%On Generalizations of the Stirling Number Triangles,
%Wolfdieter Lang February, 11 2000, update May 31 2000
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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
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\bf {Abstract}
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{\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),ドル which also represents the 
triangle of signless Lah numbers. 
Certain associated number triangles, denoted by $s2(k;n,m)$ and $s1(k;n,m),ドル 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,円\Dx{},ドル 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,円d_{x})^n\speq \sum_{m=1}^n,円 S2(n,m),円x^m,円d_{x}^{\ m}\ \ , \ \ 
n\in {\bf N},\label{(1.1)} 
\end{equation}
where the derivative operator $d_{x}\equiv \Dx{}{},ドル and $E_{x}$ is the {\sl Euler}
operator satisfying $E_{x},円x^k\speq k,円x^k$. A recursion relation can be derived from 
eq.~\ref{(1.1)} by considering 
$x,円d_{x}(x,円d_{x})^{n-1},ドル 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,円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,円 x,ドル $\lambda \in \bf R_{+},ドル is $E_{x}$. In order to exhibit these numbers within this framework consider first \begin{eqnarray} e^{c,円 x,円 d_{x}}&\; =\; & \sum_{n=0}^{\infty},円\frac{c^n}{n!}\ E_{x}^{\ n} \; = \; 1+\sum_{n=1}^{\infty},円\frac{c^n}{n!}\sum_{m=1}^{n},円 S2(n,m),円 x^m,円d_{x}^{\ m} \label{(1.3)}\\ &\; = \;& 1+\sum_{m=1}^{\infty}\left ( \sum_{n=m}^{\infty}\frac{c^n}{n!},円 S2(n,m)\right ),円 x^m,円d_{x}^{\ m} \; =\; 1+\sum_{m=1}^{\infty},円 G2_{m}(c),円 x^m,円 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!},円(G2(c))^m,ドル with $G2(c)=exp(c)-1$; therefore we obtain \begin{equation} e^{c,円x,円d_{x}}\speq \sum_{m=0}^{\infty}\frac{1}{m!},円(G2(c),円x)^m,円d_{x}^{\ m} \speq :e^{(exp(c)-1)x,円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,ドル and move all operators $d_x$ to the right-hand side, ignoring the usual commutation rule $[d_{x},x]\equiv d_{x},円x-x,円d_{x} = 1$. For example, $:(x,円d_{x})^m:\speq x^m,円d_{x}^m.$) 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,円x,円d_{x}},円 f(x)\speq :e^{(exp(c)-1)x,円d_{x}}:,円f(x)\speq f(x+(e^c-1)x)\ =\ f(x^{\prime}) \ \ , \ \label{(1.5)} \end{equation} with $x^{\prime}=e^c,円x$. 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,円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 $E_{k;x}\equiv x^k,円d_{x}$ 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,円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,ドル after separation of variables, we obtain the equation $(x^{\prime})^{1-k}\spm x^{1-k}\speq (1-k),円c$. Setting \begin{equation} x^{\prime}\speq (1+g(k;c;x)),円x \label{(1.8)} \end{equation} we have \begin{equation} 1+g(k;c;x)\speq \bigl{(}1-(k-1),円c,円x^{k-1}\bigr{)}^{-\frac{1}{k-1}}\ . \label{(1.9)} \end{equation} Therefore \begin{equation} e^{c,円x^k,円d_{x}},円f(x)\speq f(x^{\prime})\speq f \left( (1-(k-1),円c,円x^{k-1})^{-\frac{1}{k-1}},円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),ドル 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,ドル \begin{equation} e^{c,円x^k,円d_{x}}\speq :e^{g(k;c;x),円x,円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,円d_{x})^n\speq \sum_{m=1}^n,円 S2(k;n,m),円 x^{m+(k-1),円n},円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),円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},円 S2(k;n,m),円\frac{x^n}{n!} \ , \label{(1.14)} \end{equation} satisfies \begin{equation} G2(k;m;x)\speq \frac{1}{m!},円(G2(k;x))^m \label{(1.15)} \end{equation} for $k \neq 1,ドル with \begin{equation} G2(k;x)\speq (k-1),円 g2(k;\frac{x}{k-1})\ , \label{(1.16)},円 \end{equation} where \begin{equation} g2(k;y)\spdef \sum_{n=1}^{\infty},円 s2(k;n,1),円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},円 \frac{m!}{n!},円 S2(k;n,m)\ . \label{(1.18)} \end{equation} These number triangles, or lower triangular infinite-dimensional matrices, ${\bf s2}(k),ドル which are here only defined for $k\in {\bf Z}\setminus \{1\},ドル obey the recursion relation \begin{equation} s2(k;n,m)\speq \frac{k-1}{n},円\left[ (k-1)(n-1)+m\right],円 s2(k;n-1,m)\spp \frac {m}{n},円 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},円 s2(k;n,m),円 y^n\speq (g2(k;y))^m\ , \label{(1.21)} \end{equation} with \begin{equation} g2(k;y)\speq y,円c2(1-k;y), \label{(1.22)} \end{equation} and, for $l\in {\bf Z}\setminus \{0\},ドル \begin{equation} c2(l;y)\speq \frac{1-(1-l^2,円y)^{\frac{1}{l}}}{l,円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\},ドル eq.~\ref{(1.16)} now yields \begin{equation} G2(k;x)\speq -1\spp (1+(1-k),円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},円 S2(k;n,m),円 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},円 \binom n p ,円 S2_{p}(k;x),円 S2_{n-p}(k;y) \speq \sum_{p=0}^{n},円 \binom n p ,円 S2_{p}(k;y),円 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)}),ドル 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ドル-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},円 S2(-k;n,m)\ \ ,\ \ k\in \N0. \label{(1.27)} \end{equation} The e.g.f. for column $m$ of the triangle ${\bf S2}p(-&#124;k&#124;)$ is \begin{equation} G2p(-&#124;k&#124;;m;x)\speq \frac{1}{m!},円(G2p(-&#124;k&#124;;x))^m \label{(1.28)} \end{equation} with \begin{equation} G2p(-&#124;k&#124;;x)\speq -,円 G2(-&#124;k&#124;;-x) \speq 1\spm (1-(&#124;k&#124;+1),円x)^{\frac{1}{&#124;k&#124;+1}} . \label{(1.29)},円 \end{equation} Eqs.~\ref{(1.19)} and ~\ref {(1.20)} will be seen to imply that the ${\bf s2}(-&#124;k&#124;)$ 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,円d_{x}^{\ n} \speq \sum_{m=1}^{n} S1(n,m),円 (x,円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},円 S1(n,m) \ \ . \label{(1.32)} \end{equation} Their recurrence formula is \begin{equation} S1p(n+1,m)\speq n,円S1p(n,m)\spp S1p(n,m-1)\ \ , \label{(1.33)} \end{equation} with $S1p(1,1)=1,ドル $S1p(n,0)=0$ 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},円d_{x}^{\ n}\speq \sum_{m=1}^{n} S1(k;n,m),円 x^{(k-1)(n-m)},円(x^k,円d_{x})^m ,円\ \ {\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},円 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),円m+n-1],円S1(k;n-1,m)\spp S1(k;n-1,m-1)\ \ , \label{(1.37)} \end{equation} with $S1(k;1,1)=1,ドル $S1(k;n,0)=0$ 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},円\frac{m!}{n!},円 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),円m+n-1\bigr{)},円 s1(k;n-1,m) +\frac{m}{n},円 s1(k;n-1,m-1),円 \label{(1.39)} \end{equation} with $s1(k;n,m)= 0$ for $n<m,ドル $s1(k;1,1) = 1$ 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},円 s1(k;n,m),円 y^n \speq (g1(k;y))^m\ , \label{(1.40)} \end{equation} for $k\in {\bf Z}\setminus \{1\},ドル with \begin{equation} g1(k;y)\speq y,円 c1(k-1;y),\label{(1.41)} \end{equation} and, for $l \in {\bf Z}\setminus \{0\},ドル \begin{equation} c1(l;y)\speq \frac{-1\spp (1-l,円 y)^{-l}}{l^2,円y}\ .\label{(1.42)} \end{equation} Hence ${\bf s1}(k)$ is, like ${\bf s2}(k),ドル 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},円 c1_{n}^{(l)},円y^n\ ,\ \ c1_{n}^{(l)}\speq \binom {n+l} {l-1} ,円 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,ドル $c1(l;y)$ becomes a polynomial in $y$; {\it e.g.} $c1(-2;y)=1+y,ドル 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,ドル \begin{equation} c1_{n}^{(-l)}\speq {\binomial{l}{n+1}},円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}},円(k-1)^{n-2}& \mbox{for $k=2,3,..., {\rm and} \ n\in \Na$}\\ \\ {\binomial{&#124;k&#124;+1}{n}},円(&#124;k&#124;+1)^{n-2}& \mbox{for $-k\in \N0,円 ,,円n= 1,2,... &#124;k&#124;+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,...,ドル \begin{eqnarray} G1p(k;m;x) &\spdef& \sum_{n=m}^{\infty},円 S1p(k;n,m),円 \frac{x^n}{n!}\ , \label{(1.46)}\\ &\speq & \frac{1}{m!},円 \Bigl{[}(k-1),円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 $G1p(1;m;x)\speq \frac{1}{m!},円\bigl{(}-ln(1-x)\bigr{)}^m$. From eqs.~\ref{(1.47)}, ~\ref{(1.41)} and ~\ref{(1.42)}, \begin{equation} G1p(k;1;x)\speq (k-1),円 g1(k;\frac{x}{1-x})\speq \frac{1}{k-1},円 (-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 ${\bf S1}(-&#124;k&#124;)$ is, from eqs.~\ref{(1.36)} and ~\ref{(1.46)}, $G1(-&#124;k&#124;;m;x)\speq (-1)^m,円G1p(-&#124;k&#124;;m;-x),ドル hence \begin{equation} G1(k;1;x)\speq \frac{1}{1+&#124;k&#124;},円(-1\spp (1+x)^{1+&#124;k&#124;})\ \ \ \text{for} \ \ -k\in \N0\ . \label{(1.49)} \end{equation} For the signed matrix ${\bf S1}(-&#124;k&#124;)$ with elements defined by $S1s(-&#124;k&#124;;n,m):= (-1)^{n-m},円S1(-&#124;k&#124;,n,m)$ the e.g.f. of the $m-$th column is $G1s(-&#124;k&#124;;m;x)\speq (-1)^m,円 G1(-&#124;k&#124;;m;-x),ドル {\it i.e.} $G1s(-&#124;k&#124;;x),円 \equiv ,円 G1s(-&#124;k&#124;;1;x)\speq (1-(1-x)^{1+&#124;k&#124;})/(1+&#124;k&#124;)$ for $k\in \N0$. Tables 3 and 4 give the {\it EIS} A-numbers of some of the number triangles ${\bf s1}(k),ドル ${\bf S1}(k)$ 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},円 S1(k;n,m),円 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)}),ドル where $\overline{G1(k;t)}\speq G2(k;t)$ is given, for $k\neq 1,ドル in eq.~\ref{(1.24)}. Also $\overline{G1(1;t)}= G2(1;t)= exp(t)-1$ is obtained in the limit as 1ドル-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\},ドル \begin{eqnarray} r2(k;x)& \speq& \frac{g2(k;x)}{1-g2(k;x)}\speq \frac{-1\spp [1-(1-k)^2,円 x]^{\frac{1}{1-k}}} {k\spm [1-(1-k)^2,円 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),円 x]^{k-1}}{(1+(1-k)^2),円[1-(k-1),円 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,円[-1+(1-(k-1),円x)^{\frac{1}{1-k}}]\spm 1\ \ , \label{(1.53)}\\ & & \nonumber \\ R2p(-&#124;k&#124;;x)& \speq& e^{G2p(-&#124;k&#124;;x)}\spm 1\speq exp,円 [1-(1-(1+&#124;k&#124;),円 x)^{\frac{1}{1+&#124;k&#124;}}]\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},円[-1+(1-x)^{\frac{1}{k-1}}]\Bigr{)} \spm 1\ \ , \label{(1.55)}\\ & & \nonumber \\ R1(-&#124;k&#124;;x)& \speq& e^{G1(-&#124;k&#124;;x)}\spm 1\speq exp \Bigl{(}\frac{1}{1+&#124;k&#124;} [-1+(1+x)^{1+&#124;k&#124;}]\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),ドル 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 $(x^k,円d_{x})^n\speq x^k,円d_{x},円(x^k,円d_{x})^{n-1}$ 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 $\{x^m,円d_{x}^{\ m}\}_{1}^{n}$.\ \ \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),ドル 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ドル-k\in \Na$ the first term in eq.~\ref{(1.19)} becomes negative if and only if $(1-k),円(n-1)<m$ 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,円y],円g2^{\prime}(k;m,y)\spm m,円(k-1),円g2(k;m,y)\spm m ,円 g2(k;m-1,y) \speq 0\ \ , \ \ && \label{(2.1)}\\ g2(k;m,0)=0\ ,\ m\in \Na; \ \ g2^{\prime}(k;m,y)&#124;_{y=0}= 0\ ,\ m\in \{2,3,...\}\ ;\ g2^{\prime}(k;1,y)&#124;_{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 $y,円{\frac{d\ }{dy}},円\sum_{n=m}^{\infty},円n,円s2(k;n,m),円 y^n$ 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,ドル $g2(k;y):= g2(k;1,y)$ satisfies the first order differential equation \begin{equation} [1-(k-1)^2,円y],円g2^{\prime}(k;y)\spm (k-1),円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\},ドル \begin{equation} g2(k;y)\speq {\frac{1}{[1-(k-1)^2,円y]^{\frac{1}{k-1}}}},円 {\frac{1-[1-(k-1)^2,円y]^{\frac{1}{k-1}}}{k-1}}\ =:\ {\frac{y}{[1-(k-1)^2,円y]^{\frac{1}{k-1}}}},円 c2(k-1;y)\ . \label{(2.4)} \end{equation} {\it Proof}: Standard integration of a first order inhomogeneous differential equation of the form $g^{\prime}(y)+ f(y),円 g(y) = k(y)$. \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,円x]^{\frac{1}{l}}}{l,円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,円x)^{\frac{1}{l}},ドル one can write $g2(k;y)\speq y,円c2(1-k;y),ドル as stated in eq.~\ref{(1.22)}. \vspace*{+.1in} \noindent Consider the expansion 1ドル/(1-l^2,円x)^{1/l}\sspeq \sum_{n=0}^{\infty},円 b_{n}^{(l)} ,円 x^n\ ,ドル where $b_{n}^{(l)}\sspeq l^n,円(\prod_{j=1}^{n},円(j,円l\sspp 1 \sspm l))/n!$ and $b_{0}^{(l)}\sspeq 1,ドル $n\geq 1$. 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 ${\bf s2}(1),円=,円{\bf 1}$ we take $g2(1;y),円=,円y$. \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),ドル defined in eq.~\ref{(1.14)}, is $G2(k;m;x)\speq \frac{1}{m!},円(G2(k;1;x))^m,ドル $m\in \Na,ドル with $G2(k;1;x)\spequiv G2(k;x)\speq (k-1),円 g2(k;{\frac{x}{k-1}})$ 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),ドル and then use eq.~\ref{(1.21)}. For the ordinary Stirling numbers, {\it i.e.} for $k=1,ドル the stated result is well-known \cite{AS}. \eop \vspace*{+.1in} \noindent {\bf Lemma 8:} For $k\in {\bf Z}\setminus \{1\},ドル \begin{equation} e^{c,円x^k,円d_{x}}\speq \sum_{0}^{\infty}{\frac{1}{m!}},円\Bigl{[} g2(k;{\frac{c}{k-1}},円x^{k-1}),円(k-1),円x\Bigr{]}^m,円d_{x}^{\ m}\speq :e^{g(k;c;x),円x,円d_{x}}:\ , \label{(2.6)} \end{equation} where $g(k;c;x):= g2(k;{\frac{c}{k-1}},円x^{k-1}),円(k-1),ドル 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),ドル using eq.~\ref{(1.18)} (and remembering that $k\neq 1$) we find the o.g.f. $g2(k;m;{\frac{c}{k-1}},円x^{k-1})$ 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,ドル 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 $exp,円({\bf B}),円 x^l,円exp(-{\bf B})$ for $l\in \N0,ドル setting the operator ${\bf B}=,円c,円E_{k;x}=c,円x^k,円d_{x}$ for $k\in {\bf Z}$ and the commutator $[E_{k;x},x^l],円=,円 l,円x^{l+k-1}$. For $k=1$ we find $exp(c,円x,円d_{x}),円x^l,1円 \sspeq (exp(c),円x)^l,円exp(c,円x,円d_{x}),1円\sspeq (exp(c),円x)^l$. For $k\neq 1$ we first obtain $exp(c,円E_{k;x}),円x^l,円exp(-c,円E_{k;x}) \sspeq \sum_{n=0}^{\infty}{\frac{1}{n!}}(l/(k-1))_{n},円(c,円(k-1),円x^{k-1})^n$ using the rising factorial (or {\it Pochhammer}) symbol $(\nu)_{n}:= \nu,円(\nu +1)\cdots (\nu +n-1)$. This implies $exp(c,円x^k,円d_{x}),円x^l,1円\sspeq [(1+g(k;c;x)),円x]^l,1円$ with 1ドル+g(k;c;x)$ given in eq.~\ref{(1.9)}. The 1ドル$ 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),円z^n/n!,ドル is given by \begin{equation} {\cal G}2(k;z,x)\speq e^{x,円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 ],円[x^m],円 e^{x,円G2(k;z)}\ , \label{(2.8)} \end{equation} where $[y^k],円f(y)$ 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 $n!,円F_{n}(x)$ 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 $E_{n}(x) \sspeq \sum_{n=1}^{m}J(n,m),円x^n$ 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,ドル is equivalent to the functional equation \begin{equation} {\cal G}2(k;z,x+y)\speq {\cal G}2(k;z,x),円{\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. $r(x)\sspdef \sum_{n=1}^{\infty},円 r_{n},円 x^n$ of the row sums $r_{n}\sspdef \sum_{m=1}^{n},円s(n,m)$ 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},ドル 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. $R(x)\sspdef \sum_{n=1}^{\infty},円 R_{n},円 x^n/n!$ of the row sums $R_{n}\sspdef \sum_{m=1}^{n},円S(n,m)$ 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),ドル resp. ${\bf S2p}(k),ドル 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),ドル resp. $G2p(-k;x),ドル from eq.~\ref{(1.24)}, resp. eq.~\ref{(1.29)}.\ \ \eop \section{${\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),ドル are defined by \begin{equation} x^n,円d_{x}^{\ n}\speq \sum_{m=1}^{n},円S1(k;n,m),円x^{-m(k-1)},円(x^k,円 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,ドル {\it i.e.} ${\bf S1}(k)$ is a lower triangular matrix. \vspace*{+.1in} \noindent {\bf Lemma 13:}\hskip 4cm ${\bf S2}(k),円\cdot {\bf S1}(k) \speq {\bf 1},ドル or \begin{equation} \sum_{m=p}^{n},円 S2(k;n,m),円S1(k;m,p)\speq \delta_{n,p} ,円 \label{(3.2)} \end{equation} for fixed $k\in \bf Z,ドル $n\in \Na$ and $p\in \Na,ドル 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,ドル 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},ドル $n\in \Na$ and $x\neq 0,ドル \begin{equation} {\cal O}_{x}(k;n)\spdef x^{-(k-1)n},円(x^k,円d_{x})^n \speq \sum_{p=1}^{n},円 \delta(k;n,p),円{\cal O}_{x}(k;p)\ , \label{(3.3)} \end{equation} with $\delta(k;n,p):= \sum_{m=p}^{n},円 S2(k;n,m),円S1(k;m,p)$. 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 ${\frac{1}{p!}},円 {\cal O}_{x}(k;p)$ for fixed $k\in {\bf Z}$ and $p=1,...,n$ to the monomials $x^q,ドル for $q=1,..,n$. The linear independence is then inferred from the non-singularity of the $n\times n$ matrix $A_{q,p}(k)\:={\frac{1}{p!}},円 \prod_{j=0}^{p-1}(q+j(k-1))$. In fact, $Det,円 {\bf A}(k)=+1$ 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),円\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 $\{x^p,円d_{x}^{\ p}\}_{1}^{n}$. \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 $x^{n},円 d_{x}^{\ n}\speq (x,円d_{x}-(n-1)),円x^{n-1},円d_{x}^{\ n-1}$ and insert eq.~\ref{(3.1)} in both sides of this identity. After differentiation, remembering the triangularity of ${\bf S1}(k),ドル 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),ドル 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ドル-k\in \Na$ this follows from the fact that $s1(k;n-1,m)\speq 0$ if $n-1,円>,円 (1-k),円m,ドル {\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,ドル 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,ドル 
{\it i.e.} $s1(k;n-1,m)=0$ for $n-1,円>,円 (1-k),円m,ドル leads to a vanishing 
second term in the $s1(k;n-1,m+1)$ recurrence for all $n-1,円>,円 
(1-k),円m,円+,1円.$ Therefore, $s1(k;(1-k),円(m+1),円+,円 1,m+1)$ will be zero because 
the coefficient of the first term of this recurrence vanishes and the second 
term is absent since $(1-k),円(m+1),円>,円(1-k),円m$. 
Then $s1(k;n-1,m+1)$ vanishes recursively for all 
$n-1,円\geq,円(1-k),円(m+1),円+,1円$.\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),円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,ドル the first-order linear differential-difference equation
\begin{eqnarray}
[1-(k-1),円y],円g1^{\prime}(k;m;y)\spm m,円(k-1)^2,円g1(k;m;y)\spm m ,円
g1(k;m-1;y)
\speq 0\ \ , \ \ && \label{(3.6)}\\ 
g1(k;m;0)=0\ ,\ m\in \Na; \ \ g1^{\prime}(k;m;y)&#124;_{y=0}= s1(k;1,1),円 
\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,ドル which results in a standard linear 
inhomogeneous differential equation for $g1(k;y):= g1(k;1;y),ドル namely
\begin{equation}
[1-(k-1),円y],円g1^{\prime}(k;y)\spm (k-1)^2,円g1(k;y)\spm 1\speq 0 ,,円
 \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},ドル and $c1(-l;x)$ is a polynomial in $x$ for $l\in \Na$. For example,
$c1(-3;x)\sspeq 1\sspp 3,円x\sspp 3,円x^2.$ 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},ドル the signed matrix ${\bf s2s}(k)$ 
by $s2s(k;n,m):= (-1)^{n-m},円 s2(k;n,m)$.
Then eq.~\ref{(3.2)} implies
\begin{equation}
 {\bf s2s}(k),円\cdot,円{\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 
$c2(l;y)\sspeq \sum_{n=0}^{\infty},円 c2^{(l)}_{n},円y^n$
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,円\prod_{j=1}^{n},円 (j,円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),ドル defined in eq.~\ref{(1.36)} for $k\in \Na,ドル is $G1p(k;m;x)\speq \frac{1}{m!},円
(G1p(k;1;x))^m,ドル $m\in \Na,ドル with 
$G1p(k;1;x)\spequiv G1p(k;x)\speq (k-1),円 g1(k;{\frac{x}{k-1}})$ for 
$k=2,3,...$ and $G1p(1;1;x)\spequiv G1p(1;x)\speq -,円ln(1-x)$.
\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),ドル $k> 1$: 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 -,円\overline{G2(k;-x)},ドル 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},円 S1(k;n,m),円 x^m\ ,ドル $n\in \Na,ドル and 
$S1_{0}(k;x):=1$ is
\begin{equation}
{\cal G}1(k;z,x):=\sum_{n=0}^{\infty}S1_{n}(k;x),円z^n/n! 
\speq e^{x,円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,ドル 
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 ],円[x^m],円 e^{x,円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,ドル
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,ドル 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),円
{\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 ${\bf S1}(-&#124;k&#124;)$ 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),ドル 
resp. ${\bf S1}(-&#124;k&#124;),ドル 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),ドル resp. $G1(-&#124;k&#124;;x),ドル 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),ドル $k\in \Na,ドル resp. 
${\bf S1s}(-&#124;k&#124;)$ triangles.
Here Lemma 12 applies with the e.g.f.s $G1(k;x),ドル resp. 
$G1s(-&#124;k&#124;;x),ドル 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 
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\bibitem{Dr} N. Dragon: {\it \htmladdnormallink{Konforme Transformationen}{http://www.itp.uni-hannover.de/~dragon/Group.html}}, ps.gz file: \\
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\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)},ドル $\bf{k\neq 1}$\ \ \ \ $\bf{s2(1)}:={\bf 1}$ }
\end{center}
\begin{center}
\begin{tabular}{&#124;c&#124;c&#124;c&#124;c&#124;}\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ドル$) \\
&&&\\ \hline
&&&\\
 2 & \seqnum{A007318}($n-1,m-1$) (Pascal) & \seqnum{A000012} & \seqnum{A000079} (powers of 2ドル$) \\
&&&\\ \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}{&#124;c&#124;c&#124;c&#124;c&#124;}\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ドル$) \\
&&&\\ \hline
&&&\\	
 1 & \seqnum{A008277} (Stirling 2nd kind) & \seqnum{A000012} (powers of 1ドル$) & \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)},ドル ${\bf k\neq 1}$ \ \ \ ${\bf s1(1):= \bf 1}$}
\end{center}
%\vspace{2mm}
\begin{center}
\begin{tabular}{&#124;c&#124;c&#124;c&#124;c&#124;}\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,ドル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ドル$) \\
&&&\\ \hline
&&&\\
 2 & \seqnum{A007318}($n-1,m-1$) (Pascal) & \seqnum{A000012} (powers of 1) & \seqnum{A000079} (powers of 2ドル$) \\
&&&\\ \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}{&#124;c&#124;c&#124;c&#124;c&#124;}\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ドル$) \\
&&&\\ \hline
&&&\\	
 1 & \seqnum{A008275} (unsigned Stirling 1st kind) & \seqnum{A000142}($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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