lemma 11 and the $g2(k;x)$ result from Lemma 6, 
eq.~\ref{(2.4)}.\ \ \eop

\vspace*{+.1in}
\noindent
{\bf Proposition 3:} E.g.f. for the sequence of row sums of the ${\bf S2}(k)$ and ${\bf S2p}(k)$ triangles.
For $k\in {\bf \N0}$ the e.g.f. of the sequence of row sums of 
the nonnegative lower triangular matrix ${\bf S2}(k)$, 
resp. ${\bf S2p}(k)$, defined from eq.~\ref{(1.13)}, resp. eq.~\ref{(1.27)}, 
is given by eq.~\ref{(1.53)}, resp. eq.~\ref{(1.54)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Lemma 12 and $G2(k;x)$, resp. $G2p(-k;x)$, from eq.~\ref{(1.24)},
resp. eq.~\ref{(1.29)}.\ \ \eop    
\section{${\bf k-}$Stirling numbers of the first kind}\
\hskip 1cm $k-$Stirling numbers of the first kind can be defined as the 
elements of the (infinite-dimensional, lower triangular) inverse  
matrix ${\bf S1}(k)$ to the matrix ${\bf S2}(k)$ formed from the $k-$Stirling numbers of the second kind.

\vspace*{+.1in}
\noindent
{\bf Definition 3:} $k-${\it Stirling numbers of the first kind},
$S1(k;n,m)$, are defined by
\begin{equation}
x^n\,d_{x}^{\ n}\speq \sum_{m=1}^{n}\,S1(k;n,m)\,x^{-m(k-1)}\,(x^k\,
d_{x})^m\ \ ,\  \text{for}\ k\in {\bf Z},\ n\in \Na\ . \label{(3.1)}
\end{equation} 
Note that this equation is obtained from eq.~\ref{(1.34)} by 
multiplication by $x^{-n(k-1)}$ on the left.
Therefore the equations
are equivalent for every $k$ provided $x\neq 0$.
We set $S1(k;n,m)\speq 0$ if $n<m$, {\it i.e.} ${\bf S1}(k)$ is a lower triangular 
matrix.

\vspace*{+.1in}
\noindent
{\bf Lemma 13:}\hskip 4cm ${\bf S2}(k)\,\cdot {\bf S1}(k) \speq {\bf 1}$, or
\begin{equation}
\sum_{m=p}^{n}\, S2(k;n,m)\,S1(k;m,p)\speq \delta_{n,p} \,
 \label{(3.2)}
\end{equation} 
for fixed $k\in \bf Z$, $n\in \Na$ and $p\in \Na$, where $\delta_{n,p}$ is the 
Kronecker symbol.

\vspace*{+.1in}
\noindent
{\it Proof}: Insert eq.~\ref{(3.1)} with $n\to m$ and $m\to p$ 
into the defining eq.~\ref{(1.12)} for the $S2(k;n,m)$ numbers, 
and then extend the $p-$sum from $m$ to $n$, using lower triangularity of 
each matrix ${\bf S1}(k)$. After interchanging the summations over 
$m$ and $p$ we find, for all $k\in {\bf Z}$, $n\in \Na$ and $x\neq 0$,
\begin{equation}
{\cal O}_{x}(k;n)\spdef x^{-(k-1)n}\,(x^k\,d_{x})^n \speq \sum_{p=1}^{n}\,
\delta(k;n,p)\,{\cal O}_{x}(k;p)\ , \label{(3.3)}
\end{equation}
with $\delta(k;n,p):= \sum_{m=p}^{n}\, S2(k;n,m)\,S1(k;m,p)$. 
Since the operators $\{{\cal O}_{x}(k;p)\}_{p=1}^{n}$ acting on functions 
$f\in C^{n}$ are a linearly independent\footnote{This 
linear independence 
can be proved by applying the differentiation operators ${\frac{1}{p!}}\,
{\cal O}_{x}(k;p)$ for fixed $k\in {\bf Z}$ and $p=1,...,n$ to the 
monomials $x^q$, for $q=1,..,n$. 
The linear independence is then inferred from the non-singularity of the 
$n\times n$ matrix $A_{q,p}(k)\:={\frac{1}{p!}}\, 
\prod_{j=0}^{p-1}(q+j(k-1))$. In fact, $Det\, {\bf A}(k)=+1$ for each $k\in
{\bf Z}$ and $n\in \Na$.}, eq.~\ref{(3.3)} implies $\delta(k;n,p)=
\delta_{n,p}$ for each $k$.\hskip 1cm \eop

\vspace*{+.1in}
\noindent
Similarly, one finds 
\begin{equation}
{\bf S1}(k)\,\cdot {\bf S2}(k) \speq {\bf 1}\   \label{(3.4)}
\end{equation}
after inserting eq.~\ref{(1.12)} with $n\to m,\ m\to p$ into eq.~\ref{(3.1)}.
Now we compare coefficients of the operators 
$\{x^p\,d_{x}^{\ p}\}_{1}^{n}$.

\vspace*{+.1in}
\noindent 
{\bf Lemma 14:} The $k-$Stirling numbers 
of the first kind  satisfy the recurrence given in  eq.~\ref{(1.37)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Use $x^{n}\, d_{x}^{\ n}\speq (x\,d_{x}-(n-1))\,x^{n-1}\,d_{x}^{\ n-1}$ 
and insert eq.~\ref{(3.1)} in both sides of this identity. After 
differentiation, remembering the triangularity of ${\bf S1}(k)$, we compare
coefficients of the linearly independent operators 
$\{{\cal O}_{x}(k;m)\}_{m=1}^{n}$ defined in eq.~\ref{(3.3)}. \ \ \eop

\vspace*{+.1in}
\noindent
{\bf Note 8:} It is obvious from the recurrence ~\ref{(1.37)} together with the 
initial conditions
that all $S1(k;n,m)$ are integers for $k\in {\bf Z}$.

\vspace*{+.1in}
\noindent
{\bf Definition 4:}  $s1(k;n,m)$.
The {\it associated $k-$Stirling numbers of the first kind}, $s1(k;n,m)$, 
are defined for $k\in {\bf Z}\setminus \{1\}$ by eq.~\ref{(1.38)}.

\vspace*{+.1in}
\noindent
{\bf Lemma 15:} The numbers $s1(k;n,m)$ satisfy the recurrence given in
eq.~\ref{(1.39)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Rewrite the recurrence relation  eq.~\ref{(1.37)} for $S1(k;n,m)$ with
$k\neq 1$. The lower triangularity of the matrix ${\bf s1}(k)$ is inherited
from ${\bf S1}(k)$. \hskip 1cm \eop

\vspace*{+.1in}
\noindent
{\bf Note 9:} For $k=1$ eq.~\ref{(1.39)} gives the 
unit matrix ${\bf s1}(1)= {\bf 1}$. This will be used as the definition of
${\bf s1}(1)$.
{\bf Lemma 16:} Nonnegativity of ${\bf s1}(k)$.
The entries of the lower triangular matrix ${\bf s1}(k)$ are nonnegative 
for each $k$.

\vspace*{+.1in}
\noindent
{\it Proof}: If $k-1\geq 0$ this follows from
eq.~\ref{(1.39)}. For $1-k\in \Na$ this follows from the fact that 
$s1(k;n-1,m)\speq 0$ if $n-1\, >\, (1-k)\,m$, {\it i.e.} if the coefficient of
the first term in the recurrence eq.~\ref{(1.39)} is negative.
This will be shown
by induction on $m$.
For $m=1$ the assertion
is true because only the first term in the recurrence is present, and since 
$s1(k,2-k,1)=0$, due to the vanishing coefficient of the first term in its 
recursion, the recurrence shows that $s1(k;n-1,1)$ vanishes for 
$n-1= 2-k,3-k,...\ $ (if $n-1=2-k$ the multiplier in the first recursion term 
vanishes). Assuming the assertion holds for given $m \ge 1$, 
{\it i.e.} $s1(k;n-1,m)=0$ for $n-1\, >\, (1-k)\,m$, leads to a vanishing 
second term in the $s1(k;n-1,m+1)$ recurrence  for all $n-1\, >\, 
(1-k)\,m\,+\,1.$ Therefore, $s1(k;(1-k)\,(m+1)\,+\, 1,m+1)$ will be zero because 
the coefficient of the first term of this recurrence vanishes and the second 
term is absent since $(1-k)\,(m+1)\,>\,(1-k)\,m$. 
Then $s1(k;n-1,m+1)$ vanishes recursively for all 
$n-1\,\geq\,(1-k)\,(m+1)\,+\,1$.\hskip 1cm \eop

\vspace*{+.1in}
\noindent
{\bf Lemma 17:} The o.g.f. for the $m$-th column of ${\bf s1}(k)$
(see eqs.~\ref{(1.40)}, ~\ref{(1.41)} and ~\ref{(1.42)}).
${\bf s1}(k)$ is a Bell matrix (see Note 7 for this name), 
{\it i.e.} the o.g.f. for the sequence $\{s1(k;n,m)\}_{n=1}^{\infty}$ is given by $g1(k;m;y)\sspeq 
(g1(k;1;y))^m$ and
\begin{equation}
g1(k;y)\spdef g1(k;1;y) \speq 
{\frac {-1\sspp (1-(k-1)\,y)^{-(k-1)}}{(k-1)^2}}\ \ 
\text{for}\ \ k\in  {\bf Z}\setminus \{1\}\ .  
\label{(3.5)}
\end{equation} 
Since we have set ${\bf s1}(1)=\bf 1$ we take $g1(1;y)=y$.

\vspace*{+.1in}
\noindent
{\it Proof}: From the recurrence relation eq.~\ref{(1.39)} we find, for 
$k\in \bf Z$, the first-order linear differential-difference equation
\begin{eqnarray}
[1-(k-1)\,y]\,g1^{\prime}(k;m;y)\spm m\,(k-1)^2\,g1(k;m;y)\spm m \,
g1(k;m-1;y)
\speq 0\ \ , \  \ && \label{(3.6)}\\ 
g1(k;m;0)=0\ ,\ m\in \Na; \  \ g1^{\prime}(k;m;y)|_{y=0}= s1(k;1,1)\, 
\delta_{m,1}= \delta_{m,1} . 
 &&\label{(3.7)}
\end{eqnarray}
The prime denotes differentiation with respect to $y$. The $y=0$ conditions
follow from the definition of $g1(k;m;y)$ in eq.~\ref{(1.40)}. 
Eq.~\ref{(3.6)} is solved using $g1(k;m;y)
\sspeq (g1(k;1;y))^m$, which results in a standard linear 
inhomogeneous differential equation for $g1(k;y):= g1(k;1;y)$, namely
\begin{equation}
[1-(k-1)\,y]\,g1^{\prime}(k;y)\spm (k-1)^2\,g1(k;y)\spm 1\speq 0 \,,
 \label{(3.8)}
\end{equation} 
with the initial condition $g1(k;0)=0$.
The solution is given by equation
eq.~\ref{(3.5)} (cf. eq.~\ref{(1.41)}, \ref{(1.42)}).\ \ \eop

\vspace*{+.1in}
\noindent
{\bf Note 10:} Generalized {\it EIS} 
\seqnum{A001792} sequences.
Analogous to the
generalized Catalan numbers generated by $c2(l;y)$ of 
eq.~\ref{(1.23)} (see Note 4), we can use $c1(l;y)$ defined in 
eq.~\ref{(1.42)} as the o.g.f. for sequences 
$\{c1^{(l)}_{n}\}_{n=0}^{\infty}$. 
We find that
$c1(1;y)=1/(1-y)$ generates {\it EIS} \seqnum{A000012}
(powers of 1), $c1(2;y)$ is the 
o.g.f. for the sequence
\seqnum{A001792}($n$).
The {\it EIS} A-numbers for the sequences
for $l=k-1$ are found in the second column of Table 3 for $l=1,...,5$ 
and $l=-1,...,-6$. See also {\it EIS} \seqnum{A053113}.

In order to have $g1(1;y)=y$ we set
$c1(0;y)\equiv 1$ (see eq.~\ref{(1.41)}). An explicit expression for 
$c1^{(l)}_{n}$ with $l\in \Na$ is given in eq.~\ref{(1.43)}.
Also $c1^{(0)}_{n}= \delta_{n,0}$, and $c1(-l;x)$ is a polynomial in $x$ for $l\in \Na$. For example,
$c1(-3;x)\sspeq 1\sspp 3\,x\sspp 3\,x^2.$ The triangle of coefficients in
these polynomials can be found as {\it EIS} 
\seqnum{A049323}
(increasing powers of $x$), or
\seqnum{A033842}
(decreasing powers of $x$).
The explicit form for these coefficients 
is given in eq.~\ref{(1.44)}.

\vspace*{+.1in}
\noindent
{\bf Lemma 18:} The entries of the matrix ${\bf s1}(k)$ are integers for all $k\in {\bf Z}$.

\vspace*{+.1in}
\noindent
{\it Proof}: The first column of ${\bf s1}(k)$ consists of integers since 
$c1(k-1;y)$ generates the integers $c1^{(k-1)}_{n}$ given explicitly in 
eqs.~\ref{(1.43)} and ~\ref{(1.44)}, and $g1(k;y)$ is given 
by eq.~\ref{(1.41)}
(see Lemma 17).
The case $k=1$ is trivial.
Since ${\bf s1}(k)$ is an ordinary convolution triangle (or Bell matrix)
it is sufficient to prove that the first column consists of integers.\hskip 1cm \eop 

\vspace*{+.1in}
\noindent
{\bf Lemma 19:} The entries of the matrix ${\bf s2}(k)$ are integers for all $k\in {\bf Z}$.

\vspace*{+.1in}
\noindent
{\it Proof}: Once this has been established, all entries of $s2(k;n,m)$
are nonnegative integers by Lemma 3. 
For the proof we first substitute eqs. \ref{(1.18)} and \ref{(1.38)} into eq.~\ref{(3.2)}.
Define, for $k\in {\bf Z}$, the signed matrix ${\bf s2s}(k)$ 
by $s2s(k;n,m):= (-1)^{n-m}\, s2(k;n,m)$.
Then eq.~\ref{(3.2)} implies
\begin{equation}
 {\bf s2s}(k)\,\cdot\,{\bf s1(k)}\speq {\bf 1}\ .  \label{(3.9)}
\end{equation}
Using the fact that the $s1(k;n,m)$ are integers from the previous 
lemma (from Lemma 16 they are even known to be nonnegative) 
this equation allows us to carry out the proof recursively.
We omit the details.~\eop

\vspace*{+.1in}
\noindent
{\bf Note 11:} Using Lemmas 16 and 19, eqs.~\ref{(1.21)} and 
~\ref{(1.22)} show that 
$c2(l;y)\sspeq \sum_{n=0}^{\infty}\, c2^{(l)}_{n}\,y^n$
defined in eq.~\ref{(1.23)} generates positive integers for all 
$l\in {\bf Z}\setminus \{0\}$. Their explicit form is given by
\begin{equation}\label{E77}
c2^{(l)}_{n} \speq l^n\,\prod_{j=1}^{n}\, (j\,l-1)/(n+1)! ~.
\end{equation} 
By definition $c2(0;y)\sspdef 1$.

\vspace*{+.1in}
\noindent
{\bf Lemma 20:} The e.g.f. for the $m-th$ column sequence of the 
unsigned  $k-$Stirling triangle of the first kind, ${\bf S1p}(k)$, defined in eq.~\ref{(1.36)}  for $k\in \Na$, is $G1p(k;m;x)\speq \frac{1}{m!}\,
(G1p(k;1;x))^m$, $m\in \Na$, with 
$G1p(k;1;x)\spequiv G1p(k;x)\speq (k-1)\, g1(k;{\frac{x}{k-1}})$ for 
$k=2,3,...$ and $G1p(1;1;x)\spequiv G1p(1;x)\speq -\,ln(1-x)$.

\vspace*{+.1in}
\noindent
{\it Proof:} For $k \geq 2$ substitute $S1p(k;n,m)$ from eqs.~\ref{(1.36)} and 
~\ref{(1.38)} into the definition of $G1p(k;m;x)$ given in  
eq.~\ref{(1.46)}. In this way the o.g.f. $g1(k;m;y)$ appears in the 
desired form. The result for the ordinary unsigned Stirling numbers 
($k=1$) is well-known \cite{AS}. \hskip 1cm \eop

\vspace*{+.1in}
\noindent
{\bf Note 12:} Explicit form for $G1p(k;m;x)$, $k> 1$:  eq.~\ref{(1.48)} and 
Lemma 20.
Equation \ref{(1.48)} follows from
the o.g.f. $g1(k;m;y)$ in eqs.~\ref{(1.40)} and 
~\ref{(3.5)}.
This shows that 
$G1p(k;1;x)\speq -\,\overline{G2(k;-x)}$, the negative compositional 
inverse of $G2(k;-x)$ of eq.~\ref{(1.24)}. Inverse Jabotinsky matrices
like ${\bf S2}$ and ${\bf S1}$ ({\it cf}. eqs.~\ref{(3.2)} and ~\ref{(3.4)}) 
have first column e.g.f.'s which are inverse to each other 
in the compositional sense \cite {Knuth}.

\vspace*{+.1in}
\noindent
{\bf Lemma 21:} Row polynomials for ${\bf S1}(k)$.
For $k\in {\bf Z}$ the e.g.f. of the row polynomials 
$S1_{n}(k;x)\spdef \sum_{m=1}^{n}\, S1(k;n,m)\, x^m\ $, $n\in \Na$, and 
$S1_{0}(k;x):=1$ is
\begin{equation}
{\cal G}1(k;z,x):=\sum_{n=0}^{\infty}S1_{n}(k;x)\,z^n/n!  
\speq e^{x\,G1(k;z)} \ , \label{(3.11)}
\end{equation} 
where $G1(k;z)\sspeq (-1\sspp (1+z)^{1-k})/(1-k)$ for $k\neq 1$, 
and $G1(1;z)\sspeq ln(1+z)$ are the e.g.f.s for the first 
$(m=1)$ column sequences of the triangular matrices ${\bf S1}(k)$ .

\vspace*{+.1in}
\noindent
{\it Proof}: Analogous to that of Lemma 9.

\vspace*{+.1in}
\noindent
{\bf Note 13:}  $S1(k;n,m)\speq 
\left[ {\frac{z^n}{n!}} \right ]\,[x^m]\, e^{x\,G1(k;z)}\ $ 
(cf. Note 7).

\vspace*{+.1in}
\noindent
{\bf Proposition 5:} Exponential convolution property of the $S1_{n}(k;x)$
polynomials.
The row polynomials $S1_{n}(k;x)$ defined in Lemma 21 for $n\in \N0$,
satisfy for each $k\in {\bf Z}$ the exponential 
(or binomial)
convolution  property shown in eq.~\ref{(1.26)} with $S2$ replaced 
everywhere by $S1$.

\vspace*{+.1in}
\noindent
{\it Proof}: For fixed $k$, compare the coefficients of $z^n/n!$ on both 
sides of the identity ${\cal G}1(k;z,x+y)\speq {\cal G}1(k;z,x)\,
{\cal G}1(k;z,y)\ $.\ \ \eop\pbn
{\bf Note 14:} In the notation of the umbral calculus ({\it cf.} \cite{Roman}) 
the polynomials $S1_{n}(k;x)$ are called associated
polynomial (or Sheffer) sequences for 
$(1,\overline{G1(k;t)}\speq G2(k;t))$.
For $k \neq 1$
$G2(k;t)$ is given in  
eq.~\ref{(1.24)}.
Also $\overline{G1(1;t)}\sspeq G2(1;t)\sspeq exp(t)-1 $. 


\vspace*{+.1in}
\noindent
{\bf Proposition 6:} O.g.f. for row sums of ${\bf s1}(k)$ triangles.
For $k\in {\bf Z}\setminus \{1\}$ the o.g.f. of the sequence of 
row sums of the lower triangular matrix ${\bf s1}(k)$ is given by 
eq.~\ref{(1.52)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Lemma 11 and the $g1(k;x)$ result in Lemma 17.\ \ \eop

\vspace*{+.1in}
\noindent{\bf Proposition 7:} E.g.f. of the sequence of row sums of ${\bf S1p}(k)$ and ${\bf S1}(-|k|)$ triangles.
For $k\in {\bf \N0}$ the e.g.f. of the sequence of row sums of 
the nonnegative lower triangular matrix ${\bf S1p}(k)$, 
resp. ${\bf S1}(-|k|)$, defined in eq.~\ref{(1.36)}, resp. eq.~\ref{(1.37)}, 
is given by eq.~\ref{(1.55)}, resp. eq.~\ref{(1.56)}.

\vspace*{+.1in}
\noindent
{\it Proof}: Lemma 12 and $G1p(k;x)$, resp. $G1(-|k|;x)$, from 
Lemma 20, {\it i.e.} eq.~\ref{(1.48)}, resp. eq.~\ref{(1.49)}.\ \ \eop \pbn  
{\bf Note 15:} Row-sums of signed ${\bf S1}(k)$, $k\in \Na$, resp. 
${\bf S1s}(-|k|)$ triangles.
Here Lemma 12 applies with the e.g.f.s $G1(k;x)$, resp. 
$G1s(-|k|;x)$, given in the first line after eq.~\ref{(3.11)}, resp. 
in the paragraph after eq.~\ref{(1.49)}. \pbn\pbn
\section *{\bf Acknowledgements}
The author would like to thank Stefan Theisen for a conversation at a very early 
stage of this work (Note 6, case $k=1$). Thanks go also to Norbert Dragon
who pointed out his web-pages (ref. \cite{Dr}). This work has its origin in 
an exercise in the author's 1998/1999 lectures on conformal field theory 
({\it Konforme Feldtheorie}, Blatt 1, Aufgabe 2, available as a ps.gz file under
\htmladdnormallink{http://www-itp.physik.uni-karlsruhe.de/${\ \tilde{}}\ $wl/Uebungen.html}
{http://www-itp.physik.uni-karlsruhe.de/~wl/Uebungen.html}).
%http://www-itp.physik.uni-karlsruhe.de/${\ \tilde{}}\ $wl/Uebungen.html).\pbn\pbn  
\begin{thebibliography}{99}
\bibitem{AS} M. Abramowitz and I. A. Stegun: 
{\it Handbook of Mathematical Functions}, Dover, 1968.
%\bibitem{Dr} N. Dragon: {\it \htmladdnormallink{Konforme Transformationen}{http://www.itp.uni-hannover.de/${\ \tilde{}}\ $dragon/Group.html}}, ps.gz file: \\

\bibitem{Dr} N. Dragon: {\it \htmladdnormallink{Konforme Transformationen}{http://www.itp.uni-hannover.de/~dragon/Group.html}}, ps.gz file: \\
http://www.itp.uni-hannover.de/${\ \tilde{}}\ $dragon/Group.html, and references
given there.

\bibitem{GKP} R.L. Graham, D.E. Knuth, and O. Patashnik:  {\it Concrete Mathematics}, Addison-Wesley, Reading MA, 1989.
\bibitem{Knuth} D. E. Knuth: Convolution polynomials, 
{\it The Mathematica J.}, {\bf 2.1} (1992), 67--78.
\bibitem{Riordan ICA} J. Riordan: {\it An Introduction to Combinatorial 
Analysis}, Wiley, New-York, 1958.
\bibitem{Rogers} D. G. Rogers: Pascal triangles, Catalan numbers and renewal 
arrays, {\it Discrete Math.} {\bf 22} (1978), 301--310.
\bibitem{Roman} S. Roman: {\it The Umbral Calculus}, Academic Press, New York, 1984
\bibitem{S} L.W. Shapiro: A Catalan triangle, {\it Discrete Math.}
{\bf 14} (1976), 83--90.
\bibitem{SGWW} L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson: The Riordan group, {\it Discrete Appl. Math.}
{\bf 34} (1991), 229--239.
\bibitem{Sloane} N. J. A. Sloane and S. Plouffe: {\it The Encyclopedia of Integer 
Sequences}, Academic Press, San Diego, 1995.
\bibitem{Sloane2}
N. J. A. Sloane (2000), {\it \htmladdnormallink{The On-Line Encyclopedia of Integer Sequences}{http://www.oeis.org}},
published electronically at {\tt http://www.oeis.org}.
\end{thebibliography}

% comment out this brace when single-spacing
%}


\newpage
%%%%%%%%%%%%%%%%%%%%% start of 4 tables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\large {\bf Table 1: Associated k-Stirling number triangles of the 
second kind}}
\end {center}
\begin{center}  
{\large  $\bf{s2(k)}$, $\bf{k\neq 1}$\ \ \ \ $\bf{s2(1)}:={\bf 1}$ }
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
&&&\\
$k$& A-number of  & A-number of  & A-number of  \\ 
 & triangle & sequence of first column & sequence of row sums\\
&&&\\ \hline\hline
$\vdots$ &&&\\ \hline
&&&\\
-5 & \seqnum{A049224}
& \seqnum{A025751}
(Gerard) & 
\seqnum{A025759}
(Gerard) \\
&&&\\ \hline
&&&\\
-4 & \seqnum{A049223} 
& \seqnum{A025750} (Gerard) & \seqnum{A025758} (Gerard) \\
&&&\\ \hline
&&&\\
-3 & \seqnum{A049213}
& \seqnum{A025749}
(Gerard) 
& \seqnum{A025757}
(Gerard)\\
&&&\\ \hline
&&&\\
-2 & \seqnum{A048966} & \seqnum{A025748} (Gerard) & \seqnum{A025756} (Gerard)\\
&&&\\ \hline
&&&\\
-1 & \seqnum{A033184} (Catalan) & \seqnum{A000108}($n-1$)  & \seqnum{A000108} (Catalan) \\
&&&\\ \hline
&&&\\
 0 & \seqnum{A023531} ($\bf 1$ matrix) & \seqnum{A000007}($n-1$) & \seqnum{A000012} (powers of $1$)  \\
&&&\\ \hline
&&&\\
 2 & \seqnum{A007318}($n-1,m-1$) (Pascal) & \seqnum{A000012} & \seqnum{A000079} (powers of $2$) \\
&&&\\ \hline
&&&\\
 3 & \seqnum{A035324} & \seqnum{A001700}($n-1$) & \seqnum{A049027} \\
&&&\\ \hline
&&&\\
 4 & \seqnum{A035529} & \seqnum{A034171}($n-1$) & \seqnum{A049028} \\
&&&\\ \hline
&&&\\
 5 & \seqnum{A048882} & \seqnum{A034255}($n-1$) & \seqnum{A048965} \\
&&&\\ \hline
&&&\\
 6 & \seqnum{A049375} & \seqnum{A034687} & \seqnum{A039746} \\
&&&\\ \hline
$\vdots$&&&\\
\hline
\end{tabular}
\end{center}
\newpage
%%%%%%%%%%%%%%%%%%%%%% start of table 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\large {\bf Table 2: k-Stirling number triangles of the second kind }}
\end{center}
\begin{center}
{\large  ${\bf S2(k), k=0,1,2, ...,}$ \hskip 1cm${\bf S2p(k), k=0,-1,-2,... }$}
\end{center}
%\vspace{2mm}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
&&&\\
$k$ & A-number of  & A-number of  & A-number of  \\ 
 & triangle & sequence of first column & sequence of row sums\\
&&&\\ \hline\hline
$\vdots$ &&&\\ \hline
&&&\\
-5 & \seqnum{A013988} & \seqnum{A008543}($n-1$) (Keane) & \seqnum{A028844}  \\
&&&\\ \hline
&&&\\
-4 & \seqnum{A011801} & \seqnum{A008546}($n-1$) (Keane) & \seqnum{A028575}  \\
&&&\\ \hline
&&&\\
-3 & \seqnum{A000369} & \seqnum{A008545}($n-1$) (Keane) & \seqnum{A016036} \\
&&&\\ \hline
&&&\\
-2 & \seqnum{A004747} & \seqnum{A008544}($n-1$) (Keane) & \seqnum{A015735}\\
&&&\\ \hline
&&&\\
-1 & \seqnum{A001497}($n-1,m-1$) (Bessel) & \seqnum{A001147}($n-1$) (double factorials)  
& \seqnum{A001515} (Riordan) \\
&&&\\ \hline
&&&\\
 0 & \seqnum{A023531} ($\bf 1$ matrix) & \seqnum{A000007}($n-1$) & \seqnum{A000012} (powers of $1$)  \\
&&&\\ \hline
&&&\\	
 1 & \seqnum{A008277} (Stirling 2nd kind) & \seqnum{A000012} (powers of $1$) & \seqnum{A000110} (Bell)  \\
&&&\\ \hline
&&&\\
 2 & \seqnum{A008297} (unsigned Lah) & \seqnum{A000142} (factorials) & \seqnum{A000262} (Riordan)  \\
&&&\\ \hline
&&&\\
 3 & \seqnum{A035342} & \seqnum{A001147} (2-factorials)& \seqnum{A049118} \\
&&&\\ \hline
&&&\\
 4 & \seqnum{A035469} & \seqnum{A007559} (3-factorials) & \seqnum{A049119} \\
&&&\\ \hline
&&&\\
 5 & \seqnum{A049029} & \seqnum{A007696} (4-factorials) & \seqnum{A049120} \\
&&&\\ \hline
&&&\\
 6 & \seqnum{A049385} & \seqnum{A008548} (5-factorials) & \seqnum{A049412} \\
&&&\\ \hline
$\vdots$&&&\\
\hline
\end{tabular}
\end{center}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%% start of table 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\large {\bf Table 3: Associated k-Stirling number triangles of the 
first kind}}
\end {center}
\begin{center}  
{\large ${\bf s1(k)}$, ${\bf k\neq 1}$ \ \ \ ${\bf s1(1):= \bf 1}$}
\end{center}
%\vspace{2mm}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
&&&\\
$k$& A-number of  & A-number of  & A-number of  \\ 
 & triangle & sequence of first column & sequence of row sums\\
&&&\\ \hline\hline
$\vdots$ &&&\\ \hline
&&&\\
-5 & \seqnum{A049327} & \seqnum{A049323}(5,$m$) & \seqnum{A049351} \\
&&&\\ \hline
&&&\\
-4 & \seqnum{A049326} & \seqnum{A049323}(4,$m$) & \seqnum{A049350} \\
&&&\\ \hline
&&&\\
-3 & \seqnum{A049325} & \seqnum{A049323}(3,$m$) & \seqnum{A049349} \\
&&&\\ \hline
&&&\\
-2 & \seqnum{A049324} & \seqnum{A049323}(2,$m$) & \seqnum{A049348}\\
&&&\\ \hline
&&&\\
-1 & \seqnum{A030528} & \seqnum{A019590}=\seqnum{A049323}($1,m$)  & \seqnum{A000045}($n+1$) (Fibonacci) \\
&&&\\ \hline
&&&\\
 0 & \seqnum{A023531} ($\bf 1$ matrix) & \seqnum{A000007}($n-1$)=\seqnum{A049323}(0,$m$) & \seqnum{A000012} (powers of $1$)  \\
&&&\\ \hline
&&&\\
 2 & \seqnum{A007318}($n-1,m-1$) (Pascal) & \seqnum{A000012} (powers of 1) & \seqnum{A000079} (powers of $2$) \\
&&&\\ \hline
&&&\\
 3 & \seqnum{A030523} & \seqnum{A001792} & \seqnum{A039717} \\
&&&\\ \hline
&&&\\
 4 & \seqnum{A030524} & \seqnum{A036068} & \seqnum{A043553} \\
&&&\\ \hline
&&&\\
 5 & \seqnum{A030526} & \seqnum{A036070} & \seqnum{A045624} \\
&&&\\ \hline
&&&\\
 6 & \seqnum{A030527} & \seqnum{A036083} & \seqnum{A046088} \\
&&&\\ \hline
$\vdots$&&&\\
\hline
\end{tabular}
\end{center}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%start of Table 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\large {\bf Table 4: k-Stirling number triangles of the first kind }}
\end{center}
\begin{center}
{\large  ${\bf S1p(k), k=0,1,2, ...,}$ \hskip 1cm$ {\bf S1(k), k=0,-1,-2,... }$ }
\end{center}
%\vspace{2mm}
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
&&&\\
$k$& A-number of  & A-number of  & A-number of  \\ 
 & triangle & sequence of first column & sequence of row sums\\
&&&\\ \hline\hline
$\vdots$ &&&\\ \hline
&&&\\
-5 & \seqnum{A049411} & \seqnum{A008279}(5,$n-1$) (numbperm) & \seqnum{A049431}  \\
&&&\\ \hline
&&&\\
-4 & \seqnum{A049424} & \seqnum{A008279}(4,$n-1$) (numbperm) & \seqnum{A049427}  \\
&&&\\ \hline
&&&\\
-3 & \seqnum{A049410} & \seqnum{A008279}(3,$n-1$) (numbperm) & \seqnum{A049426} \\
&&&\\ \hline
&&&\\
-2 & \seqnum{A049404} & \seqnum{A008279}(2,$n-1$) (numbperm) & \seqnum{A049425}\\
&&&\\ \hline
&&&\\
-1 & \seqnum{A049403} & \seqnum{A008279}(1,$n-1$) (numbperm)  
& \seqnum{A000085} \\
&&&\\ \hline
&&&\\
 0 & \seqnum{A023531} ($\bf 1$ matrix) & \seqnum{A000007}($n-1$) & \seqnum{A000012} (powers of $1$)  \\
&&&\\ \hline
&&&\\	
 1 & \seqnum{A008275} (unsigned Stirling 1st kind) & \seqnum{A000142}($n-1$) & \seqnum{A000142} 
(factorials) \\
&&&\\ \hline
&&&\\
 2 & \seqnum{A008297} (unsigned Lah) & \seqnum{A000142} (factorials) & \seqnum{A000262} (Riordan)  \\
&&&\\ \hline
&&&\\
 3 & \seqnum{A046089} & \seqnum{A001710}($n+1$) (Mitrinovic$^2$) & \seqnum{A049376} \\
&&&\\ \hline
&&&\\
 4 & \seqnum{A035469} & \seqnum{A001715}($n+2$) (Mitrinovic$^2$) & \seqnum{A049377} \\
&&&\\ \hline
&&&\\
 5 & \seqnum{A049353} & \seqnum{A001720}($n+3$) (Mitrinovic$^2$) & \seqnum{A049378} \\
&&&\\ \hline
&&&\\
 6 & \seqnum{A049374} & \seqnum{A001725}($n+4$) (Mitrinovic$^2$) & \seqnum{A049402} \\
&&&\\ \hline
$\vdots$&&&\\
\hline
\end{tabular}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%% end of tables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\vspace*{+.5in}
\centerline{\rule{5.4in}{.01in}}

\vspace*{+.1in}
\noindent
{\small
(Concerned with sequences
\seqnum{A000007},
\seqnum{A000012},
\seqnum{A000045},
\seqnum{A000079},
\seqnum{A000085},
\seqnum{A000108},
\seqnum{A000110},
\seqnum{A000142},
\seqnum{A000262},
\seqnum{A000369},
\seqnum{A001147},
\seqnum{A001497},
\seqnum{A001515},
\seqnum{A001700},
\seqnum{A001710},
\seqnum{A001715},
\seqnum{A001720},
\seqnum{A001725},
\seqnum{A001792},
\seqnum{A004747},
\seqnum{A007318},
\seqnum{A007559},
\seqnum{A007696},
\seqnum{A008275},
\seqnum{A008277},
\seqnum{A008279},
\seqnum{A008297},
\seqnum{A008543},
\seqnum{A008544},
\seqnum{A008545}, 
\seqnum{A008546},
\seqnum{A008548},
\seqnum{A011801},
\seqnum{A013988},
\seqnum{A015735},
\seqnum{A016036},
\seqnum{A019590},
\seqnum{A023531},
\seqnum{A025748}, 
\seqnum{A025748}-\seqnum{A025755},
\seqnum{A025749},
\seqnum{A025750},
\seqnum{A025751},
\seqnum{A025756},
\seqnum{A025757}, 
\seqnum{A025758},
\seqnum{A025759},
\seqnum{A028575},
\seqnum{A028844},
\seqnum{A030523},
\seqnum{A030524},
\seqnum{A030526}, 
\seqnum{A030527},
\seqnum{A030528},
\seqnum{A033184},
\seqnum{A033842},
\seqnum{A034171},
\seqnum{A034255},
\seqnum{A034687},
\seqnum{A035323}, 
\seqnum{A035324},
\seqnum{A035342},
\seqnum{A035469},
\seqnum{A035529},
\seqnum{A036068}, 
\seqnum{A036070},
\seqnum{A036083},
\seqnum{A039717},
\seqnum{A039746},
\seqnum{A043553},
\seqnum{A045624},
\seqnum{A046088},
\seqnum{A046089},
\seqnum{A048882},
\seqnum{A048965}, 
\seqnum{A048966},
\seqnum{A049027},
\seqnum{A049028}, 
\seqnum{A049029},
\seqnum{A049118},
\seqnum{A049119},
\seqnum{A049120},
\seqnum{A049213},
\seqnum{A049223},
\seqnum{A049224},
\seqnum{A049323},
\seqnum{A049324},
\seqnum{A049325},
\seqnum{A049326},
\seqnum{A049327}, 
\seqnum{A049348}, 
\seqnum{A049349},
\seqnum{A049350},
\seqnum{A049351},
\seqnum{A049353},
\seqnum{A049374},
\seqnum{A049375},
\seqnum{A049376},
\seqnum{A049377},
\seqnum{A049378},
\seqnum{A049385},
\seqnum{A049402},
\seqnum{A049403},
\seqnum{A049404}, 
\seqnum{A049410}, 
\seqnum{A049411},
\seqnum{A049412},
\seqnum{A049424},
\seqnum{A049425},
\seqnum{A049426},
\seqnum{A049427},
\seqnum{A049431}, and
\seqnum{A053113}.)
}

\centerline{\rule{5.4in}{.01in}}

\vspace*{+.1in}
\noindent
Received February 11, 2000;
published in Journal of Integer Sequences September 13, 2000;
minor editorial changes November 30, 2000; fixed OEIS links August 11
2012.

\centerline{\rule{5.4in}{.01in}}

\vspace*{+.1in}
\noindent
Return to \htmladdnormallink{Journal of Integer Sequences home
page}{http://www.oeis.org}.

\centerline{\rule{5.4in}{.01in}}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%% end of file %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 


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