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\begin{center}
\vskip 1cm{\LARGE\bf The Generalized Stirling and Bell Numbers \\
\vskip .1in Revisited}
\vskip 1cm
\large Toufik Mansour\\
Department of Mathematics\\
University of Haifa \\
31905 Haifa \\
Israel\\
\href{mailto:toufik@math.haifa.ac.il}{\tt toufik@math.haifa.ac.il}\\
\ \\
Matthias Schork\\
Camillo-Sitte-Weg 25 \\
60488 Frankfurt \\
Germany\\
\href{mailto:mschork@member.ams.org}{\tt mschork@member.ams.org}\\
\ \\
Mark Shattuck\\
Department of Mathematics\\
University of Tennessee \\
Knoxville, TN 37996\\
USA \\
\href{mailto:shattuck@math.utk.edu}{\tt shattuck@math.utk.edu}\\
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\begin{abstract}
The generalized Stirling numbers $\mathfrak{S}_{s;h}(n,k)$ introduced
recently by the authors are shown to be a special case of the three
parameter family of generalized Stirling numbers
$S(n,k;\alpha,\beta,r)$ considered by Hsu and Shiue.  From this
relation, several properties of $\mathfrak{S}_{s;h}(n,k)$ and the
associated Bell numbers $\mathfrak{B}_{s;h}(n)$ and Bell polynomials
$\mathfrak{B}_{s;h|n}(x)$ are derived.  The particular case $s=2$ and
$h=-1$ corresponding to the meromorphic Weyl algebra is treated
explicitly and its connection to Bessel numbers and Bessel polynomials
is shown. The dual case $s=-1$ and $h=1$ is connected to Hermite
polynomials. For the general case, a close connection to the Touchard
polynomials of higher order recently introduced by Dattoli et al. is
established, and Touchard polynomials of negative order are introduced
and studied. Finally, a $q$-analogue $\mathfrak{S}_{s;h}(n,k|q)$ is
introduced and first properties are established, e.g., the
recursion relation and an explicit expression.  It is shown that the
$q$-deformed numbers $\mathfrak{S}_{s;h}(n,k|q)$ are special cases of
the type-II $p,q$-analogue of generalized Stirling numbers introduced
by Remmel and Wachs, providing the analogue to the undeformed case
($q=1$). Furthermore, several special cases are discussed explicitly,
in particular, the case $s=2$ and $h=-1$ corresponding to the
$q$-meromorphic Weyl algebra considered by Diaz and Pariguan.
\end{abstract}

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