\documentclass[10pt]{amsart}
\usepackage[cp1251]{inputenc}
\usepackage[english,russian]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}

\def\udcs{???.?} %Here the author places classificators of the paper according to Russian classification system
\def\mscs{??X??} %Here the author places  classificators according to the AMS classification list.
\setcounter{page}{144}

\usepackage{tikz-cd}
\usepackage{enumerate}

\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}
\newtheorem{example}{Example}
\newtheorem{remark}{Remark}

\def\logo{{\bf\huge S\raisebox{0.2ex}{\hspace{0.55ex}\raisebox{0.05ex}e\hspace{-1.65ex}$\bigcirc$}MR}}

\def\semrtop
     {
  \vbox{
     \noindent\logo
     \hspace{80mm}\raisebox{1ex}{ISSN 1813-3304 }

     \vspace{5mm}

     \begin{center}
     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
%     {\small Sobolev Institute of Mathematics SB RAS}
     \end{center}
     \vspace{-3mm}
     \noindent
     \begin{tabular}{c}
     \hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa} \\
     \hline\hline
     \end{tabular}

     \vspace{1mm}
     {\flushleft\it Том 16, стр. 144--144 (2019) \hspace{65mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
                                                                                    %filled by Editorial board
     \newline
         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
  }%\vbox
}
\begin{document}
\renewcommand{\refname}{References}
\renewcommand{\proofname}{Proof.}
\renewcommand{\figurename}{Fig.}


\thispagestyle{empty}

\title[Laguerre Expansions of $C-$regularized semigroups Functions]{Laguerre Expansions of $C-$regularized semigroups Functions}

\author{{Y. Bajjou}}%
\address{Youssef Bajjou
		\newline\hphantom{iii}Department of mathematics and computer science,
	\newline\hphantom{iii} Dhar El mahraz Faculty of Sciences,
	\newline\hphantom{iii} Sidi Mohamed Ben Abdellah University,
	\newline\hphantom{iii} 	F\`es Atlas, Morocco}%
\email{youssefbajjou2017@gmail.com}%

\author{{A. Blali}}%
\address{Aziz Blali
\newline\hphantom{iii} Department of mathematics,
\newline\hphantom{iii} \'Ecole Normale Sup\'erieure,
\newline\hphantom{iii} Sidi Mohamed Ben Abdellah University,
\newline\hphantom{iii} B.P. 5206 Bensouda-F\`es, Morocco}%
\email{aziz.blali@usmba.ac.ma}%

\author{{A. El Amrani}}%
\address{Abdelkhalek El Amrani
	\newline\hphantom{iii} Department of mathematics and computer science,
	\newline\hphantom{iii} Dhar El mahraz Faculty of Sciences,
	\newline\hphantom{iii} Sidi Mohamed Ben Abdellah University,
	\newline\hphantom{iii} 	 F\`es Atlas, Morocco}%
\email{abdelkhalek.elamrani@usmba.ac.ma}%

\thanks{\sc Bajjou, Y.,  Blali, A., El Amrani, A.
Laguerre Expansions of $C-$regularized semigroups Functions}
\thanks{\copyright \ 2023 Bajjou, Y.,  Blali, A., El Amrani, A.}
\thanks{\rm The work is supported by RFFI (grant 03-01-11111)}
\thanks{\it Received  January, 1, 2015, published  March, 1,  2015.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. }
The aim of this paper is to approximate the exponentially bounded $C-$regularized semigroups function by the Laguerre series, recalling the notions and the results used. \medskip

\noindent{\bf Keywords:} Laguerre functions, $C-$regularized semigroup, $C_{0}-$semigroup.
 \end{quote}
}


\section{ Introduction and preliminaries}
The series expansion of Laguerre orthogonal polynomials have been an
important tool in mathematical physics, in problems involving the
integration of Helmholtz's equation in prabolic coordinates, in the
theory of the Hydrogen atom, in the theory of propagation of
electomagnetic waves along transmission lines terminated by a
lumped inductance [\ref{H}]. The study of sufficient conditions for
the convergence of Laguerre series has been the subject of numerous
works, for more details see [\ref{aa}], [\ref{ab}], [\ref{ac}],
[\ref{H}] and [\ref{ag}].\\In 2014, Abadias  and Miana  studied in
their article [\ref{af}] the Laguerre expansion of
$C_{0}$-semigroups and Resolvent Operators. In this
work we will be interested in Laguerre expansion of $C$-regularized semigroups function, starting with reminding the notations, concepts and results used.\\
Throughout this paper $E$ denotes a non-trivial complex Banach
space, $\mathfrak{F}(E,F)$ denotes the set of all applications from
$E$ to another Banach space $F,$ $B(E)$ denotes the space of all
bounded linear operators from $E$ into itself, and $L^{1}_{loc}(E)$
the set of all $f\in \mathfrak{F}(\mathbb{R},E)$ locally integrable.
For a closed linear operator $A$ on $E,$ $\mathcal{D}(A),$ $R(A)$
and $\rho(A)$ denote its domain, range and resolvent set,
respectively. $\mathcal{D}(A)$ equipped with the graph norm
$\parallel x\parallel_{\mathcal{D}(A)}=\parallel
x\parallel_{E}+\parallel Ax\parallel_{E}$  becomes a Banach space.
Throughout this paper, $C\in B(E)$ will be an injective operator.
The $C$-resolvent set of $A,$ denoted by $\rho_{C} (A),$ is defined
by $\rho_{C} (A):=\{\lambda\in\mathbb{C}\mid R(C)\subseteq R(\lambda
I-A)~~ and~~ \lambda I-A\  is\  injective \} $ and if
$\lambda\in \rho_{C} (A)$ then we denote by $R_{C}(\lambda
,A)=(\lambda I-A)^{-1}C$ the $C-$resolvent.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Laguerre functions and Laguerre expansions on Banach spaces} 

For all $n\in\mathbb{N},$ and arbitrary real $\alpha > -1$  the
classical Laguerre polynomial is defined by Rodrigues
formula:$$(\forall x\in
\mathbb{R})~~\phi_{n,\alpha}(x)=e^{x}\frac{x^{-\alpha}}{n!}\frac{d^{n}}{dx^{n}}(e^{-x}x^{n+\alpha}),$$
$\phi_{n,\alpha}$ is a polynomial with the degree $n,$ the same
parity as $n,$ whose highest monomial degree is
$\frac{(-1)^{n}}{n!}x^{n}$ and have real coefficients. Furthemore,
they verify the following condition of orthogonality wiht respect to
gamma density $x\longmapsto x^{\alpha}e^{-x}$ on $[0,+\infty[$: $$\int_{\mathbb{R}^{+}}\phi_{n,\alpha}(x)\phi_{m,\alpha}(x)x^{\alpha}e^{-x}dx=\delta_{n,m}\frac{\Gamma(n+\alpha+1)}{n!},$$
where $\delta_{n,m}$ is the Kronecker delta. We also have recurrence
relations, differential equations and the estimates:
\begin{equation}\label{Hn}
%\begin{equation}\label{estimaa}
%(\exists c>0)~(\forall (n_{0},t)\in\mathbb{N}\times\mathbb{R} : ~~t^{2}\leq 2(2n_{0}+1))~(\forall n\geq n_{0}) \  \\
(\forall x\in\mathbb{R}^{+})~~(\exists c_{x}>0)~~(\exists
n_{0}\in\mathbb{N})~~(\forall n\geq n_{0}), \mid
\phi_{n,\alpha}(x)\mid\leq c_{x}n^{\frac{\alpha}{2}}.
%(\frac{e^{\frac{t^{2}}{2}}\sqrt{2^{n}n!}}{n^{\frac{1}{12}}}).
\end{equation}
For more details of the classical theory of orthogonal polynomials
see [\ref{H}], [\ref{aa}], [\ref{ab}], [\ref{ac}], [\ref{af}] and [\ref{ag}].\\
The Laguerre functions on $\mathbb{R}^{+}$ are defined by:
\begin{equation}
\varphi_{n,\alpha}(x)=\sqrt{\frac{n!}{\Gamma(n+\alpha+1)}}\phi_{n,\alpha}(x)x^{\frac{\alpha}{2}}e^{-\frac{x}{2}}=\frac{1}{\sqrt{n!\Gamma(n+\alpha+1)}}x^{\frac{-\alpha}{2}}e^{\frac{x}{2}}\frac{d^{n}}{dx^{n}}(e^{-x}x^{n+\alpha}),
\end{equation}
$(\varphi_{n,\alpha})_{n\in\mathbb{N}}$ is an orthonormal basis in
the Hilbert space $L^{2}(\mathbb{R}^{+})$ and satisfies some
recurrence relations, equalities and inequalities. For more details see
[\ref{aa}],~[\ref{ab}],~[\ref{af}],~[\ref{ac}] and [\ref{ag}].\\
For $n\in\mathbb{N}$ and arbitrary real $\alpha>-1$, we denote by
$\psi_{n,\alpha},$ the function on $\mathbb{R}^{+}$ defined by:
\begin{equation}
(\forall x\in \mathbb{R}^{+})~~
\psi_{n,\alpha}(x)=\frac{n!}{\Gamma(n+\alpha+1)}x^{\alpha}e^{-x}\phi_{n,\alpha}(x)=\frac{1}{\Gamma(n+\alpha+1)}\frac{d^{n}}{dx^{n}}(x^{n+\alpha}e^{-x}),
\end{equation}
$\psi_{n,\alpha}$ satisfies recurrence relations and differential
equations, for example:
\begin{equation}\label{hnn}
\left( \forall \left(  n,m\right) \in \mathbb{N}^{2}\right) ~
\psi_{n,\alpha}^{(m)}=\psi_{n+m,\alpha-m}.
\end{equation}
And the following useful inequality:
\begin{equation}\label{hn}
(\forall n\geq 1)~~\parallel \psi_{n,\alpha}\parallel_{1}\leq
\frac{c_{\alpha}}{n^{\frac{\alpha}{2}}}.
\end{equation} For more details see   [\ref{af}] and [\ref{H}]. \\
 The most important property of the family
$(\psi_{n,\alpha})_{n\in\mathbb{N}}$ is that if $f
:\mathbb{R}^{*}_{+}\rightarrow E$ be a differentiable function such
that $\int_{0}^{+\infty}e^{-t}t^{\alpha}\parallel f(t)\parallel^{2}
dt<+\infty ,$ then the series $\displaystyle\sum_{n\in
\mathbb{N}}c_{n}(f)\phi_{n,\alpha}(t)$ converges pointwise to $f$ on
$\mathbb{R}^{*}_{+},$ where
\begin{center}$c_{n}(f)=\int_{0}^{+\infty}\psi_{n,\alpha}(t)f(t)dt .$\end{center}
For more details see [\ref{H}], [\ref{ab}] and [\ref{ac}].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$C$-regularized semigroup}
A family of operators $(T(t))_{t\geq 0}$ in $B(E)$ is called
exponentially bounded $C-$regularized semigroup or exponentially
bounded $C$-semigroup on $E,$ if
\begin{enumerate}
\item $T(t + s)C=T(t)T(s)\ \ for \ all\  t,s\in
\mathbb{R}^{+}.\label{idefsemi}$
\item $T(0) = C .$
\item The function $t\mapsto T(t)x$ is continuous on $\mathbb{R}^{+}$ for any $x\in E .$
\item $\left( \exists M\geq 0\right)
\ \left( \exists\omega\geq 0\right): \ \   \parallel T(t)\parallel\leq
Me^{\omega t},\ for\ all\ t\geq 0 $ (exponentially bounded condition).
\end{enumerate}
Its generator $W$ is defined by
$$\mathcal{D}(W) = \left\{x\in E : \displaystyle \lim_{s\rightarrow
0^{+}} \frac{T(s)x - Cx }{s} \mbox{ exists in R(C) } \right\}$$ and
$$\left( \forall x\in \mathcal{D}(W)\right) ~~\displaystyle Wx = C^{-1}\displaystyle \lim_{s\rightarrow 0^{+}} \frac{T(s)x - Cx }{s}.$$
In all that follows, $(T(t))_{t\geq 0} \subset B(E)$ is
exponentially bounded $C-$regularized semigroup on $E$ with
generator $(W,\mathcal{D}(W))$ such that
\begin{equation}\label{expborn}
\left(\exists M> 0\right) ~~\left( \exists\omega\geq
0\right): ~~\left(\forall t\geq 0 \right)
\parallel T(t)\parallel\leq Me^{\omega t}.
\end{equation}
We present some known facts about $C-$semigroup and its generator,
which will be used in the sequel (see [\ref{ak}],
[\ref{balagrishnane}], [\ref{caa}], [\ref{ak}], [\ref{ch}],
[\ref{fracpowofoperat}], [\ref{cl}], [\ref{caaaa}], [\ref{caaaaaaa}]
and [\ref{ca}] for more details):
\begin{itemize}
\item By the property $1,$  we conclude that
$T(t)T(s)=T(s)T(t)$ for all $t,s\geq 0,$ this means that $T(t)x\in\mathcal{D}(W)$ and $WT(t)x=T(t)Wx,$ for
all $t\geq 0$ and $x\in D(W).$
\item $\int_{0}^{t}T(s)xds\in \mathcal{D}(W)$ and $W\int_{0}^{t}T(s)xds=T(t)x-Cx$ for every $x\in E$ and $t\geq 0,$  which implies that for each $x\in\mathcal{D}(W),~~u:=T(.)x$ is of class $C^{1}(\mathbb{R}^{+},E)$ and solves the Abstract Cauchy problem 
$$((ACP(W,x,0)_{1}) \left\{
  \begin{array}{ll}
    u'(t) = Wu(t),&\hbox{} t\in \mathbb{R^{+}} \\
      u(0)=Cx . & \hbox{}
  \end{array}
\right.$$
\item $\displaystyle \lim_{h\rightarrow
0^{+}} \frac{1}{h}\int_{t}^{t+h}T(s)xds=T(t)x$ for all $x\in E$ and
$t\geq 0,$ this means that $W$ is a closed linear operator with
$R(C)\subset \overline{D(W)}.$
\item The $C-$resolvent operator $R_{C}(\lambda,W)$ is analytic in the
$C-$resolvent set $\rho_{C}(W)$ and
\begin{equation}\label{resolvset1}
\frac{d^{n}}{d\lambda^{n}}(R_{C}(\lambda,W))=(-1)^{n}n!(\lambda I -W))^{-n-1}C \ 
for \  all \ n\in\mathbb{N}.
\end{equation}
\item  $W=C^{-1}WC,~(\omega , +\infty)\subset
\rho_{C}(W)~and ~R_{C}(\lambda,W)x=\int_{0}^{+\infty}e^{-\lambda
t}T(t)xdt $ for $\lambda>\omega$ and  $x\in E.$
For every $\lambda>\omega$ and $n\in\mathbb{N},$ $R(C)\subset
D((\lambda I-W)^{-n})$ and
\begin{equation} \label{resolvset2}
(\lambda
I-W)^{-n}Cx=\frac{1}{(n-1)!}\int_{0}^{+\infty}t^{n-1}e^{-\lambda
t}T(t)x dt,
\end{equation} which implies
\begin{equation}\label{resolvset3}
\parallel (\lambda-\omega)^{n}(\lambda
I-W)^{-n}C\parallel\leq M .
\end{equation}

\item Let $\alpha>0$ and $\lambda>0.$ We have $\int_{0}^{+\infty}t^{\alpha-1}e^{-\lambda
t}dt=\frac{\Gamma(\alpha)}{\lambda^{\alpha}}$ and the family of operators $(e^{-\omega t}T(t))_{t\geq 0}$  is
uniformly bounded $C-$semigroup with the generator $W-\omega I.$ 
Then we can define the fractional power of $C$-resolvent
operator(see [\ref{balagrishnane}] for more details) as below :
\begin{equation}
(\lambda I-(W-\omega
I))^{-\alpha}Cx=((\lambda+\omega)I-W)^{-\alpha}Cx:=
\frac{1}{\Gamma(\alpha)}\int_{0}^{+\infty}t^{\alpha-1}e^{-\lambda
t}e^{-\omega t}T(t)xdt,~for ~all~x\in E.
\end{equation} \label{resolvset5}
With a simple verification, $(\lambda I-(W-\omega I))^{-\alpha}C$ is
bounded linear operator.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Main results}
\begin{theorem}\label{thme}
Let $(T(t))_{t\in\mathbb{R}^{+}}$ be an exponentially bounded
$C-$semigroup in a Banach space $E$ with generator
$(W,\mathcal{D}(W)),~q\in\mathbb{N}$ such that $q> 2$ and $\alpha>
-1 .$ Then
\begin{enumerate}
\item  For any $n\in\mathbb{N}$ and $x\in E,$ we have $$ \int_{0}^{+\infty}\psi_{n,\alpha}(t)e^{-\omega t}T(t)xdt=(\omega I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}Cx.$$
\item  For $x\in \mathcal{D}(W),$ we have :
\begin{itemize}
\item (a) 
$\left( t>0\right) \ T(t)x=e^{\omega t}\displaystyle\sum_{n=0}^{+\infty}(\omega
I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}\phi_{n,\alpha}(t)Cx.$ 
%\end{equation*}
\item (b)  For each $t>0$ there is $n_{0}\in\mathbb{N}$ such that for all integer $n$ with $n\geq n_{0} $ and $ x\in \mathcal{D}(W^{q}) ,$ we have
 $$\parallel T(t)x- e^{\omega t}\displaystyle\sum_{m=0}^{n}(\omega
I-W)^{m}((\omega+1)I-W)^{-m-\alpha-1}Cx\phi_{m,\alpha}(t)\parallel\leq\frac{c_{t,\alpha, q}\parallel,
(W-\omega I)^{q}x\parallel}{n^{\frac{q}{2}-1}}.$$
where $c_{t ,\alpha ,q}$ is a constant which depends only on $t>0,$ $\alpha $ and $q.$

\end{itemize}

\end{enumerate}
$\mathrm{Proof}.$
\end{theorem}
Throughout the proof, $\alpha$ is an arbitrary real such that
$\alpha >-1 .$
\begin{enumerate}
\item  Let $n\in\mathbb{N}\ and \  x\in E.$
The function $t\mapsto \psi_{n,\alpha}(t)T(t)x$ is continuous on
$\mathbb{R}^{+}$ and integrable in the sens of Bochner. Put
$I:=\int_{0}^{+\infty}\psi_{n,\alpha}(t)(e^{-\omega t}T(t))xdt,$ then
\begin{eqnarray*}
I&:=&\int_{0}^{+\infty}\psi_{n,\alpha}(t)(e^{-\omega t}T(t))xdt\\
 &=& \frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{+\infty}\frac{d^{n}}{dt^{n}}(t^{n+\alpha}e^{-t})(e^{-\omega t}T(t))xdt \\
 &=& \frac{1}{\Gamma(n+\alpha+1)} \left( [\frac{d^{n-1}}{dt^{n-1}}(t^{n+\alpha}e^{-t})(e^{-\omega t}T(t))x]_{0}^{+\infty}- \int_{0}^{+\infty}\frac{d^{n-1}}{dt^{n-1}}(t^{n+\alpha}e^{-t})\frac{d}{dt}(e^{-\omega t}T(t))xdt\right) \\
 &=& \frac{1}{\Gamma(n+\alpha+1)}\left( 0-\int_{0}^{+\infty}\frac{d^{n-1}}{dt^{n-1}}(t^{n+\alpha}e^{-t})(W-\omega I)(e^{-\omega t}T(t))xdt\right) \\
 &=& \frac{-1}{\Gamma(n+\alpha+1)}\left( (W-\omega I)\int_{0}^{+\infty}\frac{d^{n-1}}{dt^{n-1}}(t^{n+\alpha}e^{-t})(e^{-\omega t}T(t))x dt\right)   \\
 &=&(-1)^{n}(W-\omega I)^{n}\left(  \frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{+\infty}(t^{n+\alpha}e^{-t})(e^{-\omega t}T(t))xdt\right) \\
 &=&(\omega I-W)^{n}\left( \frac{1}{\Gamma(n+\alpha+1)}\int_{0}^{+\infty}(t^{(n+\alpha+1)-1}e^{-t})(e^{-\omega t}T(t))xdt\right) \\
 &=&(\omega I-W)^{n}(I-(W-\omega I))^{-n-\alpha-1}Cx\\
 &=&(\omega I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}Cx
\end{eqnarray*}
\item Let $x\in \mathcal{D}(W).$
\begin{itemize}
\item (a)    Let's remember that $e^{-\omega(.)}T(.)x:\mathbb{R}^{+}\rightarrow E$ is in $C^{1}(\mathbb{R}^{+},E)$ and 
\begin{eqnarray*}
  \int_{0}^{+\infty}e^{-t}t^{\alpha}\parallel e^{-\omega t}T(t)x\parallel^{2}dt &\leq & \int_{0}^{+\infty}e^{-t}t^{\alpha}\parallel e^{-\omega t}T(t)\parallel^{2}\parallel x\parallel^{2}dt \\
   &\leq& M^{2}\parallel x\parallel^{2}\int_{0}^{+\infty}e^{-t}t^{\alpha}dt\\
    &<&+\infty.
\end{eqnarray*}
Now, we apply the most important properties of
$(\psi_{n,\alpha})_{n\in\mathbb{N}}$ to get the series
$\displaystyle\sum_{n\in
\mathbb{N}}c_{n}(e^{-\omega(.)}T(.)x)\phi_{n,\alpha}(.),$ where
\begin{eqnarray*}
  c_{n}((e^{-\omega(.)}T(.))x) &=& \int_{0}^{+\infty}\psi_{n,\alpha}(t)(e^{-\omega t}T(t))xdt \\
                               &=& (\omega I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}Cx
\end{eqnarray*}
converges pointwise to $e^{-\omega (.)}T(.)x $ for $t\in
\mathbb{R}^{+}.$ Therefore 
\begin{eqnarray*}
  T(t)x &=& e^{\omega t}(e^{-\omega t}T(t)x) \\
        %&=& e^{\omega t}\displaystyle\sum_{n=0}^{+\infty}(\omega I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}Cx\phi_{n,\alpha}(t) \\
        &=& e^{\omega t}\displaystyle\sum_{n=0}^{+\infty}(\omega I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}Cx\phi_{n,\alpha}(t).
\end{eqnarray*}
\item (b) Let $t\geq 0$ and $q\in\mathbb{N}$ such that $q> 2.$\\ Since $\displaystyle\lim_{n\rightarrow +\infty}(\frac{n}{n-q})^{\frac{\alpha}{2}}=1,$ we have $$(\exists N_{1}\in\mathbb{N})~(\forall n\geq max(q+1,N_{1})),~(\frac{n}{n-q})^{\frac{\alpha}{2}}\leq
2.$$
The Riemann series $\displaystyle\sum_{m\geq
q+1}\frac{1}{(m-q)^{\frac{q}{2}}}$ is convergent, hence
$$\displaystyle\sum_{m=n+1}^{+\infty}\frac{1}{(m-q)^{\frac{q}{2}}}\displaystyle\sim_{n\rightarrow
+\infty}\frac{1}{(\frac{q}{2}-1)}\frac{1}{(n-q)^{\frac{q}{2}-1}}.$$
We have
$$\frac{1}{(\frac{q}{2}-1)}\frac{1}{(n-q)^{\frac{q}{2}-1}}\displaystyle\sim_{n\rightarrow+\infty}\frac{1}{(\frac{q}{2}-1)}\frac{1}{n^{\frac{q}{2}-1}},$$ thus
$$(\exists N_{2}\in\mathbb{N})~(\forall n\geq
max(q+1,N_{1},N_{2})),~\displaystyle\sum_{m=n+1}^{+\infty}\frac{1}{m^{\frac{q}{2}}}\leq
\frac{1}{(\frac{q}{2}-1)}\frac{1}{n^{\frac{q}{2}-1}}.$$ By (\ref{Hn}),
$$(\exists~N_{3}\in\mathbb{N})~(\exists~ c_{t}>0): ~(\forall
n\geq max(N_{1},N_{2},N_{3},q+1)) ~\mid
\phi_{n,\alpha}(t)\mid\leq c_{t}n^{\frac{\alpha}{2}}.$$ 
Put
$n_{0}=max(N_{1},N_{2},N_{3},q+1),$ then
for $n\geq n_{0}$ and $x\in D(W^{q})$
\begin{eqnarray*}
J&:=&\int_{0}^{+\infty}\psi_{n,\alpha}(t)(e^{-\omega t}T(t))xdt\\
   &=&\int_{0}^{+\infty}\frac{d^{q}}{dt^{q}}(\psi_{n-q,\alpha+q}(t))(e^{-\omega t}T(t))xdt~~(according~~
to~~(\ref{hnn}))\\
   &=&[\frac{d^{q-1}}{dt^{q-1}}(\psi_{n-q,\alpha+q}(t)) (e^{-\omega t}T(t))x]_{0}^{+\infty}-\int_{0}^{+\infty}\frac{d^{q-1}}{dt^{q-1}}(\psi_{n-q,\alpha+q}(t)) \frac{d}{dt}(e^{-\omega t}T(t))x dt\\
  % \end{eqnarray*}
  % \begin{eqnarray*}
   &=&0-\int_{0}^{+\infty}\frac{d^{q-1}}{dt^{q-1}}(\psi_{n-q,\alpha+q}(t))(W-\omega I)(e^{-\omega t}T(t))xdt\\
   &=&-(W-\omega I)\int_{0}^{+\infty}\frac{d^{q-1}}{dt^{q-1}}(\psi_{n-q,\alpha+q}(t))(e^{-\omega t}T(t))xdt \\
   &=&(\omega I-W)\int_{0}^{+\infty}\frac{d^{q-1}}{dt^{q-1}}(\psi_{n-q,\alpha+q}(t))(e^{-\omega t}T(t))xdt \\
   &=&(\omega I-W)^{q}\int_{0}^{+\infty}\psi_{n-q,\alpha+q}(t)(e^{-\omega t}T(t))xdt\\
   &=&\int_{0}^{+\infty}\psi_{n-q,\alpha+q}(t)(e^{-\omega t}T(t))(\omega I-W)^{q}xdt .
\end{eqnarray*}
So
 \begin{eqnarray*}
\hspace*{-1cm}\parallel(\omega I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}Cx\parallel&=&\parallel\int_{0}^{+\infty}\psi_{n,\alpha}(t)(e^{-\omega t}T(t))xdt\parallel\\
\hspace*{-1.25cm}&=&\parallel\int_{0}^{+\infty}(\psi_{n-q,\alpha+q}(t)(e^{-\omega t}T(t))(W-\omega I)^{q}xdt\parallel\\
\hspace*{-1.25cm}&\leq&\int_{0}^{+\infty} \hspace*{-0.2cm}\mid\psi_{n-q,\alpha+q}(t)\mid \ \parallel e^{-\omega t}T(t)\parallel dt\parallel(W-\omega I)^{q}x\parallel\\
\hspace*{-1cm}&\leq&\parallel(W-\omega I)^{q}x\parallel M\parallel\psi_{n-q,\alpha+q}\parallel_{1}\\
\hspace*{-1.25cm}&\leq&\parallel(W-\omega
I)^{q}x\parallel\frac{c_{\alpha+q}}{(n-q)^{\frac{\alpha+q}{2}}}\ (according\ 
to\ (\ref{hn})).
\end{eqnarray*}
On the other hand, we are looking to estimate the quantity
\begin{equation}\label{rem}
e^{\omega t}\parallel \displaystyle\sum_{m=0}^{+\infty}(\omega
I-W)^{m}((\omega+1)I-W)^{-m-\alpha-1}Cx\phi_{m,\alpha}(t)-\displaystyle\sum_{m=0}^{n}(\omega
I-W)^{m}((\omega+1)I-W)^{-m-\alpha-1}Cx\phi_{m,\alpha}(t)\parallel.
\end{equation}
We have
 \begin{eqnarray*}
  (\ref{rem})&\leq&\displaystyle\sum_{m=n+1}^{+\infty}\parallel e^{\omega t}(\omega I-W)^{m}((\omega+1)I-W)^{-m-\alpha-1}Cx\phi_{m,\alpha}(t)\parallel \\
&\leq & e^{\omega t}\displaystyle\sum_{m=n+1}^{+\infty}\parallel(\omega I-W)^{m}((\omega+1)I-W)^{-m-\alpha-1}Cx\parallel~~ \mid \phi_{m,\alpha}(t)\mid \\
&\leq &e^{\omega t}\displaystyle\sum_{m=n+1}^{+\infty}\parallel(W-\omega I)^{q}x\parallel\frac{c_{\alpha+q}}{(m-q)^{\frac{\alpha+q}{2}}}c_{t}m^{\frac{\alpha}{2}}\\
%\end{eqnarray*}
%\begin{eqnarray*}
&\leq &\parallel(W-\omega I)^{q}x\parallel e^{\omega t}c_{t}c_{\alpha+q}\displaystyle\sum_{m=n+1}^{+\infty}\frac{m^{\frac{\alpha}{2}}}{(m-q)^{\frac{\alpha+q}{2}}}\\
&\leq &\parallel(W-\omega I)^{q}x\parallel c'_{t}c_{\alpha+q}\displaystyle\sum_{m=n+1}^{+\infty}\frac{(m-q)^{\frac{\alpha}{2}}}{(m-q)^{\frac{\alpha+q}{2}}}\frac{m^{\frac{\alpha}{2}}}{(m-q)^{\frac{\alpha}{2}}}\\
&\leq &\parallel(W-\omega I)^{q}x\parallel c'_{t}c_{\alpha+q}\displaystyle\sum_{m=n+1}^{+\infty}\frac{(m-q)^{\frac{\alpha}{2}}}{(m-q)^{\frac{\alpha+q}{2}}}(\frac{m}{m-q})^{\frac{\alpha}{2}}\\
&\leq &\parallel(W-\omega I)^{q}x\parallel c'_{t}c_{\alpha+q}\displaystyle\sum_{m=n+1}^{+\infty}\frac{1}{(m-q)^{\frac{q}{2}}}\times 2\\
&\leq &\parallel(W-\omega I)^{q}x\parallel
c'_{t}c_{\alpha+q}\frac{1}{(\frac{q}{2}-1)}\frac{1}{n^{\frac{q}{2}-1}},
\end{eqnarray*}
where
$$
\parallel T(t)x-e^{\omega t}\displaystyle\sum_{m=0}^{n}\left( \omega
I-W\right) ^{m}\left( \left( \omega+1\right) I-W\right) ^{-m-\alpha-1}Cx\phi_{m,\alpha}\left( t\right) \parallel\leq\frac{c_{t,\alpha , q}\parallel
\left( W-\omega I\right) ^{q} x \parallel} {n^{\frac{q}{2}-1}}.
$$
\end{itemize}
\end{enumerate}

\begin{theorem}\label{th}
Let $(T(t))_{t\in\mathbb{R}^{+}}$ be an exponentially bounded
$C-$semigroup in Banach space $E$ with generator
$(W,\mathcal{D}(W))$ and $ \alpha\in\mathbb{R}.$ %such that $\alpha > 0.$
\begin{enumerate}
\item  For $x\in E, $ $\alpha > 0$ and $n\in\mathbb{N},$ we have $$\int_{0}^{+\infty}\psi_{n,\alpha}(t)R_{C}(t,W-\omega
I)xdt=\int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{-\omega
s}T(s)xds.$$
\item For $x\in E $ and $\alpha > 1,$  we have $$R_{C}(t,W-\omega
I)x=\displaystyle\sum_{n=0}^{+\infty}\left( \int_{0}^{+\infty}\frac{s^{n}e^{-\omega
s}}{(s+1)^{n+\alpha+1}}T(s)xds\right) \phi_{n,\alpha}(t), \ for\ all\  t>0 .$$
\end{enumerate}
$\mathrm{Proof}.$
\end{theorem}
\begin{enumerate}
\item Let $x\in E,$  $\alpha > 0$ and $n\in \mathbb{N}. $ \\ For all $t>0, \ t+\omega\in \rho_{C}(W),$ so the function $t \longmapsto 
R_{C}(t,W-\omega I)=R_{C}(t+\omega,W)$ is analytic in
$\mathbb{R}^{*}_{+}.$ \\ In the other hand, by equation (\ref{resolvset3}), we know that $\parallel R_{C}(t+\omega,W)\parallel\leq \frac{M}{t}$ for all $t>0,$ so
\begin{eqnarray*}
\int_{0}^{+\infty}\parallel\psi_{n,\alpha}(t)R_{C}(t,W-\omega
I)x\parallel dt &=&\int_{0}^{+\infty}\mid\psi_{n,\alpha}(t)\mid\parallel(R_{C}(t,W-\omega I)x\parallel dt\\
&\leq &\int_{0}^{+\infty}\frac{n!M}{\Gamma(n+\alpha+1)t}e^{-t}t^{\alpha}\mid\phi_{n,\alpha}(t)\mid dt\\
&\leq & \frac{n!M}{\Gamma(n+\alpha+1)}\int_{0}^{+\infty}e^{-t}t^{\alpha-1}\mid\phi_{n,\alpha}(t)\mid  dt\\
&<&+\infty .
\end{eqnarray*}
Put
$H:=\Gamma(n+\alpha+1)\int_{0}^{+\infty}\psi_{n,\alpha}(t)R_{C}(t,W-\omega I)xdt , $ then 
\begin{eqnarray*}
  H &=&\Gamma(n+\alpha+1)\int_{0}^{+\infty}\psi_{n,\alpha}(t)R_{C}(t,W-\omega I)xdt\\
   &=&\int_{0}^{+\infty}\frac{d^{n}}{dt^{n}}(e^{-t}t^{n+\alpha})R_{C}(t,W-\omega I)xdt \\
   &=&\underbrace{[\frac{d^{n-1}}{dt^{n-1}}(e^{-t}t^{n+\alpha}) R_{C}(t,W-\omega I)x]_{0}^{+\infty}}_{=0}-\int_{0}^{+\infty}\frac{d^{n-1}}{dt^{n-1}}(e^{-t}t^{n+\alpha}) \frac{d}{dt}(R_{C}(t,W-\omega I)x)dt \\
   &=&[-\frac{d^{n-2}}{dt^{n-2}}(e^{-t}t^{n+\alpha}) \frac{d}{dt}(R_{C}(t,W-\omega I)x)]_{0}^{+\infty} \\
    & +& (-1)^{2}\int_{0}^{+\infty}\frac{d^{n-2}}{dt^{n-2}}(e^{-t}t^{n+\alpha})\frac{d^{2}}{dt^{2}}(R_{C}(t,W-\omega I)x)dt \\
   &=&(-1)^{n}\int_{0}^{+\infty}e^{-t}t^{n+\alpha}\frac{d^{n}}{dt^{n}}(R_{C}(t,W-\omega I )x )dt \ \ (integtration \ by \ parts)\\   
   &=&(-1)^{n}\int_{0}^{+\infty}e^{-t}t^{n+\alpha}(-1)^{n}n!( (t+\omega )I - W)^{-n-1} C xdt \\
   &=&\int_{0}^{+\infty}e^{-t}t^{n+\alpha}(n!)\frac{1}{n!}\int^{+\infty}_{0}s^{n}e^{-ts}e^{-\omega s}T(s)xdsdt\\
   &=&\int_{0}^{+\infty}s^{n}e^{-\omega s}T(s)x\left( \int_{0}^{+\infty}t^{n+\alpha}e^{-t}e^{-ts}dt\right) ds \ (Fubini's \ Theorem)\\
      &=&\int_{0}^{+\infty}s^{n}e^{-\omega s}T(s)x\left( \int_{0}^{+\infty}t^{n+\alpha}e^{-(s+1)t}dt\right)  ds \\
   &=&\int_{0}^{+\infty}s^{n}e^{-\omega s}T(s)x\left( \int_{0}^{+\infty}\frac{u^{n+\alpha}}{(s+1)^{n+\alpha}}e^{-u}\frac{du}{s+1}\right) ds~~(u=(s+1)t) \\
   &=&\int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{-\omega s}T(s)x\left( \int_{0}^{+\infty}u^{n+\alpha}e^{-u}du\right) ds \\
   &=&\int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{-\omega s}T(s)x\Gamma(n+\alpha+1)ds\\
   &=&\Gamma(n+\alpha+1)\int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{-\omega s}T(s)xds,
\end{eqnarray*}
hence the result.
\item Let $x\in E$ and $\alpha > 1 .$\\
The function $t\longmapsto R_{C}(t,W-\omega I)x=R_{C}(t+\omega,W)x$
is differentiable in $\mathbb{R}^{*}_{+}$ (because it's analytic
in $\mathbb{R}^{*}_{+}$), and
\begin{eqnarray*}
\int_{0}^{+\infty}t^{\alpha}e^{-t}\parallel R_{C}(t,W-\omega I)x\parallel^{2}dt&=&\int_{0}^{+\infty}t^{\alpha}e^{-t}\parallel((t+\omega) I-W)^{-1}Cx\parallel^{2}dt\\
&\leq&\int_{0}^{+\infty}t^{\alpha}e^{-t}\frac{M^{2}}{t^{2}}\parallel x\parallel^{2}dt\ (equation\ (\ref{resolvset3}))\\
&\leq & M^{2}\parallel x\parallel^{2}\int_{0}^{+\infty}t^{\alpha-2}e^{-t}dt\\
&= & M^{2}\parallel x\parallel^{2}\Gamma(\alpha-1)\ (because\ \alpha>1)\\
&<&+\infty.
\end{eqnarray*}
Thus, the series $\displaystyle\sum_{n\in
\mathbb{N}}c_{n}(R_{C}(.,W-\omega I)x)\phi_{n,\alpha}$ converges
pointwise to $R_{C}(.,W-\omega I)x$ on $\mathbb{R}^{*}_{+},$ where
\begin{eqnarray*}
c_{n}\left( R_{C}\left( .,W-\omega I\right) x\right) &=&\int_{0}^{+\infty}\psi_{n,\alpha}\left( t\right) R_{C}\left( t,W-\omega I\right) x dt\\
&=&\int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{-\omega s}T(s)xds\ \left( according \ to\  Theorem \ \ref{th}(1)\right) .
\end{eqnarray*}
 Therefore,  
  $$ \left( \forall t>0\right) \ \ R_{C}(t,W-\omega
I)x=((t+\omega)I-W)^{-1}Cx= \displaystyle\sum_{n=0}^{+\infty} \left( \int_{0}^{+\infty}\frac{s^{n}e^{-\omega
s}}{(s+1)^{n+\alpha+1}}T(s)xds\right) \phi_{n,\alpha}(t). $$
\end{enumerate}

\begin{example}
Let $m:\mathbb{R}\rightarrow \mathbb{R}^{-}$ be an even measurable
function. In the Banach space $L^{1}(\mathbb{R}),$ we consider the
family $T:=(T(t))_{t\geq 0}\subset \mathfrak{F}(L^{1}(\mathbb{R}))$
defined by $\forall t\geq 0,~%T:\mathbb{R}^{+}\rightarrow
%B\left(L^{1}(\mathbb{R})\right), t\mapsto
T(t):L^{1}(\mathbb{R})\rightarrow L^{1}(\mathbb{R}),f\mapsto
T(t)(f):\mathbb{R}\rightarrow\mathbb{R},s\mapsto
T(t)(f)(s)=e^{t.m(s)}f(-s)$.
%\cos\begin{pmatrix}t\sqrt{-m(s)}\end{pmatrix}f(-s).$\\
Clearly, $(T(t))_{t\geq 0}\subset B(L^{1}(\mathbb{R})).$ If we put
$T(0)=C,$ then the family of operators $(T(t))_{t\geq 0}$ is
uniformly bounded $C$-regularized semigroup with generator $(W,\mathcal{D}(W))$ defined by
$$W:\mathcal{D}(W)=\{f\in L^{1}(\mathbb{R})/ \ m.f\in
L^{1}(\mathbb{R})\}\rightarrow L^{1}(\mathbb{R}), \ f\mapsto
W(f)=m.f .$$
Theorem \ref{thme} gives for $f\in \mathcal{D}(W)$ and $s,t\in
\mathbb{R}^{+}:$
\begin{eqnarray*}
  T(t)(f)(s) &=&e^{\omega t}\displaystyle \sum_{n=0}^{+\infty}(\omega
  I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}C(f)(s)\phi_{n,\alpha}(t)\\
  &=&e^{0\times t}\displaystyle\sum_{n=0}^{+\infty}(0\times
  I-W)^{n}((0+1)I-W)^{-n-\alpha-1}C(f)(s)\phi_{n,\alpha}(t)\\
&=&\displaystyle\sum_{n=0}^{+\infty}(-m(s))^{n}(1-m(s))^{-n-\alpha-1}C(f)(s)\phi_{n,\alpha}(t).
\end{eqnarray*}
So
$T(t)(f)=\displaystyle\sum_{n=0}^{+\infty}\phi_{n,\alpha}(t)(-m)^{n}(1-m)^{-n-\alpha-1}C(f).$\\
Theorem \ref{th} gives 
$$R_{C}(t,W)(f)(.)=\displaystyle \sum_{n=0}^{+\infty}\left( \int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{sm(.)}ds\right) C(f)(.)\phi_{n,\alpha}(t).$$
\end{example}

\begin{example}
The space $X=c_{0}= \left\lbrace (x_{k})_{k\in\mathbb{N}}\in
\mathbb{C}^{\mathbb{N}}~~tq~~ \displaystyle\lim_{k\mapsto +\infty}x_{k}=0\right\rbrace ,$
equipped with the norm
$\parallel(x_{k})_{k\in\mathbb{N}}\parallel_{\infty}=\displaystyle\max_{k\in\mathbb{N}}\mid
x_{k}\mid$ becomes a Banach space. For each $n\in \mathbb{N},$ let
$e_{n}=(\delta_{n,k})_{k\in\mathbb{N}}$ be an element of $X.$ Since for
all $x=(x_{k})_{k\in\mathbb{N}}\in
X, \ x=\displaystyle\sum_{k=0}^{+\infty}x_{k}e_{k},$ we have
$X=span\{e_{n}/n\in \mathbb{N}\}.$ Considering the family of
operators $(T(t))_{t\geq0}$ defined by:
\begin{equation*}
for \ all \ t\in \mathbb{R}^{+}, \   \ for \ all\ 
x \ \left( =\displaystyle\sum_{k=0}^{+\infty}x_{k}e_{k}\right) \in X
, \ T(t)x=\displaystyle\sum_{k=0}^{+\infty}e^{-k^{2}t}x_{k}e_{k}
\end{equation*}
$(T(t))_{t\geq 0}$ is a unifomly bounded $C_{0}-$semigroup ($\parallel T(t)\parallel \leq 1$) with generator
$(W,\mathcal{D}(W))$ such that
$\mathcal{D}(W)=\{x=(x_{k})_{k\in\mathbb{N}}\in X /
(k^{2}x_{k})_{k\in\mathbb{N}}\in X\}$ and $\left( \forall x\in
\mathcal{D}(W)\right) \  Wx=\displaystyle\sum_{k=0}^{+\infty}-k^{2}x_{k}e_{k}.$
Theorem \ref{thme} gives for  $x=(x_{k})_{k\in\mathbb{N}}\in
\mathcal{D}(W)$ and for all $t\in \mathbb{R}^{+}:$
\begin{eqnarray*}
  T(t)(x) &=&e^{\omega t}\sum_{n=0}^{+\infty}(\omega
  I-W)^{n}((\omega+1)I-W)^{-n-\alpha-1}(x)\phi_{n,\alpha}(t)\\
  &=&e^{0\times t}\sum_{n=0}^{+\infty}(0\times
  I-W)^{n}((0+1)I-W)^{-n-\alpha-1}(x)\phi_{n,\alpha}(t)\\
&=&\sum_{n=0}^{+\infty}(k^{2n}(1+k^{2})^{-n-\alpha-1}x_{k})_{k\in\mathbb{N}}\phi_{n,\alpha}(t)\\
&=&\sum_{n=0}^{+\infty}\sum_{k=0}^{+\infty}k^{2n}(1+k^{2})^{-n-\alpha-1}x_{k}e_{k}\phi_{n,\alpha}(t).
\end{eqnarray*}
Theorem \ref{th} gives 
$$R_{C}(t,W)x=\displaystyle \sum_{n=0}^{+\infty}\displaystyle\sum_{k=0}^{+\infty}\left( \int_{0}^{+\infty}\frac{s^{n}}{(s+1)^{n+\alpha+1}}e^{-k^{2}s}ds\right)  x_{k}e_{k}\phi_{n,\alpha}(t).$$

\end{example}
\vspace*{0.25cm}
\textbf{Acknowledgments} \  
The authors are greatly indebted to the editor and the referee for many valuable
comments and suggestions which improved the initial draft of this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{-1.5pt}


\begin{thebibliography}{10}
\bibitem{AF}\label{af}
\textsc{L. Abadias And P. J. Miana,} \emph{$C_{0}-$ semigroups and
resolvent operators approximated by Laguerre expansions}, Journal of
Approximation Theory 213 (2017), 1-22.
%ArXiv:1311. 7542v2 [math. FA] 32oct 2014.
%\bibitem{DA}\label{da}
%\textsc{W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, }
%Vector-valued Laplace Transforms and Cauchy Problems, Convoluted
%$C-$cosine functions and semigroups. Birkh$\ddot{A}$auser Verlag,
%Basel, (2001).
\bibitem{AG}\label{ag}
\textsc{R. Askey,} \emph{Mean convergence of expansions in Laguerre
and Hermite series}, Amer. J. Math. 87 (1965), 695-708.
%\bibitem{psmg}\label{psmg}
%\bibitem{CAAA}\label{caaa}
%\textsc{R. Delaubenfels, S. Guozheng, S. Wang, R. Aftabizadeh}
%Regularized semigroups, Existence Familles And the abstract Cauchy
%Problem, Diferential and Integral Equations, Volume 8, p. 1477-1496
%(1995).
\bibitem{AB}\label{ab}
\textsc{R. Delaubenfels,} \emph{$C$-semigroups and the Cauchy
problem}, J. Funct. Anal. 111 (1993), 44-61.
\bibitem{qdelaubenfiels}\label{qdelaubenfiels}
\textsc{R. Delaubenfels,} \emph{Existence families, functional
calculi and evolution equations}, Lecture Notes in Math, 1570,
Springer-Verlag, 1994.
\bibitem{balagrishnane}\label{balagrishnane}
\textsc{R. O. Fattorini,} \emph{A note on fractional derivatives of
semigroups and cosine functions}, Pacific journal of mathematics,
Vol. 109, No 2, 1983.
\bibitem{CAA}\label{caa}
\textsc{M. Kaltenback, F. L. Schwenninger,} \emph{Generalisations of
Semigroups of Operators in the View of Linear Relations}, Analysis
und Scientic Computing der Technischen Universitat Wien, Matr. Nr.
0625539 Wahringer Strasse 125/15-1180 Wien. Vienna University of
Technology.
%\bibitem{CAAAAAA}\label{caaaaaa}
%\textsc{M. Kaltenbavk, F. Leopold,} Generalisations of Semigroups of
%Operators in the View of Linear Relations, Analysis und Scientific
%Computing der Technischen Universitat Wien.
\bibitem{ak}\label{ak}
\textsc{M. Kostic,} \emph{$(a,k)-$Regularized $C-$Resolvent
Families: Regularity and Local Properties}, Hindawi Publishing
Corporation Abstract and Applied Analysis, Volume 2009, Article ID
858242, 27 pages doi:10. 1155/2009/858242.
%\bibitem{CF}\label{cf}
%\textsc{M. Kostic,} $(a, k)-$Regularized $C-$Resolvent Families:
%Regularity and Local Properties, Hindawi Publishing Corporation
%Abstract and Applied Analysis Volume 2009, Article ID 858242 (2009).
\bibitem{H}\label{H}
\textsc{N. N. LEBEDEV,} \emph{Special function and their
applications,} Prentice-hall, INC, 1965.
%\bibitem{CE}\label{ce}
%\textsc{F. Li, J. H. Liu,} Note on multiplicative perturbation of
%local $C-$regularized cosine functions with nondensely defined
%generators, Electronic Journal of Qualitative Theory of Differential
%Equations, No. 57, 1-12 (2010).
\bibitem{CH}\label{ch}
\textsc{M. Li, Q. Zheng, J. Zhang,} \emph{Regulrized resolvent
famillies}, Taiwanese Journal of Mathematics, vol.11, no.1 (2007),
117-133.
\bibitem{FR}\label{fracpowofoperat}
\textsc{C. Martinez, M. Sanz, L. Marco,} \emph{Fractional powers of
operators}, J. Math. Soc. Japan, Vol. 40, No. 2, 1988.
\bibitem{CL}\label{cl}
\textsc{M. Mosallanezhad, M. Janfada,} \emph{On Mixed
$C-$Semigroups of Operators on Banach Spaces}, Filomat 30:10
(2016), 2673-2682.
\bibitem{CAAAA}\label{caaaa}
\textsc{P. Preda, A. Pogan, C. Preda,} \emph{The Perron Problem For
$C-$semigroups}, Math. J. Okayama Univ. 46 (2004), 141-151.
%\bibitem{CN}\label{co}
%\textsc{Sheng, H. Wang,} \emph{Properties of subgenerators of
%$C$-Regularezed semigroups}, proceedings of the American
%Mathematical Society, Volume 126, Number 2, p. 453-460 (1998).
\bibitem{AC}\label{ac}
\textsc{J. A. Shohat,} \emph{On the developement of functions in
series of orthogonal polynomials}, The university of Pennsyvania, 1935.
\bibitem{AA}\label{aa}
\textsc{G. Szego,}\emph{ Orthogonal polynomials}, PubLished by the
American Mathematical Society Providence, Rhode Island, 1939.
%\bibitem{AD}\label{ad}
%\textsc{H. Skovgaard,}Asymptotic forms of hermite
%polynomials.Departement of Mathematics California in scitute of
%Technology Pasadenea (1959).
%\bibitem{AE}\label{ae}
%\textsc{Stanford S.Bonan, Deak S. Clark,} Estimates of the Hermite
%and the Freud Polynomials, Journal of approximation theory 63,
%210-244 (1990).
%Journal of Mathematical
%Analysis and Applications, UniversidaddeZaragoza, 50009 Zaragoza,
%Spain (2015).
%\bibitem{AH}\label{ah}
%\textsc{Ervand Kogbetliantz ,} Recherches sur la sommabilite des
%series d'Hermite, Annales scientifiques de l'E.N.S. $3^{e}$ serie,
%tome 49 (1932).
\bibitem{CAAAAAAA}\label{caaaaaaa}
\textsc{Q. Zheng,} \emph{Controllability of a class of linear
systems in Banach spaces}, Proc. Amer. Math. Soc. 123 (1995),
1241-1251. MR 95e:93011. Analysis und Scientific Computing der
Technischen Universitat Wien.
\bibitem{CA}\label{ca}
\textsc{Q. Zheng, Y. Zhao,} \emph{An analyticity criterion for
regularezed semigroups}, Proceedings of the american Mathematical
society, Volume 130, Number 8, S0002-9939(02)06219-6 (2002).
%\bibitem{CB}\label{cb}
%\textsc{Chung-Cheng Kuo,} Perturbations of local $C-$cosine
%functions, Stud. Univ. Babes-Bolyai Math. 65, No.4, 585-597 (2020).
%\bibitem{CD}\label{cd}
%\textsc{Q. Zheng,} Matrices of operators and regularized cosine
%functions, J. Math. Anal. Appl. 315, 68-75 (2006).
%\bibitem{CI}\label{ci}
%\textsc{Chung-Cheng Kuo, Sen-Yen Shaw,} $C-$Cosine Functions and the
%Abstract Cauchy Problem, Journal of Mathemtical Analysis and
%Applications 210, 632-646 (1997).
%\bibitem{CJ}\label{cj}
%\textsc{Valentin Keyantuo, Carlos Lizama, Pedro J. Miana ,} Algebra
%homomorphisms defined via convoluted semigroups and cosine
%functions, Journal of Functional Analysis 257.3454-3487 (2009).
%\bibitem{CK}\label{ck}
%\textsc{Miao Li, Quan Zheng, Jizhou Zhang,} Regularezed Resolvent
%famillies, 2000, Taiwanese journal of mathematics V.1, No.1,
%pp.117-133 (2007).
%\bibitem{CM}\label{cm}
%\textsc{Fang Li, Jin Liang, Ti-Jun Xiao,Jun Zhang,} On Perturbation
%of Convoluted-Regularized Operator Families,Hindawi Publishing
%Corporation Journal of Function Spaces and Applications, Volume
%2013, Article ID 579326.
%\bibitem{CR}\label{cr}
%\textsc{Jizhou, Zhang,} Regularezed cosine Functions and Polynomials
%of Group Generators, Calculi, Scientia iranica, vol 4, Nos.1,2, pp
%12-18 (1997).
	
\end{thebibliography}

\end{document}

