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\begin{center}
\vskip 1cm{\LARGE\bf 
Integer Sequences Connected with \\
\vskip .1in
Extensions of the Bell Polynomials
}
\vskip 1cm
\large
Pierpaolo Natalini \\
Universit\`a degli Studi Roma Tre\\
Dipartimento di Matematica e Fisica \\
Largo San Leonardo Murialdo, 1\\
00146 - Roma\\
Italy\\
\href{mailto:natalini@mat.uniroma3.it}{\tt natalini@mat.uniroma3.it} \\
\ \\
Paolo Emilio Ricci\\
International Telematic University UniNettuno \\
Corso Vittorio Emanuele II, 39 \\
00186 - Roma \\
Italy \\
\href{mailto:paoloemilioricci@gmail.com}{\tt paoloemilioricci@gmail.com}
\end{center}

\vskip .2 in

\begin{abstract}
The {\it Encyclopedia of Integer Sequences} includes some
sequences that are connected with the Bell numbers 
and that have a 
particular combinatorial meaning. In this article, we find a general 
meaning for framing sequences, including the above mentioned ones.
Furthermore, by using Laguerre-type derivatives, we derive the Laguerre-type 
Bell numbers of higher order, showing, as a by-product, that it is 
possible to construct new integer sequences which are not included
in the {\it Encyclopedia}.
\end{abstract}

\section{Introduction}

The Bell polynomials \cite{be} are a mathematical tool for representing the
$n$th derivative of a composite function. They are strictly related to
partitions \cite{an,b-c,rio}.  They are also applied in 
many different frameworks such as the Blissard problem \cite[p.\ 46]{rio}, 
the representation of Lucas polynomials of the first and second kind \cite{dc-r}, 
the construction of representation formulas for the Newton sum rules 
for the zeros of polynomials \cite{i-n-r1,i-n-r2}, the recurrence 
relations for a class of Freud-type polynomials \cite{be-r},
the representation formulas for the symmetric functions of a countable set 
of numbers generalizing the classical algebraic Newton-Girard formulas \cite{c-r}.  
In particular, Cassisa and Ricci \cite{c-r} found reduction formulas for the 
{\em orthogonal invariants} of a strictly positive compact operator, 
obtaining, as a by-product, the so-called Robert formulas \cite{rob}.

Recently Kataria and Vellaisamy \cite{k-v1,k-v2} obtained connections of Bell polynomials with another important class of polynomials known as Adomian polynomials.

Some generalized forms of Bell polynomials can be found in the literature,
(see e.g., \cite{b-n-r,fu,n-r1,n-r2,n-r4,no-r,r-s}). For instance, some papers 
\cite{b-n-r,no-r} are concerned with the multi-dimensional case.

The authors \cite{n-r1} introduced the higher 
order Bell polynomials and their main properties and they \cite{n-r4} used 
some recursion formulas to compute, by means of the program Mathematica$^\copyright$, 
the $r$th order complete Bell polynomials $B^{[r]}_n$ (for $r=2,3,4,5$) and 
the relevant Bell numbers $b^{[r]}_n$.

In this article, after briefly recalling this theory, we show how to merge
a family of integer sequences, already known in the 
{\it Encyclopedia of Integer Sequences} \cite{s-p},
into a general framework.

Furthermore, we construct new sequences of integer numbers, not included 
in the {\it Encyclopedia}, related to the Laguerre-type Bell polynomials. 
To this aim, we recall, in Section 6, a short introduction to the Laguerre 
derivative.


\section{Recalling the Bell polynomials}

By considering the composite function 
$\Phi(t):= f(g(t))$ of functions $x=g(t)$ and $y=f(x)$, defined 
in suitable intervals of the real axis and $n$  times 
differentiable with respect to the relevant independent variables
and by using the notation
\begin{eqnarray}
\label{2.1} \Phi_h:= D_t^h \Phi(t), \qquad f_h:=
D_x^hf(x)|_{x=g(t)}, \qquad g_h:= D_t^h g(t) \, ,
\end{eqnarray}
and
\begin{eqnarray}
\label{2.2} \left(\left[ f,g
\right]_n\right):=(f_1,g_1;f_2,g_2;\dots;f_n,g_n) \, ,
\end{eqnarray}
the Bell polynomials are defined as follows:
\begin{eqnarray}
\label{2.3} Y_n \left( \left[ f,g \right]_n \right) := \Phi_n.
\end{eqnarray}

Inductively, by letting
$$
[g]_n:= (g_1,g_2,\dots, g_n),
$$
we can write
\begin{eqnarray}
\label{2.4}
Y_n \left( \left[ f,g \right]_n \right) =
\sum_{k=1}^{n}A_{n,k}([g]_n)f_{k},
\end{eqnarray}
where the coefficient $A_{n,k}$, for any $k=1, \ldots,n$,
is a polynomial in $g_1, g_2,\ldots, g_n$, 
homogeneous of degree $k$ and {\em isobaric} of weight $n$
(i.e., it is a linear combination of monomials
$g_1^{k_1}g_2^{k_2}\cdots g_n^{k_n}$ whose weight is constantly
given by $k_1+2k_2+\ldots +nk_n =n$).

\begin{proposition}
The Bell polynomials satisfy the recurrence relation
\begin{eqnarray}
\label{2.5} \left\{
\begin{array}{l}
      Y_0\left( \left[ f,g \right]_0 \right):= f_1 \\
      Y_{n+1} \left( \left[ f,g \right]_{n+1} \right) =
      \displaystyle \sum_{k=0}^n  {n \choose k} 
      Y_{n-k}\left (\left[ f_1,g \right]_{n-k} \right) g_{k+1},
\end{array} \right.
\end{eqnarray}
where
\begin{eqnarray*}
\left(\left[ f_1,g
\right]_{n-k}\right):=(f_2,g_1;f_3,g_2;\dots;f_{n-k+1},g_{n-k}).
\end{eqnarray*}
\end{proposition}
\vspace{1ex}

The Fa\`a di Bruno formula \cite{fa,rom,r-r} gives an explicit 
expression for the Bell polynomials, however, as this formula makes use 
of partitions, it is not useful in practice for computing higher order 
Bell polynomials, whereas this can be done in a easy way by means of the 
following recursion formula for the coefficients  $A_{n,k}$ in equation 
(\ref{2.4}) which are known as partial Bell polynomials.

\begin{theorem} We have, for all integers $n$,
\begin{eqnarray}
\label{2.6}
      A_{n+1,1} = g_{n+1}, \qquad A_{n+1,n+1} = g_1^{n+1}.
\end{eqnarray}

Furthermore, for all $k=1,2,\dots, n-1$, the
$A_{n,k}$ coefficients can be computed by the recurrence relation
\begin{eqnarray}
\label{2.7}
      A_{n+1,k+1}([g]_{n+1}) =
      \d \sum_{h=0}^{n-k}  {n \choose h} 
      A_{n-h,k}([g]_{n-h}) g_{h+1}.
\end{eqnarray}
\end{theorem}


\begin{definition}
The complete Bell polynomials are defined by
\begin{eqnarray}
\label{2.8}
    B_n([g]_n) = Y_n(1, g_1; 1, g_2; \dots 1, g_n)
    = \d \sum_{k=1}^n A_{n,k}([g]_n),
\end{eqnarray}
and the Bell numbers by
\begin{eqnarray}
\label{2.9}
    b_n = Y_n(1, 1; 1, 1; \dots ; 1, 1)
    = \d \sum_{k=1}^n A_{n,k}(1,1, \dots,1).
\end{eqnarray}
\end{definition}

\section{Bell polynomials of order $r$}

Bernardini et al. \cite{b-n-r} introduced the 
multi-dimensional Bell polynomials of higher order. Here, 
we briefly recall this extension of the classical Bell polynomials 
only in the one-dimensional case.

Consider $\Phi(t):= f(\varphi^{(1)}(\varphi^{(2)}( \cdots
(\varphi^{(r)}(t)))))$, i.e., the composition of functions 
$x^{(r)}=\varphi^{(r)}(t)$, \dots , $x^{(2)}=\varphi^{(2)}
\left( x^{(3)} \right) $, $x^{(1)}=\varphi^{(1)} \left(
x^{(2)} \right) $, $y=f \left( x^{(1)} \right) $ defined in
suitable intervals of the real axis, and suppose that the
functions $\varphi^{(r)}$, \dots , $\varphi^{(2)}$, 
$\varphi^{(1)}$, $f$ are $n$ times differentiable with
respect to the relevant independent variables so that, by using
the chain rule, $\Phi(t)$ can be differentiated $n$ times
with respect to $t$.

    We use the following notation
\begin{eqnarray}
\label{3.1}
\Phi_h &amp;:= &amp; D_t^h \Phi(t), \nonumber \\
f_{h} &amp;:=&amp; D_{x^{(1)}}^h f|_{x^{(1)}=\varphi^{(1)}(\cdots
(\varphi^{(r)} (t)))},
\nonumber \\
\varphi^{(1)}_h &amp;:=&amp; D_{x^{(2)}}^h
\varphi^{(1)}|_{x^{(2)}=\varphi^{(2)}(\cdots (\varphi^{(r)} (t)))},
 \\
&amp;\vdots&amp; \nonumber \\
\varphi^{(r)}_h &amp;:=&amp; D_t^h \varphi^{(r)}(t),\nonumber
\end{eqnarray}
and
$$
\left(\left[f,\varphi^{(1)}, \dots , \varphi^{(r)}\right]_n \right)
:= ( f_{1}, \varphi^{(1)}_1, \dots , \varphi^{(r)}_1; \dots ; f_{n},
\varphi^{(1)}_n, \dots , \varphi^{(r)}_n ).
$$

Then the $n$th derivative of the function $\Phi$ allows
us to define the one-dimensional Bell polynomials of order 
$r$, $Y_n^{[r]}$, as follows:
\begin{eqnarray}
\label{3.2}
Y_n^{[r]} \left(\left[f,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_n \right) := \Phi_n.
\end{eqnarray}
For $r=1$ we obtain the ordinary Bell polynomials $Y_n^{[1]}
\left(\left[f,\varphi^{(1)}\right]_n \right) = Y_n
\left(\left[f,\varphi^{(1)}\right]_n \right)$.

The first polynomials have the following explicit expressions
\begin{eqnarray}
\label{3.3}
&amp;&amp; Y_1^{[r]} \left(\left[f,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_1
\right) = f_1 \varphi^{(1)}_1 \cdots \varphi^{(r)}_1\\
{\phantom{\rule{1 pt}{20 pt}}}
&amp;&amp; Y_2^{[r]} \left(\left[f,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_2 \right) = f_2 \biggl( \varphi^{(1)}_1 \cdots
\varphi^{(r)}_1 \biggr)^2 + f_1  \varphi^{(1)}_2
\biggl(\varphi^{(2)}_1 \cdots \varphi^{(r)}_1
\biggr)^2 \nonumber \\
{\phantom{\rule{1 pt}{20 pt}}}
&amp;&amp; \qquad \qquad \qquad + f_1 \varphi^{(1)}_1 \varphi^{(2)}_2 \biggl
( \varphi^{(3)}_1 \cdots \varphi^{(r)}_1\biggr)^2 + f_1
\varphi^{(1)}_1 \varphi^{(2)}_1
 \cdots \varphi^{(r-1)}_1 \varphi^{(r)}_2 \nonumber.
\end{eqnarray}

In general, we have
\begin{eqnarray}
\label{3.4}
Y^{[r]}_{n}([f,\varphi^{(1)},\ldots,\varphi^{(r)}]_n) =
\d \sum_{k=1}^n  A^{[r]}_{n,k} ([\varphi^{(1)},\ldots,\varphi^{(r)}]_n) f_k .
\end{eqnarray}

The authors \cite{n-r1} proved the following useful 
properties for the polynomials $Y_n^{[r]}$

\begin{theorem}
For every integer $n$, the polynomials $Y_n^{[r]}$ are
expressed in terms of the Bell polynomials of lower order, by
means of the following equation
\begin{eqnarray}
\label{3.5}
&amp;&amp;Y_n^{[r]} \left(\left[f,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_n \right) = Y_n \left(\left[ f,Y^{[r-1]}
\left(\left[\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]\right)\right]_n \right), \qquad
\end{eqnarray}
where
\begin{eqnarray*}
&amp;&amp; \left(\left[ f,Y^{[r-1]} \left(\left[\varphi^{(1)},
\dots , \varphi^{(r)}\right]\right)\right]_n \right)\nonumber \\
{\phantom{\rule{1 pt}{22 pt}}}
&amp;&amp; \qquad :=\biggl( f_{1}, Y_1^{[r-1]} \left( \left[
\varphi^{(1)}, \dots , \varphi^{(r)} \right] _1 \right); \dots ;
f_{n}, Y_n^{[r-1]}\left( \left[ \varphi^{(1)}, \dots ,
\varphi^{(r)} \right] _n \right) \biggr). \nonumber
\end{eqnarray*}
\end{theorem}

\begin{theorem}
The following recurrence relation for the Bell polynomials
$Y_n^{[r]}$ holds
\begin{eqnarray}
\label{3.6}
\left\{
\begin{array}{l}
Y_0^{[r]}\left(\left[f,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_0 \right) =  f_{1} \\
{\phantom{\rule{1 pt}{24 pt}}}
Y_{n+1}^{[r]} \left(\left[f,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_{n+1} \right) =\displaystyle \sum_{k=0}^n {n
\choose k} Y_{n-k}^{[r]} \left(\left[f_1,\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_{n-k}
\right)\\
{\phantom{\rule{1 pt}{26 pt}}}
\hspace{5 cm} \times Y_{k+1}^{[r-1]} \left(\left[\varphi^{(1)},
\dots , \varphi^{(r)}\right]_{k+1} \right),
\end{array} \right.
\end{eqnarray}
\end{theorem}
where
$$
\left(\left[f_1,\varphi^{(1)}, \dots , \varphi^{(r)}\right]_{n-k}
\right) := \left( f_{2}, \varphi^{(1)}_1, \dots , \varphi^{(r)}_1;
\dots ; f_{n-k+1}, \varphi^{(1)}_{n-k}, \dots , \varphi^{(r)}_{n-k}
\right).
$$

The recurrence relation (\ref{2.6})--(\ref{2.7}) can
be generalized as follows:
\begin{theorem} For all integer $n$, we have 
\begin{eqnarray}
\label{3.7}
\begin{array}{l}
 A^{[r]}_{n+1,1} = Y^{[r-1]}_{n+1}\left(\left[\varphi^{(1)},
\dots , \varphi^{(r)}\right]_{n+1}\right) , \\
{\phantom{\rule{1 pt}{20 pt}}}
A^{[r]}_{n+1,n+1} = \left( Y^{[r-1]}_1\left( \left[ \varphi^{(1)},
\dots , \varphi^{(r)}\right]_1\right) \right) ^{n+1} = \left( \varphi^{(1)}_1
\cdots \varphi^{(r)}_1\right) ^{n+1}.
\end{array}
\end{eqnarray}
Furthermore, for all $k=1,2,\dots, n-1$, the $\emph{$r$th}$ 
order partial Bell polynomials $A^{[r]}_{n,k}$ satisfy the recursion
\begin{eqnarray}
\label{3.8}
\begin{array}{c}
A_{n+1,k+1}^{[r]} \left( \left[\varphi^{(1)}, \dots , \varphi^{(r)}\right]_{n+1} \right) =
\d \sum_{h=0}^{n-k} {n \choose h} A_{n-h,k}^{[r]} \left(\left[\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_{n-h} \right) \\
{\phantom{\rule{1 pt}{24 pt}}}
\hspace{2.3 cm} \times Y_{h+1}^{[r-1]} \left(\left[\varphi^{(1)}, \dots ,
\varphi^{(r)}\right]_{h+1} \right).
\end{array}
\end{eqnarray}
\end{theorem}
\begin{definition}
The complete Bell polynomials of order $r$, $B^{[r]}_n$, are defined by the equation
\begin{eqnarray*}
\begin{array}{c}
B^{[r]}_n \left(\left[\varphi^{(1)}, \dots , \varphi^{(r)}\right]_n \right) =
Y_n^{[r]}(1,\varphi^{(1)}_1, \dots , \varphi^{(r)}_1; \dots ; 1,\varphi^{(1)}_n, \dots ,\varphi^{(r)}_n)\\
{\phantom{\rule{1 pt}{24 pt}}}
= \d \sum_{k=1}^n A^{[r]}_{n,k} \left(\left[\varphi^{(1)}, \dots , \varphi^{(r)}\right]_n \right),
\end{array}
\end{eqnarray*}
and the $\emph{$r$th}$ order Bell numbers by
\begin{eqnarray*}
b^{[r]}_n = Y_n^{[r]}(1,1,1; \dots ; 1,1,1) = \d \sum_{k=1}^n A^{[r]}_{n,k} (1,1; \dots ; 1,1).
\end{eqnarray*}
\end{definition}


\section{Higher order Bell numbers, for  $r = 2, 3, 4, 5$}

The sequences of higher order Bell
numbers, which are presented here, appear in the {\it Encyclopedia of
Integer Sequences} \cite{s-p} under \seqnum{A144150}, arising from
a problem of combinatorial analysis and even \cite{be2,gi} as the McLaurin
coefficients of the functions 
\begin{eqnarray*}
\begin{array}{l}
\exp(\exp(\exp(x)-1)-1),\\
\exp(\exp(\exp(\exp(x)-1)-1)-1),\\
\exp(\exp(\exp(\exp(\exp(x)-1)-1)-1)-1),\\
\exp(\exp(\exp(\exp(\exp(\exp(x)-1)-1)-1)-1)-1),
\end{array}
\end{eqnarray*}
for the cases $r=2$, $r=3$, $r=4$, $r=5$, respectively, and so on
for the subsequent values of $r$. In our approach they
assume a more general meaning, as they are independent of the
functions $f,\varphi^{(1)}, \dots , \varphi^{(r)}$.

By using the recurrence relation (\ref{3.7})--(\ref{3.8}) 
and by means of the computer algebra program Mathematica$^\copyright$, 
we find the following sequences for the higher order Bell numbers $b^{[2]}_n$,
$b^{[3]}_n$, $b^{[4]}_n$, $b^{[5]}_n$, $(n=1,2,\dots, 10)$:

\begin{table}[H]
\begin{center}
\begin{tabular}{l * {4}{r}} 
\hline
 $n$ &amp; $b^{[2]}_n$ &amp;  $b^{[3]}_n$  &amp; $b^{[4]}_n$ &amp;  $b^{[5]}_n$ \\
\hline
1 &amp; 1  &amp; 1   &amp; 1  &amp; 1  \\
2 &amp; 3  &amp; 4    &amp; 5  &amp; 6  \\
3  &amp; 12  &amp; 22  &amp; 35  &amp; 51  \\
4  &amp; 60  &amp; 154   &amp; 315  &amp; 561  \\
5 &amp; 358  &amp; 1304   &amp; 3455  &amp; 7556  \\
6 &amp; 2471  &amp; 12915   &amp; 44590  &amp; 120196  \\
7 &amp; 19302  &amp; 146115   &amp; 660665  &amp; 2201856  \\
8 &amp; 167894  &amp; 1855570   &amp; 11035095  &amp; 45592666  \\
9 &amp; 1606137  &amp; 26097835   &amp; 204904830  &amp; 1051951026  \\
10 &amp; 16733779 &amp; 402215465   &amp; 4183174520 &amp; 26740775306  \\
\hline
\end{tabular}
\end{center}
\caption{Higher order Bell numbers for $ n=1,2,\dots, 10 $.}
\end{table}
The above table can be extended up to the
desired order, as the Mathematica$^\copyright$ program runs 
efficiently.


\section{Sequences of integer numbers connected with the above schemes}

Putting, for all integers $n$, $b^{[0]}_n := 1, b^{[1]}_n
:= b_n$, and reading Table $1$ by row, we find the new array
\begin{table}[H]
\begin{center}
\begin{tabular}{cccccccc}
      &amp; $k=0$ &amp;  $k=1$  &amp; $k=2$ &amp;  $k=3$  &amp;  $k=4$  &amp; $k=5$ &amp;  \\
\hline
$b^{[k]}_1$ \vline&amp; 1  &amp; 1 &amp; 1  &amp; 1 &amp; 1 &amp; 1 &amp;   \\
$b^{[k]}_2$ \vline&amp; 1  &amp; 2 &amp; 3  &amp; 4 &amp; 5 &amp; 6 &amp;   \\
$b^{[k]}_3$ \vline&amp; 1  &amp; 5  &amp; 12  &amp; 22  &amp; 35  &amp; 51 &amp;   \\
$b^{[k]}_4$ \vline&amp; 1  &amp; 15  &amp; 60  &amp; 154   &amp; 315  &amp; 561 &amp;   \\
$b^{[k]}_5$ \vline&amp; 1  &amp; 52 &amp; 358  &amp; 1304   &amp; 3455  &amp; 7556 &amp;  \\
$b^{[k]}_6$ \vline&amp; 1  &amp; 203 &amp; 2471  &amp; 12915   &amp; 44590  &amp; 120196 &amp;  \\
\hline
\end{tabular}
\end{center}
\caption{The above sequences to be read by row.}
\end{table}


The construction law of the sequence $\left( b^{[k]}_3 \right)_{k\ge0}$ (third row) is
\begin{align}
b^{[1]}_3 &amp;= b^{[1]}_2 + b^{[2]}_2 \nonumber \\
b^{[2]}_3 &amp;= b^{[2]}_2 + b^{[3]}_2 + b^{[4]}_2 \nonumber \\
&amp; \quad \vdots \label{5.1} \\
b^{[k]}_3 &amp;= b^{[k]}_2 + b^{[k+1]}_2 + \dots + b^{[2k]}_2 . \nonumber
\end{align}

Deriving a similar law for the subsequent rows is an open 
problem, since there is no connection between the meaning 
of the sequences $( b^{[k]}_3)_{k\ge0}$ and $( b^{[k]}_4 )_{k\ge0}$. 
In the {\it Encyclopedia of Integer Sequences}, the third row 
($1, 5, 12, 22, 35, 51, \ldots$) appears under \seqnum{A000326},
and $b^{[n]}_3 $ is identified, according to the above recursion
(\ref{5.1}), as the sum of $n$ integers starting from $n$, or as
the pentagonal number $n(3n-1)/2$.

The fourth row ($1, 15, 60, 154, 315, 561, \ldots)$ appears
under \seqnum{A005945} and $b^{[n]}_4 $ is the number of $n$-step
mappings with 4 inputs.

The subsequent rows are not included explicitly in the
{\it Encyclopedia}, so our approach is useful in order to merge
the above sequences into a general framework.


\section{Laguerre-type derivatives}

Dattoli and Ricci \cite{d-r1} introduced the Laguerre-type 
derivatives in connection  with a 
differential isomorphism denoted by the symbol
${\cal T}:= {\cal T}_x$, acting onto the space ${\cal A}:=
{\cal A}_x$ of analytic functions of the $x$ variable by
means of the correspondence  
\begin{eqnarray}
\label{6.1} D:= \frac{d}{dx} \; \rightarrow \; {\hat D}_L:= D x D
; \qquad  x \cdot \; \rightarrow \; {\hat D}_x^{-1}, \qquad
\end{eqnarray}
where
\begin{eqnarray*}
{\hat D}_x^{-1}f(x):= \int_0^x f(\xi) d\xi,
\end{eqnarray*}
\begin{eqnarray*}
{\hat D}_x^{-n}f(x):= \frac{1}{(n-1)!}\int_0^x (x -
\xi)^{n-1}f(\xi) d\xi,
\end{eqnarray*}
and ${\hat D}_x^{-n}$ denotes the $n$-fold R-L integration, so that
\begin{eqnarray}
\label{6.2} {\cal T}_x (x^n) = {\hat D}_x^{-n}(1):=
\frac{1}{(n-1)!}\int_0^x (x - \xi)^{n-1}d\xi = \frac{x^n}{n!}.
\qquad
\end{eqnarray}

\vspace{2ex}

According to the isomorphism ${\cal T}_x$, the exponential operator $e^x$ 
is transformed into the first Laguerre-type exponential 
$e_1(x):= \sum_{k=0}^\infty x^k/(k!)^2$ which is an
eigenfunction of the {\em Laguerre derivative operator} 
$D_L:=DxD$. We have, in fact,
\begin{eqnarray*}
{\cal T}_x (e^x) = \sum_{k=0}^\infty \frac{{\cal T}_x
(x^k) }{k!} = \sum_{k=0}^\infty \frac{x^k}{(k!)^2} = e_1(x),
\end{eqnarray*}
\begin{eqnarray*}
{\hat D}_L \; e_1(ax) = a e_1(ax), \qquad \forall
a \in {\bf C}.
\end{eqnarray*}

This result can be generalized by considering the $r$th
Laguerre-type exponential $e_r(x):= \sum_{k=0}^\infty
x^k/(k!)^{r+1}$, the $r$th Laguerre-type derivative
operator $D_{rL}:= DxDxD \cdots DxD$ (containing $r+1$ ordinary 
derivatives), and the iterated isomorphism ${\cal
T}^r$, since
\begin{eqnarray*}
{\cal T}_x^r (e^x) = \sum_{k=0}^\infty \frac{{\cal
T}_x (x^k) }{(k!)^r} = \sum_{k=0}^\infty \frac{x^k}{(k!)^{r+1}} =
e_r(x),
\end{eqnarray*}
\begin{eqnarray*}
{\hat D}_{rL} \; e_r(ax) = a e_r(ax), \qquad
\forall a \in {\bf C}.
\end{eqnarray*}

\begin{remark}
The above results show that, for every positive integer $r$, 
we can define a Laguerre-type exponential function
$e_r(x)$, satisfying an eigenfunction property, which is an
analog of the elementary property of the exponential. The Laguerre-type exponential function reduces to the classical exponential function when $r=0$, so that
we can put by definition
\begin{eqnarray*}
e_0(x):= e^x, \qquad  {\hat D}_{0L} := D.
\end{eqnarray*}
Obviously, ${\hat D}_{1L} := {\hat D}_L$.
For this reason we will refer to such functions as {\em
$L$-exponential functions}, or shortly  {\em $L$-exponentials}.
\end{remark}

Bernardini et al.~\cite{b-d-r} found several applications 
to the theory of $L$-exponential functions.

\section{Laguerre-type Bell polynomials and numbers}

The problem of constructing Bell polynomials can be extended in
the natural way to the case of the Laguerre-type derivatives.

By using the notation (\ref{2.1})--(\ref{2.2}), we
introduce the following definition
\begin{definition}
The $\emph{$n$th}$ Laguerre-type Bell polynomial of order $r$, denoted
by $_{rL}Y_n\left( x; \left[ f,g \right]_n \right)$, represents 
the $\emph{$n$th}$ Laguerre-type derivative of order $r$ of
the composite function $f(g(t))$.
\end{definition}
The authors \cite{n-r2} showed that $_{rL}Y_n$ can be
expressed as a polynomial in the independent variable $x$,
depending on $f_1,g_1;f_2,g_2;\dots;f_n,g_n$ in terms of the
classical Bell polynomials.

\vspace{1 ex}

According to a general result due to Viskov \cite{vi}, the 
Laguerre derivative satisfy
\begin{eqnarray}
\label{7.1}
\begin{array}{l}
(D_L)^n = (DxD)^n = D^nx^nD^n,
\end{array}
\end{eqnarray}
and furthermore, for any order $r$, it turns out that
\begin{eqnarray}
\label{7.2}
\begin{array}{l}
(D_{rL})^n = (DxDx \cdots DxD)^n = D^nx^nD^nx^n \cdots D^nx^nD^n.
\end{array}
\end{eqnarray}

Therefore, the authors \cite{n-r2} proved the following 
representation formula for the Laguerre-type Bell polynomials 
of the first order, denoted by $_{L}Y_n$, 
\begin{theorem}
The $_{L}Y_n$ polynomials are expressed in terms of the
ordinary Bell polynomials according to the equation
\begin{eqnarray}
\label{7.3}
\begin{array}{l}
_{L}Y_n\left( x; \left[ f,g \right]_n \right) = \d \sum_{k_=0}^n
 \d \frac{n!}{k!} {n \choose k} 
x^{k} \, Y_{n+k}\left(\left[ f,g \right]_{n+k} \right).
\end{array}
\end{eqnarray}

\end{theorem}
\begin{definition}
The $\emph{$1$st}$ order Laguerre-type Bell numbers are defined by
$$
_{L}b_n:= _{L}Y_n\left( 1; 1,1,\dots,1 \right) = \d
\sum_{k_=0}^n
 \d \frac{n!}{k!} {n \choose k} Y_{n+k} \left(1,1,\dots,1 \right).
$$
\end{definition}

The above results can be easily generalized, since
\begin{eqnarray}
\label{7.4}
\begin{array}{l}
(D_{2L})^n = (DxDxD)^n = D^nx^n \left( D^nx^nD^n \right) = \\
\phantom{\rule{1 pt}{26 pt}} = \d \sum_{k_1=0}^n \sum_{k_2=0}^n \d
\frac{n!}{k_1!} \frac{(n+k_1)!}{(k_1+k_2)!}  {n \choose k_1}
{n \choose k_2}  x^{k_1+k_2}  D^{n+k_1+k_2},
\end{array}
\end{eqnarray}
so that the last definition becomes
\begin{definition}
The $\emph{$2$nd}$ order Laguerre-type Bell numbers are defined by
$$
_{2L}b_n:= _{2L}Y_n\left( 1; 1,1,\dots,1 \right) = \d
\sum_{k_1=0}^n \sum_{k_2=0}^n \d \frac{n!}{k_1!} 
\frac{(n+k_1)!}{(k_1+k_2)!}  {n \choose k_1} {n \choose k_2} 
Y_{n+k_1+k_2} \left(1,1,\dots,1 \right)
$$
\end{definition}
and in general, for every integer $r$,
\begin{definition}
The $\emph{$r$th}$ order Laguerre-type Bell numbers are defined by
\begin{eqnarray}
\label{7.5}
\begin{array}{l}
 \hspace{1.2 cm} _{rL}b_n:=_{rL}Y_n\left( 1; 1,1,\dots,1 \right) \\
\phantom{\rule{1 pt}{26 pt}} \hspace{1 cm} = \d \sum_{k_1=0}^n
\sum_{k_2=0}^n \cdots \sum_{k_r=0}^n \d \frac{n!}{k_1!}
\frac{(n+k_1)!}{(k_1+k_2)!}  \cdots 
\frac{(n+k_1+k_2+\dots+k_{r-1})!}{(k_1+k_2+\dots+k_r)!}\\
\phantom{\rule{1 pt}{26 pt}} \hspace{1 cm} \times \d {n \choose
k_1} {n \choose k_2} \cdots {n \choose k_r} 
Y_{n+k_1+k_2+\dots+k_r}\left(1,1,\dots,1 \right).
\end{array}
\end{eqnarray}
\end{definition}

In particular, for $r=1$, $r=2$ and $r=3$, we obtain respectively
$$
\phantom{\rule{1 pt}{26 pt}} _{L}b_n= \d \sum_{k=0}^n \d \frac{(n!)^2}{(k!)^2 (n-k)!} b_{n+k},
$$
$$
\phantom{\rule{1 pt}{26 pt}} _{2L}b_n= \d \sum_{k_1=0}^n
\sum_{k_2=0}^n \d \frac{(n!)^3 (n+k_1)!}{(k_1!)^2 (k_2)!
(k_1+k_2)!(n-k_1)!(n-k_2)!} b_{n+k_1+k_2},
$$
and
\begin{eqnarray*}
\begin{array}{l}
_{3L}b_n = \d \sum_{k_1=0}^n \sum_{k_2=0}^n \sum_{k_3=0}^n \\
\phantom{\rule{1 pt}{26 pt}} \d \frac{(n!)^4 (n+k_1)!
(n+k_1+k_2)!}{(k_1!)^2 (k_2)! (k_1+k_2!) (k_3)! (k_1+k_2+k_3)!
(n-k_1)!(n-k_2)!(n-k_3)!} b_{n+k_1+k_2+k_3}.
\end{array}
\end{eqnarray*}


According to the above definitions, we compute the following 
integer sequences  relevant to the Laguerre-type Bell numbers 
of increasing order:

\begin{table}[H]
\begin{center}
\begin{tabular}{l * {0}{l}} 
\hline
$_{L}b_1 = b_1 + b_2 = 3$ \\
$_{L}b_2 = 2 b_2 + 4 b_3 + b_4 = 39$ \\
$_{L}b_3 = 6 b_3 + 18 b_4 + 9 b_5 + b_6 = 971$ \\
$_{L}b_4 = 24 b_4 + 96 b_5 + 72 b_6 + 16 b_7 + b_8 = 38140$ \\
$_{L}b_5 = 120 b_5 + 600 b_6 + 600 b_7 + 200 b_8 + 25 b_9 + b_{10} = 2126890$ \\
$_{L}b_6 = 720 b_6 + 4320 b_7 + 5400 b_8 + 2400 b_9 + 450 b_{10} + 36 b_{11} + b_{12} = 157874467$ \\
$_{L}b_7 = 14928602309$ \\
$_{L}b_8 = 1741809491235$ \\
$_{L}b_9 = 244735956424795$ \\
$_{L}b_{10} = 40624759074089022$ \\
\hline
\end{tabular}
\end{center}
\caption{Laguerre-type Bell numbers $_{L}b_n$, for $ n=1,2,\dots, 10 $.}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{l * {0}{l}} 
\hline
$_{2L}b_1 = b_1 + 3 b_2 + b_3 = 12$ \\
$_{2L}b_2 = 4 b_2 + 32 b_3 + 38 b_4 + 12 b_5 +  b_6 = 1565$ \\
$_{2L}b_3 = 36 b_3 + 540 b_4 + 1242 b_5 + 882 b_6 + 243 b_7 + 27 b_8 + b_9 = 597948$ \\
$_{2L}b_4 = 576 b_4 + 13824 b_5 + 50688 b_6 + 59904 b_7 + 30024 b_8 + 7200 b_9 + $\\
$\hspace{1.5 cm}  + 856 b_{10} + 48 b_{11} + b_{12} = 476170277$ \\
$_{2L}b_5 = 665045751420$ \\
$_{2L}b_6 = 1466218536786553$ \\
$_{2L}b_7 = 4751410403456508380$ \\
$_{2L}b_8 = 21492464638761800105545$ \\
$_{2L}b_9 = 130485451947856798889765548$ \\
$_{2L}b_{10} = 1031029017676447641584236719317$ \\
\hline
\end{tabular}
\end{center}
\caption{Laguerre-type Bell numbers $_{2L}b_n$, for $ n=1,2,\dots, 10 $}
\end{table}



\begin{table}[H]
\begin{center}
\begin{tabular}{l * {0}{l}} 
\hline
$_{3L}b_1 = b_1 + 7 b_2 + 6 b_3 + b_4  = 60$ \\
$_{3L}b_2 = 8 b_2 + 208 b_3 + 652 b_4 + 576 b_5 + 188 b_6 + 24 b_7 + b_8 = 104140$ \\
$_{3L}b_3 = 216 b_3 + 13608 b_4 + 94284 b_5 + 186876 b_6 + 149580 b_7 + 56808 b_8 +$ \\
$\hspace{1.5 cm}  + 11025 b_9 + 1017 b_{10} + 54 b_{11} + b_{12} = 811796953$ \\
$_{3L}b_4 = 17520336532435$ \\
$_{3L}b_5 = 821820979710053847$ \\
$_{3L}b_6 = 72322447015782400673157$ \\
$_{3L}b_7 = 10811794473114190401496880596$ \\
$_{3L}b_8 = 2555589594863343375850080281766604$ \\
$_{3L}b_9 = 904589617839233661347244697559337192973$ \\
$_{3L}b_{10} = 459477819180446167103818476324632065253323967$\\
\hline
\end{tabular}
\end{center}
\caption{Laguerre-type Bell numbers $_{3L}b_n$, for $ n=1,2,\dots, 10 $}
\end{table}
and so on, for the Laguerre-type Bell numbers of higher order.



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\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A10; Secondary 26A06, 11P81.

\noindent \emph{Keywords: } 
higher order Bell polynomial and number, combinatorial analysis, partition.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000326},
\seqnum{A005945}, and
\seqnum{A144150}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received  March 13 2017;
revised versions received August 24 2017; September 20 2017.
Published in {\it Journal of Integer Sequences}, October 29 2017.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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