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\begin{center}
\vskip 1cm{\Large\bf Motzkin Numbers: an Operational Point of View
}
\vskip 1cm
M. Artioli\\ ENEA --- Bologna Research Center\\
	 Via Martiri di Monte Sole,  4 \\
	 40129 Bologna \\ Italy\\
	 \href{mailto:marcello.artioli@enea.it}{\tt marcello.artioli@enea.it} \\\ \\	
	G. Dattoli \\
	 ENEA --- Frascati Research Center\\
	  Via Enrico Fermi 45 \\
	  00044 Frascati, Rome \\
	  Italy\\
	 \href{mailto:giuseppe.dattoli@enea.it}{\tt giuseppe.dattoli@enea.it} \\
\ \\ 
S. Licciardi\\
Dept.~of Mathematics and Computer Science\\
University of Catania \\
Viale A. Doria 6 \\
95125, Catania \\
Italy \\
and \\
ENEA --- Frascati Research Center\\
Via Enrico Fermi 45 \\
00044 Frascati, Rome \\
Italy \\
\href{silvia.licciardi@dmi.unict.it}{\tt silvia.licciardi@dmi.unict.it} \\
\ \\	
S. Pagnutti\\
ENEA --- Bologna Research Center\\
Via Martiri di Monte Sole,  4 \\
40129, Bologna \\
Italy \\
\href{mailto:simonetta.pagnutti@enea.it}{\tt simonetta.pagnutti@enea.it} \\
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\vskip .2 in
		
\begin{abstract}
The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers, and offers a tool to investigate previously unnoticed links with the theory of special functions and with the relevant treatment in terms of operational means. The use of umbral methods opens new directions for further developments and generalizations, which leads, e.g., to the identification of new Motzkin-associated forms.
\end{abstract}
		
	
\section{Introduction}
The telephone numbers $(T_{n})$, also called convolution numbers, provide a very well-known example of the link between special numbers and special polynomials. The $(T_{n})$ can be expressed in terms of the coefficients of
Hermite polynomials $ (h_{s}) $  \cite[pp.~85--86]{Riordan}.
Two of the present authors (M.~A. and G.~D.)
have recently pointed out \cite{Artioli} that the Padovan and Perrin numbers \cite{Perrin,Padovan} can be recognized as associated with particular values of
two-variable Legendre polynomials \cite{Dattoli}.

The Motzkin numbers plays a significant role in combinatorics (see,
e.g., Bernhart \cite{Weisstein}), Donaghey and Shapiro pointed out
fourteen different manifestations of these numbers \cite{Donaghey}.
Blasiak et al.~and Dattoli et al.~have studied
\cite{Blasiak,Lorenzutta} the connection between Motzkin numbers and a
family of hybrid polynomials and the relevant properties of Motzkin
numbers.

In a recent paper, Zhao and Qi \cite{Zhao} discussed similar results
employing the link between Motzkin and Catalan numbers.

 The hybrid polynomials are  defined as follows \cite{Lorenzutta}:
\begin{equation}\label{Kn}
P_{n}^{(q)}(x,y)=n!\sum_{r=0}^{\lfloor\frac{n}{2}\rfloor }\dfrac{x^{n-2r}y^{r}}{(n-2r)!r!(r+q)!},
\end{equation}
and the relevant generating function is
\begin{equation}\label{genKn}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}P_{n}^{(q)}(x,y)=\dfrac{I_{q}(2 \sqrt{y}\; t)}{(\sqrt{y}\;t)^{q}}e^{xt},
\end{equation}
where $I_{q}(x)$  is the modified Bessel function of the first kind of order $ q $.

Within the present framework, the Motzkin number sequence
can be specified as follows \cite{Blasiak}:
\begin{equation}\begin{split}\label{key}
& m_{n} = P_{n}^{(1)}(1,1)=\sum_{s=0}^{n}m_{n,s},\\
& m_{n,s}=\binom{n}{s}\;f_{s},\\
& f_{s} = \dfrac{s!}{\Gamma\left( \dfrac{s}{2}+2\right) \Gamma\left( \dfrac{s}{2}+1\right) }\left| \cos\left( s\dfrac{\pi}{2}\right) \right|, 
\end{split}\end{equation}
 where the coefficients $m_{n,s}$ can be represented as the triangle reported in the following table, in which $m_{n,2}$ corresponds, in the
 On-Line Encyclopedia of Integer Sequences (OEIS), to the sequence \seqnum{A000217}, $m_{n,4}$ to \seqnum{A034827}, $m_{n,6}$ to \seqnum{A000910}, and so on.
\begin{table}[H]
	\centering
\begin{tabular}{|c | c|c|c|c|c|c|c|c|c|c|} \hline
\toprule
	\multicolumn{10}{|c|}{\centering $m_{n,s}$ {coefficients}}& $ m_{n}$\; {Motzkin}  \\ \hline
	 \multirow{2}{*}{{Parameter}}  & \multicolumn{9}{c|}{$s$}&\multirow{2}{*}{$\sum_{s=0}^{n}m_{n,s}$}\\ 
	 \cmidrule(lr){3-10}
	&  & {0} & {1} & {2 } & {3} & {4} & {5} & {6} & {7}&\\  
	\midrule	
	 \multirow{9}{*}{$n$}  & {0} & 1 &  &  &  & & & & & {1} \\ 
	& {1}  & 1 & 0 &  &  & & &  & & {1} \\ 
	& {2} & 1 & 0 & 1 & & & &  & & {2} \\ 
	 & {3} & 1 & 0 & 3 & 0 &  &  & & & {4}\\ 
	 & {4} & 1 & 0 & 6 & 0 & 2 &  & & & {9}\\ 
	& {5} & 1 & 0 & 10 & 0 & 10 & 0  &  &  & {21}\\ 
	& {6} & 1 & 0 & 15 & 0 & 30 & 0 & 5 &  & {51}\\ 
	& {7} & 1 & 0 & 21 & 0 & 70 & 0 & 35 & 0  & {127}\\ 
	\bottomrule
\end{tabular}
\caption{Motzkin numbers and their  coefficients}
\label{table1}
\end{table}

 According to Eq.~\eqref{genKn}, the Motzkin numbers can also be defined as the coefficients of the following series expansion
\begin{equation}\label{genMn}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}m_{n}=\dfrac{I_{1}(2  t)}{t}e^{t}.
\end{equation}

In what follows we show how some progress in the study of the relevant
properties can be achieved by the use of the umbral notation.

\section{Motzkin numbers and the umbral calculus}
 
In order to simplify most of the algebra associated with the study of the properties of the Motzkin numbers, and to get new relevant identities,
we introduce a formalism successfully exploited elsewhere \cite{Borel} based on
of the umbral calculus \cite{Roman}. 
 
 To this aim, we note that the function
\begin{equation}\label{Tric}
C_{q}(x)=\dfrac{I_{q}(2 \sqrt{x})}{(\sqrt{x})^{q}}=\sum_{r=0}^{	\infty}\dfrac{x^{r}}{r!(q+r)!}
\end{equation}
can be cast in the form
\begin{equation}\label{TricOp}
C_{q}(x)=\hat{c}^{q} \circ e^{\hat{c}x},
\end{equation}
where $\hat{c}$  is an umbral operator defined according to
the equation
\begin{equation}\label{actOp}
\hat{c}^{\mu}=\dfrac{1}{\Gamma(\mu+1)},
\end{equation}
with $\mu$ real and not necessarily integer.

We define the following composition rule 
\begin{equation}\label{opop}
\hat{c}^{\mu} \circ \hat{c}^{\nu}=\hat{c}^{\mu+\nu},
\end{equation}
and we let $\hat{C}=\{\hat{c}^{\alpha},\alpha\in \mathbb{C}\}$ denote the set of $\hat{c}$-operators. Then  the pair $(\hat{C},\circ)$ satisfies
the Abelian group property. The mathematical foundations of the theory of $\hat{c}$-operators can be traced back to those underlying the Borel transform and have been carefully discussed \cite{Roman}.

The use of this formalism allows one to restyle the hybrid polynomials in the form
\begin{equation}\label{KnOp}
P_{n}^{(q)}(x,y)=\hat{c}^{q} \circ H_{n}(x,\hat{c}\;y),
\end{equation}
where
\begin{equation}\label{Herm}
H_{n}(x,y)=n!\sum_{r=0}^{\lfloor\frac{n}{2}\rfloor }\dfrac{x^{n-2r}y^{r}}{(n-2r)!r!}
\end{equation}
are the two-variable Hermite-Kamp\'e de F\'eri\'et  polynomials of order $ 2 $.

We can accordingly use the wealth of properties of this family of
polynomials to derive further and new relations regarding those of the
Motzkin numbers family.

Indeed, by recalling the generating function \cite{Lorenzutta}
\begin{equation}\label{genHermnl}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}H_{n+l}(x,y)=H_{l}(x+2yt,y)e^{xt+yt^{2}},
\end{equation}
we find
\begin{equation}\label{genmnl}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}m_{n+l}=\hat{c} \circ H_{l}(1+2\hat{c}t,\hat{c})e^{t+\hat{c}t^{2}},
\end{equation}
which, after using Eqs.~\eqref{opop}, \eqref{Herm}, and \eqref{TricOp},
finally yields
\begin{equation}\begin{split}\label{mu}
& \sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}m_{n+l}=\mu_{l}(t)\;e^{t},\\
& \mu_{l}(t)=l!\sum_{r=0}^{\lfloor\frac{l}{2}\rfloor }\dfrac{1}{r!}\sum_{s=0}^{l-2r}\dfrac{2^{s}}{s!(l-2r-s)!}\dfrac{I_{s+r+1}(2t)}{t^{r+1}}.
\end{split}\end{equation}

Furthermore, the same procedure and the use of the Hermite polynomial duplication formula \cite{Andrews}
\begin{equation}\label{Hd}
H_{2n}(x,y)=\sum_{r=0}^{n}\binom{n}{r}^{2}r!\;(2y)^{r}\left( H_{n-r}(x,y)\right)^{2}, 
\end{equation}
 yields the following identity for Motzkin numbers
\begin{equation}\begin{split}\label{mdupl}
 m_{2n}& =\hat{c} \circ \sum_{r=0}^{n}r!\;\binom{n}{r}^{2}(2\hat{c})^{r} \circ  H_{n-r}(1,\hat{c}) \circ H_{n-r}(1,\hat{c})=\\
& = \sum_{r=0}^{n}\binom{n}{r}^{2}2^{r}r!(n-r)!\sum_{s=0}^{\lfloor\frac{n-r}{2}\rfloor }\dfrac{m_{n-r}^{(r+s+1)}}{(n-r-2s)!s!},
\end{split}\end{equation}
where
\begin{equation}\label{mnmop}
m_{n}^{(q)}=P_{n}^{(q)}(1,1)=\hat{c}^{q} \circ H_{n}(1,\hat{c})
\end{equation}
are associated Motzkin numbers \cite{Blasiak}.

The identification of Motzkin numbers as in Eq.~\eqref{mnmop}, along with the use of the recurrences of Hermite polynomials, yields, e.g., the identities   
\begin{equation}\begin{split}\label{recmn}
& m_{n+1}^{(q)}=m_{n}^{(q)}+2\;n\;m_{n-1}^{(q+1)},\\
& m_{n+p}=\sum_{s=0}^{\min[n,p]}2^{s}s!\;\binom{p}{s}\;\binom{n}{s}M_{p-s,\;n-s,\;s},\\
& M_{p,\;n,\;t}=p!\sum_{r=0}^{\lfloor\frac{p}{2}\rfloor}\dfrac{m_{n}^{(t+r+1)}}{(p-2r)!r!},
\end{split}\end{equation}
in which the second identity has been derived
from the Nielsen formula for $H_{n+m}(x,y)$ \cite{Nielsen}.
 
 \section{Final comments}

In this paper we have shown that a fairly straightforward extension of
a formalism put forward in \cite{Blasiak} permits non-trivial progress in
the theory of Motzkin numbers. Further relations can be easily obtained
by applying the method we have envisaged as, e.g.,
 \begin{equation}\label{prmn}
\sum_{s=0}^{n}m_{n-s}\;m_{s}=2\;(n+1)\;m_{n}^{(2)}, \end{equation}
which represents a discrete self-convolution of Motzkin numbers.

 The associated Motzkin numbers, originally introduced \cite{Blasiak}, have been defined here  on purely algebraic grounds (see Eq.~\ref{mnmop}).  Strictly speaking they are not integers and therefore they are not amenable to a combinatorial interpretation; however, redefining them as 
\begin{equation}\label{tildem}
\tilde{m}_{n}^{(q)}=\dfrac{(n+q)!}{n!}P_{n}^{(q)}(1,1),
\end{equation}
we obtain for $q=2$ the sequences in OEIS  \seqnum{A014531}, while for $q=3$ the sequences \seqnum{A014532}, and so on.

A more appropriate interpretation in combinatorial terms can be obtained by following, e.g., the procedures indicated \cite{Banderier} and deserves further investigations, out of the scope of the present paper.

We have mentioned in the introduction the  theory of telephone numbers  $T(n)$ \cite[pp.~65--67]{Knuth}, whose importance in chemical graph theory has been recently emphasized \cite{Hatz}.  As is well known,
they can be expressed in terms of ordinary Hermite polynomials; however, the use of the two-variable extension is more effective. Indeed,
they can be expressed as $T(n)=H_{n}(1,\frac{1}{2})$  .

The use of Hermite polynomial properties,
such as the index duplication formula, yields
\begin{equation}\label{T2n}
T(2n)=\sum_{r=0}^{n}\;\binom{n}{r}^{2}\;r!\;T(n-r)^{2}.
\end{equation}
 The use of the Hermite numbers $h_{s}$ \cite{Germano} allows the derivation of the following additional expression
\begin{equation}\begin{split}\label{hs}
& T(n)=\sum_{s=0}^{n}t_{n,s},\\
& t_{n,s}=\binom{n}{s}\;h_{s}\left( \dfrac{1}{2}\right) ,\\
& h_{s}(y)=y^{\frac{s}{2}}\Gamma\left( \dfrac{s}{2}+2\right)f_{s}=\dfrac{y^{\frac{s}{2}}s!}{\Gamma\left( \dfrac{s}{2}+1\right) }\left| \cos\left( s\;\dfrac{\pi}{2}\right) \right|.  
\end{split}\end{equation}

 The coefficients $ t_{n,s} $
 of the telephone numbers can be arranged in the following triangle, in which
 the numbers belonging to the column
 $s=4$, $(3,15,45,105,210,378,\dots)$
  are identified, in the OEIS, with the sequence \seqnum{A050534} and the column in $s=6$, $(15, 105, 420,1260, 3150, \dots)$, with \seqnum{A240440}. 
\begin{table}[htbp]
	\centering
	\begin{tabular}{|c | c|c|c|c|c|c|c|c|c|c|} \hline
\toprule
		\multicolumn{10}{|c|}{\centering $t_{n,s}$ {coefficients}} & $T{(n)}$\; {telephone numbers} \\ \hline
		\toprule
		\multirow{2}{*}{{Parameter}}  & \multicolumn{9}{c|}{{$s$}}&\multirow{2}{*}{$\sum_{s=0}^{n}t_{n,s}$}\\ 
		\cmidrule(lr){3-10}
		&  & {0} & {1} & {2 } & {3} & {4} & {5} & {6} & {7}&\\  
		\midrule	
		\multirow{9}{*}{{$n$}}  & {0} & 1 &  &  &  & & & & &{1}\\ 
		& {1}  & 1 & 0 &  &  & & &  & & {1}\\ 
		& {2} & 1 & 0 & 1 & & & &  & & {2}\\ 
		& {3} & 1 & 0 & 3 & 0 &  &  & & & {4}\\ 
		& {4} & 1 & 0 & 6 & 0 & 3 &  & & & {10}\\ 
		& {5} & 1 & 0 & 10 & 0 & 15 & 0  &  & & {26}\\ 
		& {6} & 1 & 0 & 15 & 0 & 45 & 0 & 15 & & {76}\\ 
		& {7} & 1 & 0 & 21 & 0 & 105 & 0 & 105 & 0 & {232}\\ 
		\bottomrule
	\end{tabular}
	\caption{Telephone number coefficients}
	\label{table2}
\end{table}

 The use of the identification with two-variable Hermite polynomials opens further perspectives. By exploiting the polynomials (see \cite[pp.~85--86]{Riordan} and references therein)
\begin{equation}\label{Hnm}
H_{n}^{(m)}(x,y)=n!\sum_{r=0}^{\lfloor \frac{n}{m}\rfloor }\dfrac{x^{n-mr}y^{r}}{(n-mr)!r!},
\end{equation}
we can introduce the following generalization of telephone numbers 
\begin{equation}\label{Tnm}
T_{n}^{(m)}=H_{n}^{(m)}\left( 1,\dfrac{1}{m}\right), 
\end{equation} 
with generating function
\begin{equation}\label{gentn}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}T_{n}^{(m)}=e^{t+\frac{1}{m}t^{m}},
\end{equation}
which satisfy the recurrence
\begin{equation}\label{tnmp}
T_{n+1}^{(m)}=T_{n}^{(m)}+\dfrac{n!}{(n-m+1)!}T_{n-m+1}^{(m)}.
\end{equation}
In the case of $m=3$ the numbers $T_{n}^{(3)}= (1, 1, 1, 3, 9, 21, 81,
351, 1233,\dots)$ are identified with OEIS sequence \seqnum{A001470},
while for
$m=4$, the series $(1, 1, 1, 1, 7, 31, 91, 211, 1681, 12097, \dots)$,
corresponds to \seqnum{A118934}. For $m=5$ the associated series
appears to be \seqnum{A052501}, but should be more appropriately
identified with the coefficients of the expansion \eqref{Tnm}. Finally
the sequence $m=6$ is not reported in OEIS.

 A more detailed analysis of this family of numbers and the relevant interplay with Motzkin will be discussed elsewhere.

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\bibitem{Artioli} M. Artioli and G. Dattoli, 
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\url{https://tinyurl.com/ydhxuk2h}.

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\bibitem{Padovan} R. Padovan, Dom Hans Van Der Laan and the Plastic
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\bibitem{Dattoli} M. Artioli and G. Dattoli, Geometry of two-variable
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\textit{J. Comb. Theory A}  \textbf{23} (1977), 291--301.

\bibitem{Blasiak} P. Blasiak, G. Dattoli, A. Horzela, K. A. Penson,
and K. Zhukovsky, Motzkin numbers, central trinomial coefficients and
hybrid polynomials, \textit{J. Integer Sequences} \textbf{11} (2008),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL11/Penson/penson131.html}{Article 08.1.1}.

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(2017), 3489--3510.

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Math. Anal. Appl.} \textbf{87} (1982), 1--58.

\bibitem{Andrews}  L. C. Andrews, \textit{Special Functions for
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\textit{Mathematisk-Fysiske Meddelelser}  \textbf{1} (1918), Det. Kgl.
Danske Videnskabernes Selskab.

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Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas
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\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B99; Secondary 33C45, 05A10, 47A05, 05A40.

\noindent \emph{Keywords: } 
Motzkin number, special functions, combinatorics,
operator theory, umbral calculus.


\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000217},
\seqnum{A000910},
\seqnum{A001470},
\seqnum{A014531},
\seqnum{A014532},
\seqnum{A034827},
\seqnum{A050534},
\seqnum{A052501},
\seqnum{A118934}, and
\seqnum{A240440}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received February 26 2018;
revised versions received May 16 2018; June 2 2018; August 25 2018.
Published in {\it Journal of Integer Sequences}, August 26 2018.

\bigskip
\hrule
\bigskip

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