Berezin symbols of operators on the unit sphere of $\mathbb C^n$
Díaz-Ortíz Erik
Erik I. D\'{\i
We describe the symbolic calculus of operators on the unit sphere in the complex $n$-space $\mathbb C^n$ defined by the Berezin quantization. In particular, we derive a explicit formula for the composition of Berezin symbol and with that a noncommutative star product. In the way is necessary introduce a holomorphic spaces which admit a reproducing kernel in the form of generalized hypergeometric series.% and is generalization of the classical Fischer-Fock.
1702.05708.tex
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\newcommand{\vecz}{\mathbf{z}}
\newcommand{\vecw}{\mathbf{w}}
\newcommand{\vecu}{\mathbf{u}}
\newcommand{\dm}{\mathrm dm_{n,p}^\hbar}
\newcommand{\dz}{\mathrm d\vecz}
\newcommand{\dzb}{\mathrm d\overline\vecz}
\newcommand{\E}{\mathcal E_{n,p}}
\newcommand{\T}{{\bf T}_{n,p}}
\newcommand{\To}{{\mathcal T}}
\newcommand{\F}{{}_0\mathrm F_1}

\newcommand{\Sn}{\mathbf{S}^n}
\newcommand{\vecx}{\mathbf{x}}
\newcommand{\vecy}{\mathbf{y}}
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\newcommand{\ds}{\mathrm d\mathbf S_n}
\newcommand{\bk}{{\bf k}}
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\newcommand{\K}{{{\bf K}_{n,p}^\hbar}}
\newcommand{\Ku}{{{\bf K}_{n,-1}^\hbar}}
\newcommand{\Kz}{{{\bf K}_{\mathbf z}^\hbar}}
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\newcommand{\B}{{{\mathfrak B}_{\hbar,p}}}
\newcommand{\Be}{{{\mathfrak B}}}
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%opening
%\title{Berezin symbolic calculus on $\Sn$}
%\author{Díaz-Ortíz Erik}

\begin{document}
\title[Berezin symbols on the complex unit sphere]
{Berezin symbols of operators on the  unit sphere of $\mathbb C^n$}
\author{Erik I. D\'{\i}az-Ort\'iz}
\address{CONACYT Research Fellow – Universidad Pedagógica Nacional - Unidad 201 Oaxaca}
\email{eidiazor@conacyt.mx}

\maketitle
\tableofcontents
\begin{abstract}
We describe the symbolic calculus of  operators on the unit sphere in the complex $n$-space $\mathbb C^n$ defined by the Berezin quantization. In particular, we derive a explicit formula for the composition of Berezin symbol and with that a noncommutative star product. In the way is necessary introduce a  holomorphic spaces  which admit  a reproducing kernel in the form of generalized hypergeometric series.% and is generalization of the classical Fischer-Fock.
\end{abstract}



\section{Introduction and summary}
We start recalling some results of Berezin's theory that will be used below. See Ref. \cite{B74} for details.
\begin{comment} 
\subsection{The space $L^2(\Sn)$}
Let us consider the Hilbert space $L^2(\Sn)$ of the square integrable function with respect to the normalized surface measure $\ds(\vecx)$ on $\Sn$, where $\Sn=\{\vecx \in \mathbb C^n\;|\; \vecx \cdot \vecx =1\}$, and endowed with the usual inner product
\begin{equation}
 \langle \phi,\psi\rangle _{\Sn}= \int_{\Sn} \phi(\vecx) \overline{\psi(\vecx)} \ds(\vecx)\;,\;\hspace{0.5cm} \phi,\psi \in L^2(\Sn)\;,
\end{equation}
let us denote the $L^2(\Sn)$ norm by $||\cdot||_{\Sn}$. Let $\mathcal O$ the subspace in $L^2(\Sn)$ of all functions whose Poisson extension into the interior of $\Sn$ is holomorphic.
%\begin{definition}\label{definition O}
 %The space of analytic function $\phi$ defined on $\Sn$ such that $||\phi||_{\Sn}$ is finity is denoted by $\mathcal O$.
%\end{definition}


\subsection{Berezin's theory}
We start recalling some results of Berezin's theory of covariant symbol. See Ref. \cite{B74} for details.

In this subsection we briefly describe the Berezin symbol that will be used below. See Ref. \cite{B74} for details.
\end{comment} 

Let $H$ be a Hilbert space endowed with the inner product  $(\cdot,\cdot)$ and $M$ some set with the measure $\mathrm d \mu$. Let $\{e_\alpha \in H\;|\; \alpha \in M\}$ a family of functions in $H$ labelled by elements of $M$, such that satisfies the following properties:
\begin{enumerate}
\item[a) ] The family $\{e_\alpha\}$ is complete, this is for any $f,g \in H$ Parseval's identity is valid
\begin{displaymath}
(f,g)= \int_M (f,e_\alpha)(e_\alpha,g) \mathrm d \mu(\alpha)\;.
\end{displaymath}
\item[b) ] The map $f \to \hat f$, defining by $\hat f(\alpha)= (f,e_\alpha)$, is an embedding from $H$ into $L^2(M,\mathrm d\mu)$.
\end{enumerate}

In 1970’s, Berezin \cite{B74} introduced a general symbolic calculus for bounded linear operators on $H$. More specifically, for $A \in \mathbf B (H)$, the algebra of all bounded linear operator on $H$, the Berezin symbol (or Berezin transform) of $A$ is the function on $M$ defined by
\begin{displaymath}
\Be( A) (\alpha)= \frac{(Ae_\alpha,e_\alpha)}{(e_\alpha,e_\alpha)}\;, \hspace{0.5cm} \alpha \in M\;.
\end{displaymath}
The prototypes of the spaces $H$ are the Bergman spaces of all holomorphic functions in $L^2(M, \mathrm d\mu )$ on a bounded domain $M\subset \mathbb C^n$ with Lebesgue measure $\mathrm d\mu$, or the Segal-Bargmann spaces of all entire functions in $L^2(\mathbb C^n,\mathrm d \mu)$ for the Gaussian measure $\mathrm d\mu (z) = (2\pi)^{-n} e^{-|z|^2/2}\mathrm dz \mathrm d\overline z$, where $\mathrm dz \mathrm d\overline z$ denoting Lebesgue measure on $\mathbb C^n$. In these cases the functions $e_\alpha$ are the reproducing kernel.


Moreover, for every $A,B \in \mathbf B(H)$ and $f \in H$, we have the following formulas
\begin{align}
\Be(A+cB) & = \Be( A) + c \Be( B)\;, \mbox{ for all constant } c \in \mathbb C\;,\nonumber\\
\Be(\mathrm{Id}) & =1\;,\hspace{0.5cm} \mbox{with $\mathrm{Id}$ the identity operator }.\nonumber
\end{align}
If we suppose that the Berezin's symbol may be extended in a neighbourhood of the diagonal $M \times M$ to the function
\begin{displaymath}
\Be( A)(\alpha,\beta)= \frac{(Ae_\alpha,e_\beta)}{(e_\alpha,e_\beta)}\;,
\end{displaymath}
then we have the following formulas
\begin{align}
\Be( A^*)(\alpha,\beta) & = \overline{\Be(A)(\beta,\alpha)} \;, \hspace{0.5cm} \alpha,\beta\in M\;.\nonumber\\
\widehat{Af}(\alpha) & = \int_M \hat f(\beta) \Be(A)(\beta,\alpha) (e_\beta,e_\alpha) \mathrm d \mu(\beta)\;, \label{Eq 1}\\
\Be(AB)(\alpha,\beta) & = \int_M \Be(A)(\gamma,\beta) \Be(B)(\alpha,\gamma) \frac{(e_\alpha,e_\gamma)(e_\gamma,e_\beta)}{(e_\alpha,e_\beta)} \mathrm d\mu(\gamma)\;. \label{Eq 2}
\end{align}


The useful application of this symbolic calculus is that allows us to build a star product. 
In \cite{B74} Berezin applied this method to K\"ahler manifolds, in this case $H$ is the Hilbert space of functions in $L^2(M,\mathrm d\mu)$ which are analytic so that the embedding from $H$ into $L^2(M,\mathrm d\mu)$ is the inclusion, and the complete family $\{e_\alpha\}$ are the reproducing kernel with one variable fixed.

The main goal of the present paper is to introduce a Berezin symbolic calculus for the Hilbert space $\mathcal O$ of all functions in $L^2(\Sn)$ whose Poisson extension into the interior of $\Sn$ is holomorphic, where $\Sn=\{\vecx\in\mathbb C^n\;|\; |x_1|^2+\cdots+|x_n|^2=1\}$ and  $L^2(\Sn)$ denote the Hilbert space of the square integrable function with respect to the normalized surface measure $\ds(\vecx)$ on $\Sn$ and endowed with the usual inner product
\begin{equation}
 \langle \phi,\psi\rangle _{\Sn}= \int_{\Sn} \phi(\vecx) \overline{\psi(\vecx)} \ds(\vecx)\;,\;\hspace{0.5cm} \phi,\psi \in L^2(\Sn)\;.
\end{equation}
%let us denote the $L^2(\Sn)$ norm by $||\cdot||_{\Sn}$. }

To achieve this,  we build a family $\{e_\alpha\} $ of functions in $\mathcal O $ satisfying a), since coherent states meet this property and based of our experience in $ L^2(S^m) $, $ m = 2,3,5 $, where $ S^m =\{\vecy \in \mathbb R^{m + 1} \; | \; y_1^2 + \cdots + y_{m + 1}^2 = 1 \} $ (see Ref. \cite{D-V09}), we propose in Sec. \ref{class of holomorphic spaces} our coherent states in $ \mathcal O $ as a suitable power series of the inner product $\sum_{\ell=1}^n x_\ell \overline z_\ell/ \hbar $ regarding $ (x_1,\ldots,x_n) \in \Sn $ and $ (z_1,\ldots,z_n) \in \mathbb C^n $, where $ \hbar $ denoting the Planck's constant. In addition, to ensure b) we define a new Hilbert space $ L^2(\mathbb C^n, \dm) $, whose measure $ \dm $ is obtained such that the inner product of these coherent states is the reproducing kernel from range of mapping $ f \to \hat f $ (see Sec. \ref{section construction measure} for details of how this measure was obtained and  Eq. (\ref{measure on U}) for definition of $\dm$).

In order to prove that the coherent states and the space $L^2(\mathbb C^n,\dm)$ constructed in Sec. \ref{class of holomorphic spaces} satisfy the conditions a) and b), in Sec. \ref{space En} we prove that the mapping $ f \to \hat f $, denoted by $\U$, is an isometry from $\mathcal O$ to $\E \subset L^2(\mathbb C^n, \dm)$. In addition, in Sec. \ref{section coherent states} we prove that the unitary transformation $\U$ applied to our coherent states gives the reproducing kernel of the space $\E$, which allows us to prove that the family of coherent states form a complete system for $\mathcal O$.

From this and according to Berezin's theory, in Sec. \ref{section Berezin symbolic calculus} we describe the rules for symbolic calculus on $\mathbf B(\Sn)$, further more we obtain asymptotic expansions of Berezin's symbol of Toeplitz operator.

Finally,  in Sec. \ref{section star product} we define a star product on the algebra out of Berezin's symbol for bounded operators with domain in $\mathcal O$ and will prove that this noncommutative star-product satisfies the usual requirement on the semiclassical limit.

It should be noted, since $ H = \mathcal O $ and $ L^2 (M, \mathrm d \mu) = L^2 (\mathbb C^n, \dm) $, in our construction there is no inclusion of $ H $ into $ L^2 (M, \mathrm d \mu) $, further more, the complete family $\{ e_\alpha\} $ is not obtained by the reproducing kernel. This situation is thus slightly different from Berezin's one.


%It is immediate that $A$  is real analytic and $||A||_\infty \le ||A||$, and it is well-known that %there is a one-to-one correspondence between operator and Berezin symbol
%$A$ is uniquely determined by $\B(A)$
%\begin{displaymath}
%Af(\alpha)= \int f(\beta) \B(A)(\beta,\alpha)(e_\beta,e_\alpha) \mathrm d \beta\;,
%\end{displaymath}
%where
%\begin{displaymath}
%\B( A)(\alpha,\beta)= \frac{(Ae_\alpha,e_\beta)}{(e_\alpha,e_\beta)}
%\end{displaymath}
%is the extension of the function $\B(A)(\alpha)$ to $M \times M$.

In the present paper we  will be used throughout the text the following  basic notation.
For every $\vecz, \vecw \in \mathbb C^n$, $\vecz=(z_1,\ldots,z_n)$, $\vecw=(w_1,\ldots,w_n)$, let
\begin{displaymath}
\vecz\cdot\vecw=\sum_{\ell=1}^n z_\ell\,\overline w_\ell\;, \hspace{0.8cm} |\vecz|=\sqrt{\vecz \cdot \vecz}\;.
\end{displaymath}
Furthermore, for every multi-index $\bk=(k_1,\ldots,k_n)\in \mathbb Z_+^n$ of length $n$ where $\mathbb Z_+$ is the set of non negative integers, let \begin{displaymath}
|\bk|=\sum_{\ell=1}^n k_\ell\;, \hspace{0.8cm} \bk!=\prod_{\ell=1}^n k_\ell! \;, \hspace{0.8cm} \vecz^\bk=\prod_{\ell=1}^n z_\ell^{k_\ell}\;.
\end{displaymath}
Given $\omega\in \mathbb C$, let us denote its real and imaginary parts by $\Re(\omega)$ and $\Im(\omega)$ respectively.

%$\vecz, \vecw \in \mathbb C^n$, $\vecz=(z_1,\ldots,z_n)$, $\vecw=(w_1,\ldots,w_n)$, we will use the following standard notation: $\vecz \cdot \vecw =\sum_{j=1}^n z_j \overline w_j$, $|\vecz|=\sqrt{\vecz \cdot \vecz}$, $\vecz^\bk=z_1^{k_1}\cdots z_n^{k_n}$, where $\bk=(k_1,\ldots,k_n)$, $k_j \in \mathbb Z_+= \mathbb N \cup \{0\}$, is a multiindex and $|\bk|=k_1+\cdots +k_n$ is its length, $\bk!=k_1!\cdots k_n!$. Given $z\in \mathbb C$, let us denote its real and imaginary parts by $\Re(z)$ and $\Im(z)$ respectively.



%For $\vecz \in \mathbb C^n$, $\bk \in \mathbb N^n$ and $\ell \in \mathbb N$, let us define the function $\vartheta_{\vecz,\bk,\ell}: \Sn \to \mathbb C^n$ by the following expression:
%\begin{equation*}
% \vartheta_{\vecz,\bk,\ell}(\vecx)= \vecx^\bk (\vecx \cdot \vecz)^\ell\;.
%\end{equation*}
%Note that the functions  $\vartheta_{\vecz,\bk,\ell}$ are harmonic and homogeneous of order $|\bf|+\ell$. 
%\subsection{The space $L^2(\Sn)$}



\section{A class of holomorphic spaces} \label{class of holomorphic spaces}
In this section, we will give a description of how to define the special Hilbert space $L^2(\mathbb C^n,\dm)$. 

\subsection{Construction of measure $\dm$.}\label{section construction measure}

Based on the expression of coherent states obtained for spaces $L^2(S^m)$, with $m=2,3,5$ and $S^m=\{\vecx \in \mathbb R^{m+1}\;|\; x_1^2+\ldots+x_{m+1}^2=1\}$ (See \cite{D-V09}  for more details). We consider the coherent states on $L^2(\Sn)$ as the functions
\begin{equation}\label{first definition coherent states}
\Phi_{\vecw}(\vecx)=\sum_{\ell=0}^\infty \frac{c_\ell}{\ell!} \left(\frac{\vecx\cdot \vecw}{\hbar}\right)^\ell\;,\hspace{0.5cm} \vecx\in \Sn\;, \vecw\in \mathbb C^n\;,
\end{equation}
where the constants $c_\ell$ are defined below, which will be obtained considering that $\langle \Phi_\vecz,\Phi_\vecw\rangle_{\Sn}$ is the reproducing kernel of some Hilbert space. 







From Lemma \ref{product in Sn}, we have
\begin{align*}
\langle\Phi_\vecz,\Phi_\vecw\rangle_{\Sn} & =\Gamma(n)\sum_{\ell=0}^\infty c_\ell^2 \frac{\displaystyle\left(\frac{\vecz\cdot\vecw}{\hbar^2}\right)^\ell}{\ell!\;\Gamma(\ell+n)}\;,
\end{align*}
where $\Gamma$ denote the Gamma function. If the constants $c_\ell$ take the following values
\begin{displaymath}
c_\ell^2=\frac{C\;\Gamma(n+\ell)}{\Gamma(n+p+\ell)\Gamma(n)}\;,\hspace{0.5cm} \mbox{with }p \in \mathbb R,
\end{displaymath}
and $C$ a constant, then we have
\begin{equation}\label{help 1}
\langle\Phi_\vecz,\Phi_\vecw\rangle_{\Sn}= C\left(\frac{\vecz\cdot\vecw}{\hbar^2}\right)^{\frac{1}{2}(1-p-n)} \mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecz\cdot\vecw}}{\hbar}\right)
\end{equation}
with $\mathrm I_k$ denoting the modified Bessel function of the first kind of order $k$ (see Secs. 8.4 and 8.5 of Ref. \cite{G94} for definition and expressions for this special function). We are taking the branch of the square root function such that $\sqrt z=|z|^{1/2}\exp(\imath \theta/2)$, where $\theta= \mathrm{Arg}(z)$ and $-\pi < \theta <  \pi$. 

The Aronszajn-Moore theorem \cite{A50} states that,  every positive definite (Hermitian) function $K$ on $\mathbb C^n \times \mathbb C^n$ determines a unique Hilbert function space $H$ for which $K$ is the reproducing kernel. 

In order to construct the Hilbert space $H$ for which the right side of  Eq. (\ref{help 1}) is its reproducing kernel, let $\phi(z)=C(z/\hbar)^{-\nu/2}\mathrm I_\nu(2\sqrt z/\hbar)=\sum_{\ell=0}^\infty b_\ell z^\ell$. Define the space $H_\phi$  as the set of  all holomorphic functions in $\mathbb C^n$ equipped with inner product
\begin{equation}\label{help inner product}
(f,g)_{H_\phi} = \sum_{\ell=0}^\infty \frac{1}{b_\ell}\sum_{|\bk|=\ell} \frac{\bk!}{\ell!}f_\bk \overline{g_\bk}\;,
\end{equation}
where $f_\bk$, $g_\bk$ are Taylor's coefficients of $f$ and $g$, respectively.

Note that the function $\phi(\vecz\cdot\vecw)$ belongs to $H_\phi$ as a function of $\vecz\in \mathbb C^n$ for every fixed $\vecw\in \mathbb C^n$, and for any $f\in H_\phi$
\begin{equation*}
(f,\phi((\cdot)\cdot \vecw))_{H_\phi}= \sum_{\ell=0}^\infty \frac{1}{b_\ell}\sum_{|\bk|=\ell} \frac{\bk!}{\ell!}f_\bk \overline{\frac{b_\ell \ell!}{\bk!} \overline{\vecw^\bk}}= \sum_{\ell=0}^\infty \frac{1}{b_\ell}\sum_{|\bk|=\ell} \frac{\bk!}{\ell!}f_\bk \vecw^\bk=f(\vecw)\;.
\end{equation*}
Thus, $\phi(\vecz\cdot \vecw)$ is the reproducing kernel of $H_\phi$. From  (\ref{help inner product}) we see that
\begin{equation}\label{measure 1}
(\vecz^\bk,\vecz^\bk)_{H_\phi} = \frac{1}{b_{|\bk|}} \frac{\bk!}{|\bk|!}\;.
\end{equation}
We now assume that the inner product (\ref{help inner product}) can be expressed by
\begin{equation}\label{measure 2}
(f,g)_{H_\phi}=\int_{\mathbb C^n} f(\vecz)\overline{g(\vecz)} \omega(|\vecz|) \dz \dzb\;.
\end{equation}
Substituting the value of $b_{|\bk|}$ in Eq. (\ref{measure 1}), using Eq.  (\ref{measure 2}), polar coordinates and Lemma \ref{product in Sn} we obtain
\begin{align*}
\frac{1}{C}\hbar^{2|\bk|} \bk! \Gamma(\nu+|\bk|+1) & = (\vecz^\bk,\vecz^\bk)_{H_\phi} \\
& = \frac{2\pi^n \bk!}{(n-1+|\bk|!)} \int_0^\infty r^{2n-1+2|\bk|} \omega(r) \mathrm dr\;.
\end{align*}
So, we need find $\omega(r)$ that satisfies the equation
\begin{equation} \label{measure 3}
\int_0^\infty r^{2n-1+2|\bk|} \omega(r) \mathrm dr=\frac{\hbar^{2|\bk|}}{2C\pi^n} \Gamma(\nu+|\bk|+1) \Gamma(n+|\bk|)\;.
\end{equation}
%then one can easily verify by direct computation that the inner product in $H_\phi$ indeed has the form (\ref{measure 2}). 
Note that expression (\ref{measure 3}) becomes essentially the Mellin transform (see Refs. \cite{G94}, \cite{P-B-M90}). From this simple observation and the formula 6.561-16  of Ref \cite{G94} we immediately obtain 
\begin{equation}\label{measure}
\omega(r)=\frac{2}{C(\pi\hbar^2)^n}\left( \frac{r}{\hbar}\right)^{\nu-n+1} \mathrm K_{\nu-n+1}\left(\frac{2}{\hbar}r\right)\;,\hspace{0.5cm} \nu>-1\;,
\end{equation}
with $\mathrm K_p$ denoting the MacDonal-Bessel functions of order $p$ (see Secs. 8.4 and 8.5 of Ref. \cite{G94} for definitions and expressions for this special functions). Taking the constant $C$ such that $(1,1)_{H_\phi}=1$ we obtain 
\begin{equation}\label{constant C}
C=\Gamma(\nu +1)\;.
\end{equation}




\subsection{The space $\E$}
In the previous section we constructed a Hilbert space whose inner product can be expressed in the form (\ref{measure 2})  and its reproducing kernel is the right side of Eq. (\ref{help 1}); in this section we proves rigorously all those results. From Eqs. (\ref{measure}) and (\ref{constant C}) let us consider the following measure $\dm$ on $\mathbb C^n$
\begin{equation}\label{measure on U}
 \dm(\vecz)=\frac{1}{\Gamma(n+p)}\frac{2}{(\pi\hbar^2)^n}\left(\frac{|\vecz|}{\hbar}\right)^p \mathrm K_p\left(2\frac{|\vecz|}{\hbar}\right) \dz \dzb \;, \hspace{0.5cm} p>-n
 %\frac{4}{\Gamma(n)\Gamma(n-p)}\frac{1}{\hbar^{2n-p}}|\vecz|^{-p} \mathrm K_p\left(2\frac{|\vecz|}{\hbar}\right) \dz \dzb 
\end{equation}
with $\vecz=(z_1,\ldots,z_n)\in \mathbb C^n$ and $\dz\dzb$ denoting Lebesgue measure on $\mathrm C^n$.

Note that the measure $\dm$ is invariant under the rotation group
$\mathrm{SO}(n)$ (the group of $n \times n$ orthogonal matrices with unit determinant and real entries). The action of $\mathrm{SO}(n)$ on $\mathbb C^n$
that we are considering is the natural extension of the usual action of $\mathrm{SO}(n)$ on $\mathbb R^n$, this is: for $R \in \mathrm {SO}(n)$ defining $\mathrm T_R: \mathbb C^n \to \mathbb C^n$ by $\mathrm T_R(\vecz)=R \;\Re (\vecz)+ \imath R\; \Im(\vecz)$.

Let us consider the Hilbert space $L^2(\mathbb C^n,\dm)$ of square integrable functions on $\mathbb C^n$ with respect to the measure $\dm$ and endowed with the inner product
\begin{equation}
 (f,g)_p = \int\limits_{\vecz \in \mathbb C^n} f(\vecz)\overline{g(\vecz)} \dm(\vecz)\;,\hspace{0.5cm} f,g \in L^2(\mathbb C^n , \dm)\;.
\end{equation}

Let us denote by $||f||_p= \sqrt{(f,f)_p}$ the corresponding norm of $f \in L^2(\mathbb C^n , \dm)$.



\begin{definition}
 For $p>-n$, the space of entire functions $f$ defined on $\mathbb C^n$ such that $||f||_p$ is finite is denoted by $\E$.
\end{definition}

\begin{remark}
For $n=1$, the spaces $\mathcal E_{1,p}$ were used by Karp Dmitrii to derive an analytic continuation formula for functions on $\mathbb R^+$ (see \cite{K03}).
\end{remark}

\begin{lemma}\label{product in Enp}
 Let 
 \begin{equation*}
  \mathrm L:=\int\limits_{\mathbb C^n} \vecz^{\bk} \overline\vecz^\bm (\vecz \cdot \vecw)^s (\vecu \cdot \vecz)^\ell \dm(\vecz)\;,
 \end{equation*}
 with $\ell,s \in \mathbb Z_+$, $\bk,\bm \in \mathbb Z_+^n$, and $\vecw,\vecu \in \mathbb C^n\backslash\{0\}$. Then $\mathrm L=0$ if $|\bk|+s\ne|\bm|+\ell$, and if $|\bk|+s=|\bm|+\ell$, then 
 \begin{equation}
  \mathrm L= \hbar^{2(|\bk|+s)}(n)_{|\bk|+s}(n+p)_{|\bk|+s} \int_{\Sn} \vecx^\bk \overline \vecx^\bm (\vecx \cdot \vecw)^s (\vecu \cdot \vecx)^\ell \ds(\vecx)\;.
 \end{equation} 
\end{lemma}


\begin{proof} From Eq. (\ref{measure on U}) and expressing L in polar coordinates 
\begin{equation*}
 \mathrm L= \frac{|\Sn|2\hbar^{-(p+2n)}}{\pi^n\Gamma(n+p)}\int\limits_0^\infty r^{2n-1+|\bm|+|\bk|+\ell+s+p} \mathrm K_p(2\frac{r}{\hbar}) \mathrm dr \int\limits_{\Sn} \vecx^\bk \overline \vecx^\bm (\vecx \cdot \vecw)^s (\vecu \cdot \vecx)^\ell \ds(\vecx)\;,
\end{equation*}
where $|\Sn|=\mathrm{vol}(\Sn)$. From the formula 6.561-16 of Ref \cite{G94} and Lemma \ref{product in Sn} we conclude the proof of Lemma \ref{product in Enp} 
\end{proof}


\begin{prop} Let  $p>-n$ 
 \begin{enumerate}
  \item[a) ] The space of analytic polynomials defined on $\mathbb C^n$ is dense on $\E$.
  
  \item[b) ] For $\ell \in \mathbb Z_+$, let us denote by $W_\ell$ the space of homogeneous polynomials of degree $\ell$. Then 
  \begin{equation}\label{direct sum E}
   \E = \bigoplus_{\ell=0}^\infty W_\ell\;. 
  \end{equation}
  Further, the set $\{\Phi_{\bk,p}^\hbar\;|\; \bk \in \mathbb Z_+^n\}$ is orthonormal basis of $\E$, where $\Phi_{\bk,p}^\hbar: \mathbb C^n \to \mathbb C$ is defining by
  \begin{equation}\label{Phi}
   \Phi_{\bk,p}^\hbar(\vecz)=\frac{1}{\hbar^{|\bk|}} \sqrt{\frac{1}{(n+p)_{|\bk|}\bk!}} \;\vecz^{\bk}\;,
  \end{equation}
where, for $b> 0$, the Pochhammer symbol is given by
 \begin{equation*}
  (b)_\ell=\frac{\Gamma(b+\ell)}{\Gamma(b)}\;.
 \end{equation*}

  
  \item[c) ] The space $\E$ enjoys the property of having a reproducing kernel $\T=\T(\vecz,\vecw)$. Namely, for all $f \in \E$ we have
  \begin{equation}\label{property kernel E}
   f(\vecz)=(f,\T(\cdot,\vecz))=\int\limits_{\vecw \in \mathbb C^n} f(\vecw) \T(\vecz,\vecw) \dm(\vecw)\;,\hspace{0.5cm} \forall\vecz \in \mathbb C^n
  \end{equation}
  where the reproducing kernel is given by
  \begin{equation}\label{kernel}
   \T(\vecz,\vecw)= \Gamma(n+p) \left(\frac{\vecz \cdot \vecw}{\hbar^2}\right)^{\frac{1}{2}(-p-n+1)} \mathrm I_{n+p-1} \left(\frac{2 \sqrt{\vecz \cdot \vecw}}{\hbar}\right).
  \end{equation}
%  with $\mathrm I_k$ denoting the modified Bessel function of the first kind of order $k$ (see Secs. 8.4 and 8.5 of Ref. \cite{G94} for definition and expressions for this special function). We are taking the branch of the square root function such that $\sqrt z=|z|^{1/2}\exp(\imath \theta/2)$, where $\theta= \mathrm{Arg}(z)$ and $-\pi < \theta <  \pi$. 
Even more 
  \begin{equation}\label{kernel series}
   \T(\vecz,\vecw)= \sum_{\ell=0}^\infty \mathbf T_{n,p}^\ell(\vecz,\vecw)\;,
  \end{equation}
  where $\mathbf T_{n,p}^\ell$ are the reproducing kernel of the subspace $W_\ell$
  \begin{displaymath}
   \mathbf T_{n,p}^\ell(\vecz,\vecw)=\frac{1}{\ell! (n+p)_\ell}\left(\frac{\vecz \cdot\vecw}{\hbar^2}\right)^\ell\;.
  \end{displaymath}


  
  %\item The series $f(\vecz)=\displaystyle \sum_{\bk \in \mathbb N^n} c_\bk \vecz^\bk$ belongs to $\E$ if and only if $\displaystyle \sum_{\bk \in \mathbb N^n}$ is finite.
    
 \end{enumerate}
\end{prop}
\begin{proof} a) Is a consequence of the Stone-Weiertrass theorem.

b) Note that the space $W_\ell$ is contained in $L^2(\mathbb C^n, \dm)$. Then by a), we only need prove that the spaces $W_\ell$ and  $W_{\ell'}$ are orthogonal for $\ell \ne \ell'$, 
which is a consequence of Lemma \ref{product in Enp}.
The second part of this point too is a direct consequence of Lemma \ref{product in Enp}, since
\begin{equation}
 \left\{\frac{1}{\hbar^{\ell}} \sqrt{\frac{1}{(n+p)_{\ell}\bk!}} \;\vecz^{\bk}\;|\; \bk \in \mathbb Z_+^n\;,\;|\bk|=\ell\right\}
\end{equation}
is orthogonal basis of $W_\ell$.

c) By Eq. (\ref{direct sum E}), it suffices to show that the function $\mathbf T_{n,p}^\ell(\vecz,\vecw)$ is the reproducing kernel in $W_\ell$. That will appear as a simple consequence of Lemma \ref{product in Enp}.  
%By construction $\T$ is the reproducing kernel of $\E$. Moreover, as a simple consequence of Lemma \ref{product in Enp} we have that the function $\mathbf T_{n,p}^\ell(\vecz,\vecw)$ is the reproducing kernel in $W_\ell$.
\end{proof}


As a consequence of the existence of the reproducing kernel and the Cauchy-Schwartz inequality, we obtain the following estimate for any $f \in \E$
\begin{displaymath}
 |f(\vecz)|\le ||f||_p\;||\T(\cdot,\vecz)||\;,\hspace{0.5cm} \forall \vecz \in \mathbb C^n\;.
\end{displaymath}



\section{Unitary transform} \label{space En}

In this section we introduce a unitary transform $\U$ from $\mathcal O$ onto Hilbert space $\E$, with $n \ge 2$ and $p > -n$. In order to define this transformation, we first recall some results about the space $\mathcal O$. %\subsection{Definition of the Unitary transformation for $\mathcal O$}

For $\ell \in \mathbb Z_+$, let us define the space $V_\ell$ of homogeneous polynomials of degree $\ell$ as the vector space of restrictions to the $n$-sphere of homogeneous polynomials of degree $\ell$ defined initially on the ambient space $\mathbb C^n$. We will use the fact that $\mathcal O$ is equal to the direct sum of the space $V_\ell$ (see \cite{R08})
\begin{equation}
 \mathcal O= \bigoplus_{\ell=0}^\infty V_\ell\;.
\end{equation} 
Further, the set $\left\{\phi_{\bk}\;|\; \bk\in \mathbb Z_+^n\right\}$ is orthogonal basis of $\mathcal O$, where $\phi_{\bk}:\Sn \to \mathbb C$ is defining by
\begin{equation}\label{phi}
 \phi_{\bk}(\vecx)= \sqrt{\frac{(n)_{|\bk|}}{\bk!}}
 %\sqrt{\frac{\Gamma(|\bk|+n)}{\Gamma(n)\bk!}}
 \;\vecx^\bk\;.
\end{equation}


For every $\ell \in \mathbb Z_+$, exist a natural transformation from $V_\ell$ to $W_\ell$, this is the linear extension of the assignment $\phi_{\bk} \to \Phi_{\bk,p}^\hbar$, where $\phi_{\bk}$, $\Phi_{\bk,p}^\hbar$ are defined in Eqs. (\ref{phi}) and (\ref{Phi}) respectively, and $\bk \in \mathbb Z_+^n$ with $|\bk|=\ell$. Let us define this unitary transform by $\mathbf U_{n,p}^\ell$.



We now consider the linear extension $\U:\mathcal O \to \E$ of the operators $\mathbf U_{n,p}^\ell$ defined as follows:

Given $\Psi \in \mathcal O$ written as $\Psi=\sum_{\ell=0}^\infty \Psi_\ell$ with $\Psi_\ell \in V_\ell$, we define the partial sums $S_k(\Psi)=\sum_{\ell=0}^k \mathbf U_{n,p}^\ell\Psi_\ell$ and then
\begin{equation}\label{definition U limits}
 \U \Psi := \lim_{k \to \infty} S_k(\Psi)\;.
\end{equation}

The operator $\U$ is well defined and unitary due to the unitarity of the operators $\mathbf U_{n,p}^\ell$ and the fact that every element $f$ in the space $\E$ can be written in a unique way as $f=\lim_{k\to \infty} \sum_{\ell=0}^k f_\ell$ with $f_\ell \in W_\ell$. (See Eq. (\ref{direct sum E})).


%\begin{definition}
%We define the transform $\U$ as the linear extension of the assignment $\phi_{\bk} \to \Phi_{\bk,p}^\hbar$.  
%\end{definition}


\begin{theorem}\label{unitary}
 The operator $\U$ mapping $\mathcal O$ isometrically onto the space $\E$. Even more
 \begin{equation}\label{expression U}
  \U\Psi(\vecz)= %\langle \Psi,\K(\cdot,\vecz)\rangle _{_{\Sn}} = 
  \int_{\Sn} 
  %\overline{\K(\vecx,\vecz)} 
  \Psi(\vecx) \sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecz \cdot \vecx}{\hbar}\right)^\ell\
  \ds(\vecx)\;,\hspace{0.5cm} \forall \Psi \in \mathcal O\;,\vecz \in\mathbb C^n\;,
 \end{equation}
 where
 %\begin{displaymath}
 %\K(\vecx,\vecz)=\sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecx \cdot \vecz}{\hbar}\right)^\ell\;,
 %\end{displaymath}
 %and
 \begin{equation}\label{constants coherent states}
  (c_{\ell,p})^2= \frac{(n)_\ell}{(n+p)_\ell}\;.
 \end{equation}
 
 %\begin{equation}\label{kernel Sn serie}
 %\K(\vecx,\vecz)= \sum_{\ell=0}^\infty \sum_{|\bk|=\ell} \phi_{\bk}(\vecx) \overline{\Phi_{\bk,p}^{\hbar}(\vecz)}\;,\hspace{0.5cm} \vecx \in \Sn\;,\; \vecz \in \mathbb C^n\;.
 %\end{equation}
\end{theorem}

\begin{proof}
Note that for $\vecz \in \mathbb C^n$ fixed and $\vecx \in \Sn$, the series 
\begin{equation}\label{coherent states}
 \K(\vecx,\vecz):=\sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecx \cdot \vecz}{\hbar}\right)^\ell\
\end{equation}
is bounded on $\Sn$ and therefore it is a vector in $L^2(\Sn)$ as well. Then, by the Cauchy-Schwartz inequality, the right side of Eq. (\ref{expression U}) is a well defined function. 

Let $\ell \in \mathbb Z_+$, is not difficult show that
 \begin{equation}\label{expresion Ul integral}
  \mathbf U_{n,p}^\ell \Phi(\vecz)=\int_{\Sn} \sum_{|\bk|=\ell} \overline{\phi_{\bk}(\vecx)}\Phi_{\bk,p}^{\hbar}(\vecz) \Phi(\vecx) \ds(\vecx)\;,\hspace{0.5cm} \vecz \in \mathbb C^n\;,\hspace{0.5cm} \Phi\in V_\ell\;.
 \end{equation}
 
Even more, by Eqs (\ref{phi}) and (\ref{Phi})
\begin{align}
 \sum_{|\bk|=\ell} \overline{\phi_{\bk}(\vecx)} \Phi_{\bk, p}^{\hbar}(\vecz) & = \sum_{|\bk|=\ell} 
 \frac{1}{\hbar^\ell}\sqrt{\frac{(n)_\ell}{\bk!}} \sqrt{\frac{1}{(n+p)_\ell\bk!}}\; \overline \vecx^\bk \vecz^\bk\nonumber\\
 & = \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecz \cdot \vecx}{\hbar}\right)^\ell\;. \label{kernel Ul}
\end{align}
%Then, from Eqs. (\ref{expresion Ul integral}) and (\ref{kernel Ul})
%\begin{equation}\label{expresion Ul integral 1}
 % \mathbf U_{n,p}^\ell \Phi(\vecz)=\int_{\Sn} \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecz \cdot \vecx}{\hbar}\right)^\ell \Phi(\vecx) \ds(\vecx)\;,\hspace{0.5cm} \vecz \in \mathbb C^n\;,\hspace{0.5cm} \Phi\in V_\ell\;.
 %\end{equation}
Then, given $\Psi \in \mathcal O$, written as $\Psi=\sum_{\ell=0}^\infty \Psi_\ell$ with $\Psi_\ell \in V_\ell$, we obtain the Eq. (\ref{expression U}) from definition of $\U$ (see Eq. (\ref{definition U limits})), the integral expression the operator $\mathbf U_{n,p}^\ell$ (see Eqs. (\ref{expresion Ul integral}) and (\ref{kernel Ul})) and the dominated converge theorem.\end{proof}
 
We end this section with an inversion formula for the operator $\U$.
%\subsection{The inversion formula}
\begin{theorem}\label{inverse}
 Let $f \in \E$, then
 %Let $\{E_q\;|\;q \in  \mathbb N\}$ be an increasing sequence of bounded measure set in $\mathbb C^n$ such that $\mathbb C^n=\bigcup_{q \in \mathbb N} E_q$, $n \ge 2$. Let $f \in \E$, then
 \begin{equation*}
  (\U)^{-1} f(\vecx)= \int_{\mathbb C^n}  \sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecx \cdot \vecz}{\hbar}\right)^\ell f(\vecz) \dm(\vecz)\;,\hspace{0.5cm}\vecx \in \Sn\;.
  %\lim_{q \to \infty} \int_{E_q} \sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecx \cdot \vecz}{\hbar}\right)^\ell f(\vecz) \dm(\vecz)\;,
 \end{equation*}
% where the limit in Eq. (\ref{inverse}) is in $L^2(\Sn)$ sense.
\end{theorem}

\begin{proof} Since the set $\{\phi_\bk\;|\;\bk\in \mathbb Z_+^n\}$ is orthonormal basis of $\mathcal O$, then the reproducing kernel,  $\nrs$, from $\mathcal O$ is
\begin{displaymath}
\nrs(\vecx,\vecy)=\sum_{\ell=0}^\infty\sum_{|\bk|=\ell} \phi_\bk(\vecx) \overline{\phi_\bk(\vecy)}=\sum_{\ell=0}^\infty \frac{(n)_\ell}{\ell!}(\vecx \cdot\vecy)^\ell = {}_2F_1(n,1;1;\vecx\cdot\vecy),
\end{displaymath}
 where ${}_2F_1$ denoting the Gauss hypergeometric function (see Secs. 9.1 of Ref.  \cite{G94} for definition and expression for this special function).
 
Even more, for $\vecy \in \Sn$ fixed, we obtain from Lemma \ref{product in Sn}
\begin{equation}\label{U reproducing kernel Sn}
\U \nrs(\cdot,\vecy)(\vecz) = \K(\vecz,\vecy)\;\hspace{0.5cm} \forall \vecz\in \mathbb C^n\;.
\end{equation} 

Moreover, from the reproducing property of $\nrs$, the unitarity of the transform $\U$ and the Eq. (\ref{U reproducing kernel Sn}), we have
\begin{align*}
(\U)^{-1}f(\vecx) & = \langle (\U)^{-1}f, \nrs(\cdot,\vecx)\rangle_{\Sn}\\
& = (f,\U\nrs(\cdot,\vecx))_p\\
& = (f,\K(\cdot,\vecx))_p 
\end{align*}
\end{proof}



 
 
\section{Coherent states for $\mathcal O$}\label{section coherent states}
 
 We consider the set of functions $\mathcal K=\{\K(\cdot,\vecz)\;|\; \vecz \in \mathbb C^n\} \subset \mathcal O$ defined by Eq. (\ref{coherent states}).
%Based on experience to define coherent states, as the complex conjugate of the integral kernel defining the unitary operator $\U$, we called the functions $\K(\cdot,\vecz)$, for every $\vecz \in \mathbb C^n$ (see Eq. (\ref{coherent states})), as coherent states on $\mathcal O$. 
In  this section shows that the functions in $\mathcal K$  satisfy the conditions to define a Berezin symbolic calculus on $\mathcal O$.

First notice that the unitary operator $\U$ is a coherent states transform because its action on a function $\Phi$, in $\mathcal O$, is a function in $\E$ whose evaluation in $\vecz\in \mathbb C^n$ is equal to the $L^2(\Sn)$-inner product of $\Phi$ with the coherent states labeled by $\vecz$. This is
\begin{equation}\label{U inner product coherent states}
\U\Phi(\vecz)= \langle \Phi,\K(\cdot,\vecz)\rangle _{{\Sn}}\;.
\end{equation}
% Even more, the action the unitary transformation $\U$ 
%\begin{remark}
%From Eq. (\ref{U inner product coherent states}) and the unitarity we have that the map $\Phi \to \langle \Phi,\K(\cdot,\vecz)\rangle _{{\Sn}}$ is an embedding from $\mathcal O$ into $L^2(\mathbb C^n,\dm)$.
%\end{remark}
\begin{remark} 
 From Eq. (\ref{U inner product coherent states}) and theorem \ref{inverse} is not difficult to prove
\begin{equation*}\label{resolution of the identity for Sn}
\Psi(\vecx)= \int_{\mathbb C^n} \langle \Psi,\K(\vecz)\rangle _{_{\Sn}} \K(\vecx,\vecz)\dm(\vecz)\;,\hspace{0.5cm} \vecx\in\Sn\;,\Psi\in \mathcal O\;.
\end{equation*}
Thus the system of coherent states provides a resolution of identity for $\mathcal O$, this is the so-called reproducing property of the coherent states. 
\end{remark}

% We begin by showing the relation between the system of coherent states and the reproducing kernel of the Hilbert space $\E$.
 
%Moreover, by Eqs.  (\ref{U inner product coherent states}), (\ref{first definition coherent states}) and (\ref{help 1}) we have the following
 
 \begin{theorem}\label{U coherent states}
  The unitary operator $\U$ applied to $\K(\cdot,\vecz)$ gives the reproducing kernel of the Hilbert space $\E$, $n \ge 2$, $p>-n$. Namely, for all $\vecz ,\vecw$ in $\mathbb C^n$ we have
  \begin{displaymath}
   [\U\K(\cdot,\vecz)](\vecw)=\T(\vecw,\vecz)\;.
  \end{displaymath}
 \end{theorem}
\begin{proof} From theorem \ref{unitary}, the dominated converge theorem and Lemma \ref{product in Sn} we obtain
\begin{align}
 [\U\K(\cdot,\vecz)](\vecw) & = \int_{\Sn} \K(\vecx,\vecz) \overline{\K(\vecx,\vecw)} \ds(\vecx)\nonumber\\
 & = \sum_{\ell=0}^\infty \left(\frac{c_{\ell,p}}{\ell!\hbar^{\ell}}\right)^2 \int_{\Sn}\left(\vecx\cdot\vecz\right)^\ell \left(\vecw\cdot\vecx\right)^\ell \ds(\vecx)\nonumber\\
 & = \sum_{\ell=0}^\infty \frac{1}{(n+p)_\ell}\frac{1}{\ell!} \left(\frac{\vecw\cdot \vecz}{\hbar^2}\right)^\ell .\nonumber
\end{align}
And this is the expression for the reproducing kernel $\T$ given in Eq. (\ref{kernel series}) as an infinity series.
\end{proof}


  
% The previous theorem may have shown using theorem \ref{unitary}, the dominated converge theorem, the Lemma \ref{product in Sn} and the expression for the reproducing kernel $\T$ given in Eq. (\ref{kernel series}) as an infinity series.

The proof of some theorems below uses an estimate of the norm of a coherent state $\K(\cdot,\vecz)$  which in turn is a consequence of the following estimate for the inner product of two coherent states:

\begin{prop}\label{proposition inner product coherent states}
 Let $\vecz,\vecw \in \mathbb C^n$ fixed. Assume $\vecz \cdot \vecw \ne 0$ and $|\mathrm{Arg}(\vecz \cdot \vecw)| <  \pi$. Then for $\hbar \to 0$
 \begin{align}
  \langle \K(\cdot,\vecz),\K(\cdot,\vecw)\rangle _{{\Sn}}= & \frac{\Gamma(n+p)}{2\sqrt \pi} \left(\frac{\vecw \cdot \vecz}{\hbar^2}\right)^{\frac{1}{2}(-p-n+\frac{1}{2})} \mathrm{e}^{\frac{2}{\hbar}\sqrt{\vecw \cdot \vecz}}\biggl[1 -\nonumber\\
& \left.\frac{(n+p-\frac{3}{2})(n+p-\frac{1}{2})}{4\sqrt{\vecw\cdot\vecz}}\hbar+\mathrm O(\hbar^2)\right]\;.\label{asymptotic inner product coherent states}
 \end{align}
\end{prop}

\begin{proof} Since the transform $\U$ is a unitary transformation, we obtain from theorem \ref{U coherent states}, the reproducing property of $\T$ (see Eq. (\ref{property kernel E})) and the expression for the reproducing kernel $\T$ given in Eq. (\ref{kernel}) that
\begin{align}
\langle \K(\cdot,\vecz),\K(\cdot,\vecw)\rangle _{{\Sn}} & =\T(\vecw,\vecz)\nonumber\\
& = \Gamma(n+p) \left(\frac{\vecw \cdot \vecz}{\hbar^2}\right)^{\frac{1}{2}(-p-n+1)}\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecw\cdot\vecz}}{\hbar}\right).\label{inner product coherent states}
\end{align}
The modified Bessel function $\mathrm I_\vartheta$, $\vartheta\in\mathbb R$, has the following asymptotic expression when $|\omega| \to \infty$ (see formula 8.451-5 of Ref. \cite{G94})
\begin{equation}\label{asymptotic expression modified Bessel function}
\mathrm I_\vartheta(\omega)=\frac{\mathrm e^\omega}{\sqrt{2\pi\omega}}\sum_{k=0}^\infty \frac{(-1)^k}{(2\omega)^k}\frac{\Gamma(\vartheta+k+\frac{1}{2})}{k!\Gamma(\vartheta-k+\frac{1}{2})}\;,\hspace{0.5cm} |\mathrm{Arg}(\omega)|<\frac{\pi}{2}\;.
\end{equation}
From Eqs. (\ref{inner product coherent states}) and (\ref{asymptotic expression modified Bessel function}) we conclude the proof of this proposition.
\end{proof}

\begin{remark} 
We are mainly interested in using proposition \ref{proposition inner product coherent states} for the cases $\vecz=\vecw$ (and then $\mathrm{Arg}(\vecz\cdot\vecw) = 0$) in this paper. The case when $| \mathrm{Arg}(\vecz\cdot\vecw)| = \pi$ requires the use of an asymptotic expression valid in a different region than the one we are considering in proposition \ref{proposition inner product coherent states}. Thus if we take the branch of the square root function given by $\sqrt z = |z|^{1/2} exp(\imath\mathrm{Arg(z)}/2)$ with $0 < \mathrm{Arg}(z) < 2\pi$ then by using formula 8.451-5 in Ref. \cite{G94} we obtain the asymptotic expression,
\begin{align}
 \left\langle \K(\cdot,\vecz),\K(\cdot,\vecw)\right\rangle _{_{\Sn}}= & \frac{\Gamma(n+p)}{2\sqrt \pi} \left(\frac{\vecw \cdot \vecz}{\hbar^2}\right)^{\frac{1}{2}(-p-n+\frac{1}{2})} \nonumber\\
 & \left[\mathrm{e}^{\frac{2}{\hbar}\sqrt{\vecw \cdot \vecz}}  + \mathrm{e}^{-\frac{2}{\hbar}\sqrt{\vecw \cdot \vecz}}\mathrm{e}^{\pi\imath (n+p-\frac{1}{2})}\right] [1 + \mathrm O(\hbar)]\;.\label{other expression inner product}
\end{align}
Note that both asymptotic expressions in Eqs. (\ref{asymptotic inner product coherent states}) and (\ref{other expression inner product}) coincide up to an error of the order $\mathrm O(\hbar^\infty)$ in the common region where they are valid.
\end{remark}





From the proposition \ref{proposition inner product coherent states}, we obtain an estimate for the norm of a given coherent states
\begin{prop}
For $\vecz \in \mathbb C^n-\{0\}$
\begin{align}
|| \K(\cdot,\vecz)||_{\Sn}^2 = \frac{\Gamma(n+p)}{2\sqrt \pi} &\left(\frac{|\vecz|}{\hbar}\right)^{-p-n+\frac{1}{2}} \mathrm{e}^{\frac{2}{\hbar}|\vecz|}\nonumber\\
& \left[1 -\frac{(n+p-\frac{3}{2})(n+p-\frac{1}{2})}{4|\vecz|}\hbar+\mathrm O(\hbar^2)\right]\;.\label{norm coherent states}
\end{align}
\end{prop}



%We end this section by showing that the family $\mathcal K$ satisfies the Parseval's identity on $\mathcal O$.
We end this section showing that the family of coherent states $\mathcal K$ is complete
\begin{prop}\label{supercomplete}
The family $\{\K(\cdot,\vecz)\;|\;\vecz \in \mathbb C^n\}$ form a complete system for $\mathcal O$.
\end{prop}
 
\begin{proof} Let $\Phi,\Psi \in \mathcal O$, since the transform $\U$ is a unitary transformation   %and Eq. (\ref{U inner product coherent states}) 
we have
\begin{align*}
\langle \Phi,\Psi\rangle _{{\Sn}} & = (\U\Phi,\U\Psi)_p\\
& = \int_{\mathbb C^n} \U\Phi(\vecz)\overline{\U\Psi(\vecz)} \dm(\vecz)\\
%& = \int_{\mathbb C^n}  (\U\Phi,\T(\cdot,\vecz)) (\T(\cdot,\vecz),\U\Psi) \dm(\vecz)\\
& = \int_{\mathbb C^n} \langle \Phi,\K(\cdot,\vecz)\rangle _{{\Sn}} \langle \K(\cdot,\vecz),\Psi) \rangle _{{\Sn}}\dm(\vecz)\;,
\end{align*}
 where we have used Eq. (\ref{U inner product coherent states}).
 \end{proof}






\section{Berezin symbolic calculus}\label{section Berezin symbolic calculus}
According to Berezin's theory (see Ref. \cite{B74}), from Theorem \ref{unitary}, Eq. (\ref{U inner product coherent states}) and the proposition \ref{supercomplete}, we may consider the following
\begin{definition}
The Berezin's symbol of a bounded linear operator $A$ with domain in $\mathcal O$ is defined, for every $\vecz \in \mathbb C^n$, by
\begin{equation}\label{definition Berezin transform}
\B( A)(\vecz)=\frac{\langle A\K(\cdot,\vecz),\K(\cdot,\vecz)\rangle_{{\Sn}}}{\langle\K(\cdot,\vecz),\K(\cdot,\vecz)\rangle_{{\Sn}}}\;.
\end{equation}
\end{definition}
From Eq. (\ref{inner product coherent states}), we have that 
\begin{equation*}
||\K(\cdot,\vecz)||_{\Sn}^2=\Gamma(n+p) \left(\frac{|\vecz|}{\hbar}\right)^{-p-n+1}\mathrm I_{n+p-1}\left(\frac{2|\vecz|}{\hbar}\right)>0,
\end{equation*}
 hence the coherent states are continuous, i.e. the map $\vecz \to |\K(\cdot,\vecz)|$ is continuous. Therefore, if $A:\mathcal O\to \mathcal O$ is a bounded operator, its Berezin's symbol, can be extended uniquely to a function defined on a neighbourhood of the diagonal in $\mathbb C^n \times \mathbb C^n$  in such a way that it is holomorphic in the first factor and anti-holomorphic in the second. In fact, such an extension is given explicitly by
\begin{equation}\label{extended covariant symbol}
\B(A)(\vecz,\vecw):=\frac{\langle A\K(\cdot,\vecz),\K(\cdot,\vecw)\rangle_{{\Sn}}}{\langle\K(\cdot,\vecz),\K(\cdot,\vecw)\rangle_{{\Sn}}}\;.
\end{equation}
\begin{remark}
By Eqs. (\ref{inner product coherent states}), the extended Berezin's symbol has singularities in the zeros of the modified Bessel function $\mathrm I_{n+p-1}(z)$, which are well known (see Ref. \cite{L65} Sec. 5.13) and where $\vecw\cdot\vecz=0$. Then the extended Berezin's symbol is defined on $\mathbb C^n \times \mathbb C^n \setminus \mathcal S$, with
\begin{equation*}
\mathcal S=\left\{(\vecw,\vecz)\in \mathbb C^n \times \mathbb C^n\;|\; \vecw \cdot\vecz=0 \mbox{ or }2\sqrt{\vecw\cdot\vecz}/\hbar=\imath \lambda\right\}
\end{equation*}
where $\lambda$ is a negative real  number that satisfies  $\mathrm I_{n+p-1}(\imath\lambda)=0$.
\end{remark}


We now give the rules for symbolic calculus

\begin{prop}\label{proposition rules for Berezin}
Let $A, B$ bounded linear operators with domain in $\mathcal O$. Then for $\vecz,\vecw \in \mathbb C^n$ and $\phi\in \mathcal O$ we have
\begin{align}
\B(\mathrm{Id})& = 1\;, \mbox{with $\mathrm{Id}$ the identity operator,}\nonumber\\
\B(A^*) (\vecz,\vecw) & = \overline{\B(A)(\vecw,\vecz)}\;,\nonumber\\
\B(AB)(\vecz,\vecw) & = \frac{2}{(\pi\hbar)^n\hbar}\int\limits_{\mathbb C^n} \B(B)(\vecz,\vecu) \B(A)(\vecu,\vecw) \left[\frac{\vecu\cdot \vecz\; \vecw\cdot \vecu}{\vecw\cdot\vecz}\right]^{\frac{1}{2}(-p-n+1)}\nonumber\\
 & \hspace{0.5cm} \frac{\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecu\cdot \vecz}}{\hbar}\right)\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecw\cdot \vecu}}{\hbar}\right)}{\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecw\cdot \vecz}}{\hbar}\right)} |\vecu|^{p} \mathrm K_{p}\left(\frac{2|\vecu|}{\hbar}\right) \mathrm d\vecu \mathrm d\overline\vecu\;,\label{Eq 3}\\
 \U(A\phi)(\vecz) & = \frac{2}{(\pi\hbar)^n\hbar}\int\limits_{\mathbb C^n} \U \phi(\vecw) \B(A)(\vecw,\vecz)  \big(\vecz \cdot \vecw\big)^{\frac{1}{2}(-p-n+1)}  \nonumber\\
 &\hspace{0.5cm}\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecz\cdot \vecw}}{\hbar}\right) |\vecw|^{p} \mathrm K_{p}\left(\frac{2|\vecw|}{\hbar}\right) \mathrm d\vecw \mathrm d\overline\vecw.\label{Eq 4}
\end{align}
\end{prop}
\begin{proof}
This is a direct consequence from the formulas in Ref. \cite{B74} (see Eqs. (\ref{Eq 1}),  (\ref{Eq 2})), and the Eqs. (\ref{U inner product coherent states}) and (\ref{inner product coherent states}).
%\textcolor{red}{The proof  duplicates the standard proof of the existence of a Bergman kernel function}
\end{proof}

 



\begin{corollary}
Let $A:L^2(\Sn)\to L^2(\Sn)$ be a bounded operator, and $A^\#$ the operator on $\E$ with Schwartz kernel the function $\mathcal K_A$ defined on $\mathbb C^n \times \mathbb C^n$, by 
\begin{displaymath}
\mathcal K_A(\vecz,\vecw):=\langle A\K(\cdot,\vecw),\K(\cdot,\vecz)\rangle_{_{\Sn}}\;.
\end{displaymath}
Then
\begin{equation}
A^\#f(\vecz):=\U A(\U)^{-1} f(\vecz)\;\hspace{0.5cm}\forall f\in \E\;,\;\vecz \in \mathbb C^n.
\end{equation}
\end{corollary}
\begin{proof} Let $f \in \E$, and $\phi=(\U)^{-1} f$. From Eqs. (\ref{Eq 4}),  (\ref{extended covariant symbol}) and (\ref{inner product coherent states})
\begin{align*}
\U A (\U)^{-1} f (\vecz)= \int\limits_{\mathbb C^n} f(\vecw) \mathcal K_A(\vecz,\vecw) \dm(\vecw)\,.
\end{align*}
\end{proof}



\subsection{Asymptotic expansion of the Berezin's symbol.}
In this section, we obtain asymptotic  expansion of Berezin's symbol of Toeplitz operator.

%\textcolor{red}{Pag 12 de Berezin-Toeplitz quantization and Berezin Transform. ApPLICATIONS OF THE ASYMPTOTICS OF THE BEREZIN TRANSFORM}


Let $\Pro:L^2(\Sn) \to \mathcal O$ the orthogonal projection and $A$ pseudo-differential operator  on $\Sn$ of order zero. The \textsl{Toeplitz operator} is defined as
\begin{displaymath}
\To_A = \Pro A \Pro\;.
\end{displaymath}
For $\psi \in C^\infty(\Sn)$, with $C^\infty(\Sn)$ be the algebra of complex-valued $C^\infty$ functions on $\Sn$, let $\mathbf M_\psi$ the operator of multiplication by $\psi$. For simplicity we use notation $\To_\phi:=\To_{\mathrm M_\phi}$.

%If $A=\mathbf M_\psi$, with $\psi \in C^\infty(\Sn)$, where $C^\infty(\Sn)$ be the algebra of complex-valued $C^\infty$-functions on $\Sn$, and $\mathbf M_\psi$ is the operator of multiplication by $\psi$. We use notation $\To_A:=\To_f$.


\begin{theorem}
Let  $\bk\in \mathbb Z_+^n$ be a multi-index. Then
\begin{align*}
\B(\To_{\vecx^\bk})(\vecz,\vecw) & = \frac{\left(\displaystyle\frac{\vecw\cdot\vecz}{\hbar^2}\right)^{\frac{1}{2}(n+p-1)}}{\mathrm I_{n+p-1}\left(2\sqrt{\vecw\cdot\vecz}/\hbar\right)}\left(\frac{\vecw}{\hbar}\right)^\bk  \sum_{\ell=0}^\infty \left(\frac{\vecw\cdot\vecz}{\hbar^2}\right)^\ell \frac{c_{\ell,p}}{c_{|\bk|+\ell,p}}\\
&\hspace{0.5cm} \frac{1}{\ell!\Gamma(|\bk|+\ell+n+p)}%\left[\frac{\Gamma(n+\ell)}{\Gamma(n+p+\ell)}\frac{1}{\Gamma(n+|\bk|+\ell)\Gamma(n+p+|\bk|+\ell)}\right]^{\frac{1}{2}}\;,
\end{align*}
where $c_{\ell,p}$ was define in Eq. (\ref{constants coherent states}).
\end{theorem}
\begin{proof} From Eq. (\ref{extended covariant symbol}) and properties the orthogonal projection
\begin{align*}
\B(\To_{\vecx^\bk})(\vecz,\vecw) = \frac{\langle \vecx^{\bk}\K(\cdot,\vecz),\K(\cdot,\vecz)\rangle_{{\Sn}}}{\langle\K(\cdot,\vecz),\K(\cdot,\vecz)\rangle_{{\Sn}}}\;.
\end{align*}
Using the dominated convergence theorem, the Lemma \ref{product in Sn} and  Eq. (\ref{inner product coherent states}) we conclude the proof of this theorem.
\end{proof}

\begin{remark}
In the particular case when $p=0$, we obtain from the last theorem
\begin{align*}
\mathfrak B_{\hbar,0}(\To_{\vecx^\bk})(\vecz,\vecw) & = \left(\frac{\vecw}{\sqrt{\vecw\cdot\vecz}}\right)^\bk \frac{\mathrm I_{n+|\bk|-1}\left(2\sqrt{\vecw\cdot\vecz}/\hbar\right)}{\mathrm I_{n-1}\left(2\sqrt{\vecw\cdot\vecz}/\hbar\right)}.
\end{align*}
\end{remark}


%\textcolor{red}{Ver ecuación (4.3) del articulo BEREZIN-TOEPLITZ QUANTIZATION AND BEREZIN SYMBOLS FOR ARBITRARY COMPACT Kahler  MANIFOLDS}


\begin{theorem}\label{theorem asymptotic Berezin transform}
Let $p=0$, for any $\vecz \ne 0$ and $\Phi$ a smooth function on $\Sn$, the Berezin symbol $\mathfrak B_{\hbar,0}$ associated to the Toeplitz operator $\To_\Phi$ has the asymptotic expansion
\begin{align}
\mathfrak B_{\hbar,0}\To_\Phi(\vecz)& =\Phi(\vecz/|\vecz|)+\mathrm O(\hbar)\;.
\label{asymptotic Berezin transform}
\end{align}
\end{theorem}

\begin{proof}
First we observe from Eq. (\ref{coherent states}), since $p=0$, that $\K(\vecx,\vecz)=e^{\vecx\cdot\vecz/\hbar}$. Let $\Phi\in C^\infty(\Sn)$, from Eq. (\ref{norm coherent states}) we have
\begin{align}
\mathfrak B_{\hbar,0} \To_\Phi (\vecz) & =\frac{2\sqrt \pi}{\Gamma(n)} \left(\frac{|\vecz|}{\hbar}\right)^{n-\frac{1}{2}}\int_{\Sn} \mathrm{exp}\left(\frac{2}{\hbar} \big[\Re (\vecx\cdot\vecz)-|\vecz|\big]\right) \Phi(\vecx)\nonumber\\[0.5cm] 
& \hspace{3cm}\left(1+ \mathrm{O}(\hbar)\right) \ds(\vecx)\;.\label{Berezin approximation 1}
%\Phi(\vecx(\vtheta)) \mathrm e^{\frac{1}{\hbar}} 
\end{align}
%where $\Re(\omega)$ denote the real  part of $\omega\in \mathbb C$.

We identify $\mathbb R^{2n}$ with $\mathbb C^n$ in the usual way: for $(x_1,\ldots,x_n)\in \Sn$, let $\vecy=(y_1,\ldots,y_{2n})$ with $x_j=y_{j}+\imath y_{n+j}$. Note that $\vecy \in S^{2n-1}=\{\mathbf a \in \mathbb R^{2n}\;|\; |\mathbf a|=1\}$. Then,  the argument of the exponential function in Eq. (\ref{Berezin approximation 1}) is 
\begin{displaymath}
\frac{2}{\hbar}\;\big[\vecy \cdot(\Re(\vecz),\Im(\vecz))-|\vecz|\big]\;.
\end{displaymath}
%where $x_j=y_{j}+\imath y_{n+j}$. Note that $\vecy \in S^{2n-1}=\{\mathbf a \in \mathbb R^{2n}\;|\; |\mathbf a|=1\}$.

In order to estimate (\ref{asymptotic Berezin transform}), we define
\begin{align}
\mathcal A &: = \left(\frac{|\vecz|}{\pi\hbar}\right)^{n-\frac{1}{2}} \int\limits_{S^{2n-1}}  \Psi(\vecy)\mathrm{exp}\left(\frac{2}{\hbar} \big[\vecy\cdot(\Re(\vecz),\Im(\vecz))-|\vecz|\big]\right) \mathrm d\Omega(\vecy)\label{Auxiliar A}
\end{align}
where $\mathrm d\Omega$ is the surface measure on $S^{2n-1}$ and $\Psi$ is a smooth function on $S^{2n-1}$.

Note that given $(\Re(\vecz),\Im(\vecz))\in \mathbb R^{2n}$, there exist a rotation $R$ in $\mathrm {SO}(2n)$ such that $(\Re(\vecz),\Im(\vecz))= rR \hat{\mathbf e}_1 $ with $r=|\vecz|$ and $\hat{\mathbf e}_1$ is canonical unit vector in $\mathbb R^{2n}$. Thus we have
\begin{align}
\mathcal A& = \left(\frac{r}{\pi\hbar}\right)^{n-\frac{1}{2}} \int\limits_{\boldsymbol \omega\in S^{2n-1}}  \Psi(R\boldsymbol \omega) \mathrm{exp}\left(\frac{\imath}{\hbar}f_{\vecz}(\boldsymbol \omega)\right)  \mathrm d\Omega(\boldsymbol \omega) \label{Auxiliar A-1}
\end{align}
where $f_{\vecz}(\boldsymbol \omega)=-2\imath r\big[\omega_1-1 \big]$.

Let us introduce spherical coordinates for the variables $(\omega_1,\ldots, \omega_{2n})\in S^{2n-1}$:
\begin{align*}
\omega_1 & = \sin(\theta_{2n-1}) \cdots \sin(\theta_2)\sin(\theta_1)\;,\\
\omega_2 & = \sin(\theta_{2n-1}) \cdots \sin(\theta_2)\cos(\theta_1)\;,\\
&\hspace{0.2cm}\vdots\\
\omega_{2n-1}&=\sin(\theta_{2n-1})\cos(\theta_{2n-2})\;,\\
\omega_{2n}&= \cos(\theta_{2n-1})
\end{align*}
with $\theta_1\in(-\pi,\pi)$, $\theta_2,\theta_3,\cdots,\theta_{2n-1}\in(0,\pi)$.  

The function $f_\vecz$ appearing in the argument of the exponential function in Eq. (\ref{Auxiliar A-1}) has a non-negative imaginary part and has only one critical point  (as a function of the angles) $\boldsymbol \theta_0$ which contributes to the asymptotic given by $\theta_j=\pi/2$, $j=1,\ldots,2n-1$. In addition, since
\begin{equation*}
\left.\frac{\partial^2 f_\vecz}{\partial \theta_\ell\partial \theta_j}\right|_{\vtheta=\boldsymbol \theta_0}=2\imath r \delta_{j\ell}\;,
\end{equation*}
with $\delta_{ij}$ denoting the Kronecker symbol, then the determinate of the Hessian matrix of the function $f_\vecz$ evaluated at the critical point is equal to $(2\imath r)^{2n-1}$.

From the stationary phase method (see Ref. \cite{H90}) we obtain
\begin{align}
\mathcal A & = \left[\Psi(R\boldsymbol \omega(\vtheta))\right]_{\vtheta=\vtheta_0}+  \mathrm O(\hbar)=\Psi((\Re(\vecz),\Im(\vecz))/|\vecz|)+\mathrm O(\hbar)\;.\label{auxiliar mathcal A}
\end{align}


For $\vecx \in \Sn$, we consider $\Psi(\Re(\vecx),\Im(\vecx))=\Phi(\vecx)$. Then from Eqs. (\ref{Berezin approximation 1}), (\ref{Auxiliar A}) and (\ref{auxiliar mathcal A}) we conclude the proof of this theorem.
\end{proof}

When $p = -1$ we can give an asymptotic expression of the coherent states (see Appendix \ref{AppendixB}). This result will give us as a consequence an asymptotic expression for Berezin symbol $\mathfrak B_{\hbar,-1}$ to the Toeplitz operator. 
\begin{theorem}\label{theorem asymptotic Berezin transform p=1}
Let $p=-1$, for any $\vecz \ne 0$ and $\Phi$ a smooth function on $\Sn$, the Berezin transform $\mathfrak B_{\hbar,-1}$ associated to the Toeplitz operator $\To_\Phi$ has the asymptotic expansion
\begin{align}
\mathfrak B_{\hbar,-1}\To_\Phi(\vecz)& =\Phi(\vecz/|\vecz|)+\mathrm O(\hbar)\;.\label{asymptotic Berezin transform p=1}
\end{align}
\end{theorem}
\begin{proof}
Let us define the following regions on $\Sn$, with the constant $C$ mentioned in the Lemma \ref{asymptotic p=1} taken greater than one
\begin{equation}
W=\left\{\vecx \in \Sn\;|\; C\frac{\Re(\vecx\cdot\vecz)}{|\vecz|}\ge 1\right\}\;\;\mbox{and}\;\; V=\Sn-W\;.
\end{equation}
Note that $\vecx \in W$, implies $|\Im(\vecx\cdot\vecz)|\le |\vecz|\le C\Re(\vecx\cdot\vecz)$ and therefore we can use the asymptotic expression of coherent states (see proposition \ref{asymptotic coherent states p=1}).

Let 
\begin{equation}
A=\int\limits_{\vecx\in W} \Phi(\vecx) \frac{|\Ku (\vecx,\vecz)|^2}{||\Ku(\cdot,\vecz)||^2_{\Sn}} \ds(\vecx)\; ,\hspace{0.2cm} B=\int\limits_{\vecx\in V} \Phi(\vecx) \frac{|\Ku (\vecx,\vecz)|^2}{||\Ku(\cdot,\vecz)||^2_{\Sn}} \ds(\vecx).
\end{equation}
We affirm that the integral on V is actually $\mathrm O(h^\infty)$. To try this, from Lemma 10.1 of Ref \cite{D-V09} (specifically Eq. 10.4), the function $\Ku(\cdot,\vecz)$ has a integral expression given by
\begin{align}
\Ku(\vecx,\vecz) & = \frac{2\sqrt{n-1}}{\pi}e^{\vecx\cdot\vecz/\hbar}\int_0^1\left[\frac{\vecx\cdot\vecz}{(n-1)\hbar}(1-\omega^2)+1\right]e^{-\omega^2\vecx\cdot\vecz/\hbar}\nonumber\\
& \hspace{2cm} \frac{(1-\omega^2)^{n-2}\omega}{\sqrt{-\ln(1-\omega^2)}}\;\mathrm d\omega.\nonumber
\end{align}
From the last equation we obtain
\begin{align*}
|\Ku(\vecx,\vecz)| & \le C_1e^{\frac{|\vecz|}{\hbar}}\int\limits_0^1\left[\frac{|\vecz|}{(n-1)\hbar}+1\right]\mathrm {exp}\left(\frac{|\vecz|}{\hbar}\left[\frac{\Re(\vecx\cdot\vecz)}{|\vecz|}(1-\omega^2)-1\right]\right)\;\mathrm d\omega\nonumber
\end{align*}
for some constant $C_1$. Note that, for $\vecx \in V$
\begin{equation*}
\frac{\Re(\vecx\cdot\vecz)}{|\vecz|}(1-\omega^2)-1\le \mu \;,\mbox{ with } \mu=\frac{1}{C}-1<0\;.
\end{equation*}
Thus we get the estimate for $\vecx\in V$
\begin{align*}
|\Ku(\vecx,\vecz)| & \le C_1e^{\frac{|\vecz|}{\hbar}}e^{\mu\frac{|\vecz|}{\hbar}}\left[\frac{|\vecz|}{(n-1)\hbar}+1\right].
\end{align*}
From the norm estimate for the coherent states in Eq. (\ref{norm coherent states}) we obtain
\begin{equation*}
\frac{|\Ku(\vecx,\vecz)|^2}{||\Ku(\cdot,\vecz)||^2_{\Sn}} \le C_1 \left(\frac{|\vecz|}{\hbar}\right)^{n+\frac{3}{2}} e^{2\mu\frac{|\vecz|}{\hbar}}\left[\frac{|\vecz|}{(n-1)\hbar}+1\right]^2(1+\mathrm O(\hbar))\;.
\end{equation*}
Hence $B=\mathrm{O}(\hbar^\infty)$. On the other hand, from proposition \ref{asymptotic coherent states p=1} and Eq. (\ref{norm coherent states}) we have
\begin{align}
A & =\frac{2\sqrt \pi}{\Gamma(n)} \left(\frac{|\vecz|}{\hbar}\right)^{n-\frac{1}{2}}\int\limits_W  \frac{|\vecx\cdot\vecz|}{|\vecz|}\Phi(\vecx) \mathrm{exp}\left(\frac{2|\vecz|}{\hbar}\left[\frac{\Re(\vecx\cdot\vecz)}{|\vecz|}-1\right]\right)\nonumber\\
&\hspace{2cm}\left[1+\mathrm O(\hbar)\right]\ds\;. \label{A p=1}
\end{align}
Note that, for $\vecx \in V$, $\frac{\Re(\vecx\cdot\vecz)}{|\vecz|}-1<\frac{1}{C}-1<0$.  Thus, we can take the integral defining $A$ in Eq. (\ref{A p=1}) over the whole sphere with error $\mathrm O(\hbar^\infty)$.
From Eq. (\ref{Auxiliar A}), considering 
\begin{equation}
 \Psi(\Re(\vecx),\Im(\vecx))=\Phi(\vecx)\frac{|\vecx\cdot\vecz|}{|\vecz|}\;, \hspace{0.5cm} \vecx \in \Sn
\end{equation} 
 and (\ref{auxiliar mathcal A}) we conclude the proof of theorem.
\end{proof}

%We end this section with a simply application of the asymptotic of the Berezin's symbol obtain in the Theorems \ref{theorem asymptotic Berezin transform} and \ref{theorem asymptotic Berezin transform p=1}. First note that 









\section{The star product}\label{section star product}
%\textcolor{red}{Function theory in the unit ball. p 2, 13}
In Ref. \cite{B74}, Berezin show that the formula (\ref{Eq 3}) will allow us to define a start product (see Refs. \cite{BF78} for the standard definition of star-product) on  the algebra $\A$ out of Berezin's symbol for bounded operators with domain in $\mathcal O$ (see Eq. (\ref{definition Berezin transform})), i.e.
\begin{displaymath}
\A=\{\B(A)\;|\; A\in \mathbf B (\mathcal O)\}.
\end{displaymath}
In this section we verifies that this noncommutative star-product, which will be denoted by $*_p$, satisfies the usual requirement on the semiclassical limit, i.e. as $\hbar\to 0$
\begin{equation*}
f_1*_p f_2(\vecz)=f_1(\vecz)f_2(\vecz)+\hbar B(f_1,f_2)(\vecz)+\mathrm O(\hbar^2),\hspace{0.5cm}\vecz\in \mathbb C^n, f_1,f_2\in \A, 
\end{equation*}
where $B(\cdot,\cdot)$ is a certain bidifferential operator of first order.


\begin{theorem}\label{theorem asymptotic expansion} Let $n\ge 2$, $\vecz\in \mathbb C^n$, $\beta$ a smooth function defined on $\mathbb C^n$, $p> -n$ and $\mu,\nu \in \mathbb R$ with $\mu,\nu>\frac{1}{2}$. Then for $\hbar \to 0$
\begin{align}
\mathbf I (\vecz) &: = \int_{\mathbb C^n} \beta(\vecw) \frac{\F(\nu,\vecw\cdot\vecz/\hbar^2) \F(\mu,\vecz\cdot\vecw/\hbar^2)}{\F(\mu,|\vecz|^2/\hbar^2)}\dm(\vecw)\nonumber\\
& = \left(\frac{|\vecz|}{\hbar}\right)^{n+p-\nu}\frac{\Gamma(\nu)}{\Gamma(n+p)}
\left\{ \beta(\vecz) + \hbar\left(\frac{1}{4|\vecz|} (p-\nu+1)(p+\nu-1)\beta(\vecz)\right.\right.\nonumber\\ 
&\hspace{1cm}+ \frac{R}{2}\beta(\vecz)+g^{\bar j i}(\vecz)\left[\frac{1}{2|\vecz|^2}\big[\overline z_i(n+p-\nu)\partial_{\bar j} + z_j(n+p-\mu)\partial_{i}\big]\beta(\vecw)\right.\nonumber\\
& \hspace{1cm}\left. \left.\left.+\partial_{i}\partial_{\bar j} \beta(\vecw)+\frac{\beta(\vecz)}{|\vecz|^{n+p-\nu}} \partial_{\bar j}\partial_i \left(\frac{\xi_\vecz^{\nu,\mu}}{g}(\vecw)\right)\right]_{\vecw=\vecz} \right)\right\} + \mathrm O(\hbar^2) \label{asymptotic expansion} 
\end{align}
 with $\F$ denoting the generalized hypergeometric function (see Sec.9.14 of Ref. \cite{L65} for definition and expressions for this special function),  $\partial_i=\partial/\partial w_i$ and $\partial_{\bar j}=\partial/\partial \overline w_j$. Further, where $R$ is the scalar curvature defined by the K\"ahler metric $\mathrm d s^2 =g_{i  \bar j} \mathrm d \overline z_j \wedge \mathrm d z_i$, and $g_{i\bar j}, g^{i\bar j}, \xi_\vecz^{\nu,\mu}$, $g$ are functions described subsequently.
\end{theorem}

\begin{proof} 
Let us define the following two regions on $\mathbb C^n$
\begin{equation}
W=\left\{\vecw \in \mathbb C^n\;|\; \frac{\Re\sqrt{\vecw\cdot\vecz}}{|\vecz|}\ge \frac{1}{4}\right\}\;\;\mbox{and}\;\; V=\mathbb C^n-W\;.
\end{equation}
The integral in Eq. (\ref{asymptotic expansion}) can be written as an integral on the region $W$ plus an integral on $V$ denoted by the letters $\mathbf J_W$ and $\mathbf J_V$ respectively. The integral on $W$ is the one giving us the main asymptotic and the integral on $V$ is actually $\mathrm O(\hbar^\infty)$. In order to estimate $\mathbf J_V$, from the equality 
\begin{equation}\label{equality bessel = hypergeometric}
\Gamma(\nu +1)\left(\frac{1}{2}z\right)^{-\nu}\mathrm I_\nu(z)=\F(\nu +1,z^2/4)
\end{equation}
 (see formula 9.6.47 of Ref. \cite{A-S72} for details), and the integral representation of the function $\mathrm I_\lambda$ (see formula 8.431-1 of Ref. \cite{G94})
\begin{equation*}
\mathrm I_\lambda(z)=\frac{\left(\frac{z}{2}\right)^\lambda}{\Gamma\left(\lambda+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}\int_{-1}^1(1-t^2)^{\lambda-\frac{1}{2}}e^{\pm zt}\mathrm dt\;,\hspace{0.5cm} \Re\left(\lambda\right) > -\frac{1}{2}
\end{equation*}
we have
\begin{align*}
\left|\F\left(\nu,\frac{\vecw\cdot\vecz}{\hbar}\right)\right| &  = C \left|\int_{-1}^1 (1-t^2)^{\nu-\frac{3}{2}}e^{\frac{2}{\hbar}\sqrt{\vecw\cdot\vecz}\;t} \mathrm dt\right|\\
& \le C e^{\frac{1}{\hbar}|\vecz|} \int_{-1}^1(1-t^2)^{\nu-\frac{3}{2}}\mathrm{exp}\left({\frac{|\vecz|}{\hbar}\left[2\frac{\Re\sqrt{\vecw\cdot\vecz}}{|\vecz|}-1\right]}\right)\mathrm dt\:.
\end{align*}
for some constant $C$. Note that, for $\vecw \in V$,
\begin{equation}\label{points in V}
2\frac{\Re\sqrt{\vecw\cdot\vecz}}{|\vecz|}-1<-\frac{1}{2}\;.
\end{equation}
Thus we get the estimate for $\vecw\in V$
\begin{align*}
\left|\F\left(\nu,\frac{\vecw\cdot\vecz}{\hbar}\right)\right| & \le C e^{\frac{1}{\hbar}|\vecz|} e^{-\frac{1}{2\hbar}|\vecz|}\int_{-1}^1(1-t^2)^{\nu-\frac{3}{2}}\mathrm dt\\
& = C \mathrm B\left(\frac{1}{2},\nu-\frac{1}{2}\right) e^{\frac{1}{\hbar}|\vecz|} e^{-\frac{1}{2\hbar}|\vecz|}\;,
\end{align*}
where $\mathrm B(x,y)$ denoting the beta function (see 8.38 of Ref. \cite{G94} for definition and expression for this special function). From the equality $\F(\mu,|\vecz|^/\hbar^2)=\Gamma(\mu) |\vecz|^{-\mu+1}\mathrm I_{\mu-1}(2|\vecz|/\hbar)$ (see Eq. (\ref{equality bessel = hypergeometric})) and Eq. (\ref{asymptotic expression modified Bessel function}) we obtain for $\vecw\in V$
\begin{equation}
 \frac{|\F(\nu,\vecw\cdot\vecz/\hbar^2) |\;|\F(\mu,\vecz\cdot\vecw/\hbar^2)|}{\F(\mu,|\vecz|^2/\hbar^2)} \le C\left(\frac{|\vecz|}{\hbar}\right)^{\mu-\frac{1}{2}} e^{-\frac{1}{\hbar}|\vecz|}\left(1+\mathrm O(\hbar)\right)\;.
 % \frac{Ce^{\frac{2|\vecz|}{\hbar}}e^{\frac{2\eta}{\hbar}|\vecz|}}{\left(\frac{|\vecz|}{\hbar}\right)^{-\mu+1}e^{\frac{2|\vecz|}{\hbar}}\left(\right)}\;.
\end{equation}
Hence $\mathbf J_V=\mathrm O(\hbar^\infty)$. Let us now study the term $\mathbf J_W$, 
\begin{comment}
first we note that for $\vecw\in W$, $\mathrm{Arg}(\vecw\cdot\vecz)<\pi$. So from Eqs. (\ref{equality bessel = hypergeometric}) and (\ref{asymptotic expression modified Bessel function}) 
\begin{align}
\F\left(\nu,\frac{\vecw\cdot\vecz}{\hbar}\right) &= \Gamma(\nu)\frac{(\vecw\cdot\vecz)^{-\frac{\nu}{2}+\frac{1}{4}}}{2\sqrt \pi \hbar^{-\nu+\frac{1}{2}}} \mathrm{exp}\left(\frac{2}{\hbar}\sqrt{\vecw\cdot\vecz}\right)\biggl[1+\nonumber\\
&\hspace{0.5cm}\left.\frac{\hbar}{16}\frac{1-4(\nu-1)^2}{\sqrt{\vecw\cdot\vecz}} + E_1(\vecz,\hbar)\right]\;, \hspace{1.5cm} \vecw\in W\label{error1}
\end{align}
where the error term $E_1(\vecz,\hbar)$ in Eq. (\ref{error1}) is $\mathrm O(\hbar^2)$ uniformly with respect to $\vecw \in W$ because in such a region we have $\frac{1}{|\sqrt{\vecw\cdot\vecz}|}\le\frac{4}{|\vecz|}$. 
Furthermore from the asymptotic expression of the MacDonal-Bessel function of order $p$, $\mathrm K_p$(see formula  9.7.2 of Ref. \cite{A-S72}) we have
\begin{align}
\mathrm K_p\left(2\frac{|\vecw|}{\hbar}\right) &= 
\end{align}
\end{comment}
first we note that for $\vecw\in W$, $\mathrm{Arg}(\vecw\cdot\vecz)<\pi$. So from Eqs. (\ref{equality bessel = hypergeometric}), (\ref{asymptotic expression modified Bessel function}), the expression for the measure $\dm $ (see Eq. (\ref{measure on U})) and the asymptotic expression of the MacDonal-Bessel function of order $p$, $\mathrm K_p$(see formula  9.7.2 of Ref. \cite{A-S72}) we find that
\begin{align*}
\mathbf J_W (\vecz)  &=\int_{W} \beta(\vecw) \frac{\F(\nu,\vecw\cdot\vecz/\hbar^2) \F(\mu,\vecz\cdot\vecw/\hbar^2)}{\F(\mu,|\vecz|^2/\hbar^2)}\dm(\vecw)\nonumber\\
%\vecw|^{-p} \mathrm K_p\left(2\frac{|\vecw|}{\hbar}\right)\\
& = \frac{1}{2\pi^2}\frac{\hbar^\nu}{\hbar^{2n+p}}\frac{\Gamma(\nu)}{\Gamma(n+p)}\int_{W}\beta(\vecw) \xi_{\vecz}^{\nu,\mu}(\vecw) \chi_\vecz(\vecw)\mathrm{exp} \left(\frac{1}{\hbar} f_\vecz(\vecw)\right) \mathrm d\vecw \mathrm d \overline\vecw
\end{align*}
with
\begin{align}
\xi_{\vecz}^{\nu,\mu}(\vecw) & = |\vecz|^{\mu-\frac{1}{2}} |\vecw|^{p-\frac{1}{2}}(\vecz\cdot\vecw)^{-\frac{\mu}{2}+\frac{1}{4}}(\vecw\cdot\vecz)^{-\frac{\nu}{2}+\frac{1}{4}},\nonumber\\ 
\chi_\vecz(\vecw) & = 1 + \frac{\hbar}{16}\left[\frac{1-4(\mu-1)^2}{\sqrt{\vecz\cdot\vecw}} + \frac{1-4(\nu-1)^2}{\sqrt{\vecw\cdot\vecz}} +\frac{4p^2-1}{|\vecw|} \right.\nonumber\\
& \hspace{0.5cm}\left.+ \frac{4(\mu-1)^2-1}{|\vecz|})\right] + E(\vecz,\hbar)\;,\label{error}\\
f_\vecz(\vecw) & = 2\left(\sqrt{\vecz\cdot \vecw}+\sqrt{\vecw\cdot\vecz}-|\vecw|-|\vecz|\right) =2\left(2\Re(\sqrt{\vecz\cdot\vecw})-|\vecz|-|\vecw|\right),\nonumber
\end{align}
where the error term $E(\vecz,\hbar)$ in (\ref{error}) is $\mathrm O(\hbar^2)$ uniformly with respect to $\vecw \in W$ because in such a region we have $\frac{1}{|\sqrt{\vecw\cdot\vecz}|}\le\frac{4}{|\vecz|}$ and $\frac{1}{|\vecw|}\le \frac{4^4}{|\vecz|^3}$. 

%where we have use the asymptotic expression, when $\vartheta \to 0$, of the MacDonal-Bessel function of order $p$, $\mathrm K_p$ (see formula 9.7.2 of Ref. \cite{A-S72}).
Furthermore, from Eq. (\ref{points in V}) $f_\vecz(\vecw)\le -|\vecz| -|\vecw|<0$ for $\vecw\in V$. Thus we can take the integral defining $\mathbf J_W$ over the whole space $\mathbb C^n$ with an error $\mathrm O(\hbar^\infty)$.

Let us now write $ \mathbf J_W (\vecz)=\mathrm B(\vecz)+\hbar\; \mathrm C(\vecz) + \mathrm O(\hbar^2)$ with
\begin{align}
\mathrm B(\vecz) & = \frac{1}{2\pi^2}\frac{\hbar^\nu}{\hbar^{2n+p}}\frac{\Gamma(\nu)}{\Gamma(n+p)} \int\limits_{\mathbb C^n} \beta(\vecw) \xi_\vecz^{\nu,\mu}(\vecw) \mathrm{exp}{ \left(\frac{1}{\hbar} f_\vecz(\vecw)\right)} \mathrm d\vecw \mathrm d \overline\vecw,\label{B}\\
\intertext{and}
\mathrm C(\vecz)& = \frac{1}{32\pi^2}\frac{\hbar^\nu}{\hbar^{2n+p}}\frac{\Gamma(\nu)}{\Gamma(n+p)} \int\limits_{\mathbb C^n} \beta(\vecw)\xi_\vecz^{\nu,\mu}(\vecw) \left[
\frac{1-4(\mu-1)^2}{\sqrt{\vecz\cdot\vecw}} + \frac{1-4(\nu-1)^2}{\sqrt{\vecw\cdot\vecz}}\right.\nonumber\\
%& (n-p-\frac{3}{2})(n-p-\frac{1}{2})\right.\nonumber\\
&\hspace{1cm}\left.+\frac{4p^2-1}{|\vecw|} + \frac{4(\mu-1)^2-1}{|\vecz|}) \right] \mathrm{exp} \left(\frac{1}{\hbar} f_\vecz(\vecw)\right) \mathrm d\vecw \mathrm d \overline\vecw.\label{C}
\end{align}


For obtain the asymptotic expansion (\ref{asymptotic expansion}), we express  the integrals $\mathrm B(\vecz)$ and $\mathrm C(\vecz)$, using the stationary phase method (see Ref \cite{H90}), as $\mathrm B_1(\vecz) + \hbar \mathrm B_2(\vecz) + \mathrm O(\hbar^2)$, $\mathrm C_1(\vecz)+\mathrm O(\hbar)$ respectively.

%Then we may use the stationary phase method (see Ref \cite{H90}) in the integrals $\mathrm B(\vecz)$ and $\mathrm C(\vecz)$ to get the asymptotic expansion (\ref{asymptotic expansion star product}). 

For our purpose, note that, using Schwartz inequality,  the phase function $f_\vecz$ satisfies $f_\vecz(\vecw)\le -2(\sqrt{|\vecz|}-\sqrt{|\vecw|})^2\le 0$. Moreover $f_\vecz(\vecw)=0$ if and only if $\vecw=\mathrm e^{\imath \theta}\vecz$ for some $\theta\in [0,2\pi)$ and $\Im(\sqrt{\vecz\cdot\vecw})=0$. Therefore, $f_\vecz(\vecw)=0$ if and only if $\vecw=\vecz$. Thus $f_\vecz$ is smooth function on a neighbourhood of the critical point $\vecw=\vecz$. %Thus for $\vecw\in V$, $f_\vecz(\vecw)<0$
We also have
\begin{displaymath}
-\partial_{\bar j}\partial_i f_\vecz(\vecw)=
%-\frac{\partial^2f_\vecz}{\partial \overline w_j \partial w_i} (\vecw)= 
\frac{1}{|\vecw|}\left(\delta_{i,j}-\frac{w_j\overline w_i}{2|\vecw|^2}\right).
\end{displaymath}
Using that  fact that for any complex numbers $a_i, b_i$, $i=1,\ldots,n$,  $\mathrm{det} (\delta_{i,j}\pm a_ib_j)=1\pm \sum_\ell a_\ell b_\ell$, we obtain %the determinant of the Hessian matrix of the function $f_\vecz$ ($\mathrm{det} f_\vecz ''$) is
\begin{equation}\label{Hessian f_z}
\mathrm{det}(-\partial_{\bar j}\partial_i f_\vecz(\vecw))=
%\mathrm{det}\left[-\frac{\partial^2f_\vecz}{\partial \overline w_j \partial w_i} (\vecw)\right]=%\frac{1}{|\vecw|^n}\left( 1-\sum_{\ell=1}^n \frac{\overline w_\ell w_\ell}{2|\vecw|^2}\right)= 
\frac{1}{2|\vecw|^n}\;.
\end{equation}

From the stationary phase method, applied to the integral $\mathrm C(\vecz)$, we deduce
\begin{equation}\label{C1}
\mathrm C_1(\vecz)=\left(\frac{|\vecz|}{\hbar}\right)^{n+p-\nu}\frac{\Gamma(\nu)}{\Gamma(n+p)}\frac{1}{4|\vecz|} (p-\nu+1)(p+\nu-1)\beta(\vecz)\;.
\end{equation}

Furthermore, the computation of $\mathrm B_1(\vecz)$ and $\mathrm B_2(\vecz)$ in Eq. (\ref{B}) 
it is using theorem 3 in Ref. \cite{E00}. For this we consider the K\"{a}hler potential $\Phi(\vecw)=2|\vecw|$, and let $\Phi(\vecw,\vecu)=2\sqrt{\vecw\cdot\vecu}$ be a sesqui-analytic extension of $\Phi(\vecw)=\Phi(\vecw,\vecw)$  to a neighbourhood of the diagonal.

From it, we have
\begin{displaymath}
f_\vecz(\vecw)= \Phi(\vecz,\vecw)+\Phi(\vecw,\vecz)-\Phi(\vecw,\vecw)-\Phi(\vecz,\vecz)\;,
\end{displaymath}
and 
\begin{align*}
g(\vecw) & =\mathrm{det}\left[g_{i\overline j}(\vecw)\right],\\
g_{i\overline j} & =\partial_{\bar j}\partial_i \Phi=-\partial_{\bar j}\partial_i f_\vecz(\vecw).
%\frac{\partial^2\Phi}{\partial \overline w_j \partial w_i} =-\frac{\partial^2f_\vecz}{\partial \overline w_j \partial w_i}\;.
\end{align*}

From theorem 3 in Ref. \cite{E00} and Eq. (\ref{Hessian f_z})
\begin{align}
\mathrm B_1(\vecz) & = \frac{1}{2\pi^n}\frac{\hbar^\nu}{\hbar^{2n+p}}\frac{\Gamma(\nu)}{\Gamma(n+p)}\left[\frac{(\pi\hbar)^n}{g(\vecz)}\right]|\vecz|^{n+p-\nu}\beta(\vecz)\nonumber\\
& =\left(\frac{|\vecz|}{\hbar}\right)^{n+p-\nu}\frac{\Gamma(\nu)}{\Gamma(n+p)}\beta(\vecz)\; \label{B1}
\end{align}
and 
\begin{align}
\mathrm B_2(\vecz) & = \frac{1}{2\pi^n}\frac{\hbar^\nu}{\hbar^{2n+p}}\frac{\Gamma(\nu)}{\Gamma(n+p)}(\pi \hbar)^n\left[\mathrm L_1\left(\left.\frac{\beta(\vecw)}{g(\vecw)}\xi_\vecz^{\nu,\mu}(\vecw)\right)\right|_{\vecw=\vecz}+\frac{R}{2}\frac{\beta(\vecz)}{g(\vecz)}|\vecz|^{p-\nu}\right]\nonumber\\
\intertext{where $\mathrm L_1:= g^{\bar j i} \partial_i\partial_{\bar j} $ is the Laplace-Beltrami operator, with $g^{i\bar j}$ the coefficients of the inverse matrix of $[g_{i\bar j}(\vecw)]$, and $R:=\mathrm L_1(\log g)$ is the scalar curvature. Then}
&  = \left(\frac{|\vecz|}{\hbar}\right)^{n+p-\nu}\frac{\Gamma(\nu)}{\Gamma(n+p)}\left\{g^{\bar j i}(\vecz)\left[\partial_{i}\partial_{\bar j} \beta(\vecw)+\frac{\beta(\vecz)}{|\vecz|^{n+p-\nu}} \partial_{\bar j}\partial_i \left(\frac{\xi_\vecz^{\nu,\mu}}{g}(\vecw)\right)\right.\right.\nonumber\\
& \hspace{0.5cm}\left. \left.+\frac{1}{2|\vecz|^2}\big[\overline z_i(n+p-\nu)\partial_{\bar j} + z_j(n+p-\mu)\partial_{i}\big]\beta(\vecw)\right]_{\vecw=\vecz}+ \frac{R}{2}\beta(\vecz)\right\}\label{B2}
%&  = \left(\frac{|\vecz|}{\hbar}\right)^{n-p-\nu}\frac{\Gamma(\nu)}{\Gamma(n-p)}\left\{g^{\bar j i}\left[\left[\partial_{i}\partial_{\bar j} \beta(\vecw)\right]_{\vecw=\vecz}+ \frac{\beta(\vecz)}{|\vecz|^4}\overline z_iz_j (n-p-\mu)\right. \right.\nonumber\\
%& \hspace{1cm}+ \left.\frac{\beta(\vecz)}{|\vecz|^4}\left(\frac{1}{2}\overline z_i z_j(n-p-\mu-2)+|\vecz|^2\delta_{ij}\right)(n-p-\frac{1}{2}) \right] + \frac{R}{2}\beta(\vecz)\nonumber\\
%&  \hspace{1cm}+\left. \left[\frac{\overline z_i}{2|\vecz|^2}(n-p-\mu)\partial_{\bar j}\beta(\vecw) +\frac{ z_j}{2|\vecz|^2}(n-p-\nu)\partial_{i}\beta(\vecw) \right]_{\vecw=\vecz}\right\}\label{B2}
\end{align}
where we have use the definition  $\mathrm L_1$ and 
\begin{align*}
\left.\frac{\partial}{\partial w_i}\frac{\xi_\vecz^{\nu,\mu}}{g}\right|_{\vecw=\vecz} & = |\vecz|^{n+p-\nu-2}(n+p-\nu)\overline z_i\\
\left.\frac{\partial}{\partial \overline w_j}\frac{\xi_\vecz^{\nu,\mu}}{g}\right|_{\vecw=\vecz} & = |\vecz|^{n+p-\nu-2}(n+p-\mu) z_j
%\left(\frac{|\vecz\cdot\vecw|}{|\vecz||\vecw|}\right)^{p-n+\frac{1}{2}}\right|_{\vecw=\vecz}=\left.\frac{\partial}{\partial \overline w_j}\left(\frac{|\vecz\cdot\vecw|}{|\vecz||\vecw|}\right)^{p-n+\frac{1}{2}}\right|_{\vecw=\vecz}=0
\end{align*}
\end{proof}
We note that in Eq. (\ref{asymptotic expansion}) appears $\partial_{\bar j}\partial_i (\xi_\vecz^{\nu,\mu}/g)$, which is not difficult to prove that
\begin{align}
\left.\partial_{\bar j}\partial_i \frac{\xi_\vecz^{\nu,\mu}}{g}(\vecw)\right |_{\vecw=\vecz} & =\frac{|\vecz|^{n+p-\nu}}{|\vecz|^4}\left[\left(n+p-\frac{1}{2}\right)\big[-z_j\overline z_i+\delta_{ij}|\vecz|^2\big]\right.\nonumber\\ 
&\hspace{1cm}+z_j\overline z_i(n+p-\nu)(n+p-\mu)\biggl]\label{second derivative}
\end{align}
which results in later use.

%We end this section showing that the star product satisfies the usual requirement on the semiclassical limit. 
We remember that for $f_1,f_2\in \A$, the law of multiplication in $\A$ is
\begin{align}
\big(f_1 *_p f_2\big)(\vecz) & =\frac{2}{(\pi\hbar)^n\hbar}\int\limits_{\mathbb C^n} f_1(\vecu,\vecz) f_2(\vecz,\vecu) \left[\frac{|\vecu\cdot \vecz|}{|\vecz|}\right]^{-p-n+1}\nonumber\\
 & \hspace{0.5cm} \frac{\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecu\cdot \vecz}}{\hbar}\right)\mathrm I_{n+p-1}\left(\frac{2\sqrt{\vecz\cdot \vecu}}{\hbar}\right)}{\mathrm I_{n+p-1}\left(\frac{2|\vecz|}{\hbar}\right)} |\vecu|^{p} \mathrm K_{p}\left(\frac{2|\vecu|}{\hbar}\right) \mathrm d\vecu \mathrm d\overline\vecu\;,\label{expression star product}
\end{align}
where the functions $f_j(\vecz,\vecw)$, $j=1,2$, which are the analytic continuation of $f_j(\vecz)$  to  $\mathbb C^n\times \mathbb C^n$ (see Eq. (\ref{extended covariant symbol})).

\begin{theorem} Let $n\ge 2$, $p>-n$. The product $ *_p$ satisfies
\begin{enumerate}
\item[a) ] $f *_p 1=1*_p f= f$, for all $f \in \A$,
\item[b) ] is associative, and
\item[c) ] for $f_1, f_2 \in \A$ we have the following asymptotic expression when $\hbar \to 0$
\begin{align}
f_1 *_p f_2(\vecz) &= f_1(\vecz) f_2(\vecz) + \hbar\Bigg[g^{\bar j i}\left[\partial_{i} f_1(\vecz,\vecw)+\partial_{\bar j} f_2(\vecw,\vecz)\right]_{\vecw=\vecz} \nonumber\\
& \hspace{0.5cm}\left.+f_1(\vecz)f_2(\vecz)\left(-\frac{1}{4|\vecz|}(n-1)(n-1+2p)+\frac{R}{2}\right.\right.\nonumber\\
&\hspace{0.5cm}\left.\left.+g^{\bar j i}(\vecz)\frac{p-n+\frac{1}{2}}{2|\vecz|^4}(z_j-|\vecz|^2\delta_{ij})\right) \right]+ \mathrm O(\hbar^2)\;, \label{asymptotic expansion star product}
\end{align}
where $R$ and $g^{i\bar j}$ was defined in the theorem \ref{theorem asymptotic expansion}.% is the sacalar curvature defined by the K\"ahler metric $\mathrm ds^2=g_{i\bar j} \mathrm d\overline z_j \wedge \mathrm dz_i$ and $g^{i\barj}$ is defined in
\end{enumerate}
\end{theorem}

\begin{proof}
a) Let $f \in \A$, then $f(\vecz)=\B(A)(\vecz)$ with $A\in \mathbf B(\mathcal O)$. From Eq. (\ref{extended covariant symbol}) and proposition \ref{supercomplete} 
\begin{align*}
f *_p 1 (\vecz) & = \int\limits_{\mathbb C^n} f(\vecw,\vecz) \frac{|\langle \K(\cdot,\vecz),\K(\cdot,\vecw)\rangle _{_{\Sn}}|^2}{\langle \K(\cdot,\vecz),\K(\cdot,\vecz)\rangle _{_{\Sn}}} \dm(\vecw)\\
& = \int\limits_{\mathbb C^n} \langle A\K(\cdot,\vecw),\K(\cdot,\vecz)\rangle _{_{\Sn}} \frac{\langle \K(\cdot,\vecz),\K(\cdot,\vecw)\rangle _{_{\Sn}}}{\langle \K(\cdot,\vecz),\K(\cdot,\vecz)\rangle_{_{\Sn}}} \dm(\vecw)\\
%& = \B(A)(\vecz)\\
& = f(\vecz)\;.
\end{align*}
Analogously $1 *_p f=f$.

b) The associativity follows from the fact that the composition in the algebra of all bounded linear operator on $\mathcal O$ is associative.

c) The asymtotic expression given in the Eq. (\ref{asymptotic expansion star product}) is a direct consequence from the integral expression of star product (see Eq. (\ref{expression star product})), the Eq. (\ref{equality bessel = hypergeometric}) and theorem \ref{theorem asymptotic expansion}.
\end{proof}
%\begin{remark}
%Note that in the Eq. (\ref{asymptotic expansion star product}) we verifies that noncommutative "star-product" satisfies the usual requirement on the semiclassical limit.
%\end{remark}
%The main idea of the proof consist in showing that the restriction of the operator $\U$ from $V_\ell$ onto $W_\ell$ is unitary for each $\ell \in \mathbb Z_+$, and the corresponding linear and unitary extension to the whole space $\mathcal O$ coincides witch $\U$.

%\begin{lemma}
 %The operator $\mathbf U_{n.p}^\ell:V_\ell \to W_\ell$ is given by
 %\begin{displaymath}
  %\mathbf L_p^\ell \Psi(\vecz)=\int_{\Sn} \sum_{|\bk|=\ell} \overline{\phi_{\bk}(\vecx)}\Phi_{\bk,p}^{\hbar}(\vecz) \Psi(\vecx) \ds(\vecx)\;,\hspace{0.5cm} \vecz \in \mathbb C^n\;,\hspace{0.5cm} \Psi\in V_\ell\;.
 %\end{displaymath}
 %Then $\mathbf L_p^\ell$ is a unitary operator.
%\end{lemma}

%\begin{proof} Note that the right side of operator $\mathbf L_p^\ell$ is linear and for each $\bk \in \mathbb Z_+^n$ satisfies
%\begin{align}
% \mathbf L_p^\ell \phi_{\bk}(\vecz) & = \sum_{|\bm|=\ell} \Phi_{\bm,p}^{\hbar}(\vecz) \int_{\Sn} \overline{\phi_{\bm}(\vecx)} \phi_{\bk}(\vecx) \ds(\vecx)\nonumber\\
% & = \Phi_{\bk,p}^\hbar(\vecz)\;.
%\end{align}
%Therefore by definition of $\mathbf U_{n,p}^\ell$ we have $\mathbf L_p^\ell=\mathbf U_{n,p}^\ell$.
%\end{proof}


%By the proposition \ref{prop basis O}, we need only  prove the Lemma for functions $\Psi=\phi_{\bk}$ with $\bk \in \mathbb Z_+^n$.



%Since for $\vecz \in \mathbb C^n$ fixed and $\vecx \in \Sn$ we have
%\begin{displaymath}
% \left|\K(\vecx,\vecz)\right|\le \sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{|\vecz|}{\hbar}\right)^\ell \langle  \infty\;.
%\end{displaymath}
%(See Proposition \ref{kernel coherent states} for details of the constants $c_{\ell,p}$).

%Given $\Psi \in \mathcal O$ written as $\Psi=\sum_{\ell=0}^\infty \Psi_\ell$ with $\Psi_\ell \in V_\ell$, we define the partial sums $S_k(\Psi)=\sum_{\ell=0}^k \mathbf U_{n,p}^\ell\Psi_\ell\Psi$ everywhere on $\mathbb C^n$ by using the dominated converge theorem. This finishes the proof of theorem \ref{unitary}.
%\end{proof}


%\begin{prop}\label{kernel coherent states} Let $\vecx \in \Sn$, $\vecz \in \mathbb C^n$, then
% \begin{displaymath}
% \K(\vecx,\vecz)=\sum_{\ell=0}^\infty \frac{c_{\ell,p}}{\ell!}\left(\frac{\vecx \cdot \vecz}{\hbar}\right)^\ell\;,
% \end{displaymath}
% where
% \begin{displaymath}
%  (c_{\ell,p})^2= \frac{(n)_\ell}{(n-p)_\ell}\;.
% \end{displaymath}
%\end{prop}

%\begin{proof}












%Let 
%For $\bk \in \mathbb N^n$, let us define the function $\mathrm e_{\bk,p}^\hbar: \mathbb C^n \to \mathbb C$ by the following expression:
%\begin{equation}
% \mathrm e_{\bk,p}^\hbar(\vecz)=\frac{1}{\hbar^{|\bk|}} \sqrt{\frac{\Gamma(n-p)}{\Gamma(|\bk|+n-p)}}\; \vecz^\bk\
%\end{equation}

%From Lemma \ref{product in Enp} is not difficult show that $\{\mathrm e_{\bk,p}:\bk \in \mathbb N^n\}$ is a ortonormal base of $\E$ 
%\begin{equation*}
% \vartheta_{\vecz,\bk,\ell}(\vecx)= \vecx^\bk (\vecx \cdot \vecz)^\ell\;.
%\end{equation*}
%Note that the functions  $\vartheta_{\vecz,\bk,\ell}$ are harmonic and homogeneous of order $|\bf|+\ell$. 


\section{Appendix A. On the inner product on complex sphere}
In this appendix we study the inner product of functions on $\Sn$ from the form $\vartheta_{\vecz}^{\bk,s}(\vecx)=\vecx^\bk (\vecx\cdot \vecz)^s$ with $\vecz\in\mathbb C^n$, $\vecx \in \Sn$, $\bk \in \mathbb Z_+^n$ and $s\in \mathbb Z_+$. We introduce the following notation, for multi-indices $\bk,\bm \in \mathbb Z_+^n$ we will write $\bk \ge \bm$ if and only if $k_j\ge m_j$ for all $j=1,\ldots,n$.
\begin{lemma}\label{product in Sn}
 Let 
 \begin{equation*}
  %\langle \vartheta_{\vecz,\bk,\ell}, \vartheta_{\vecw,\bm,s}\rangle 
  F:=\langle \vartheta_{\vecz}^{\bk,s},\vartheta_{\vecz}^{\bm,\ell}\rangle_{\Sn}=\int_{\Sn} %\vartheta_{\vecz}^{\bk,s}(\vecx) \overline{\vartheta_{\vecz}^{\bm,\ell}(\vecx)}\ds(\vecx)\;.
  \vecx^\bk \overline \vecx^\bm (\vecx \cdot \vecz)^s(\vecw \cdot \vecx)^\ell \ds(\vecx)\;,
 \end{equation*}
 with $\ell,s \in \mathbb Z_+$, $\bk,\bm \in \mathbb Z_+^n$, and $\vecz,\vecw \in \mathbb C^n\backslash\{0\}$.
 Then $F=0$ if $|\bk|+s\ne|\bm|+\ell$, furthermore
 \begin{enumerate}
  \item[ a)] If $|\bk|+s=|\bm|+\ell$ and $\bm \ge \bk$, then
  \begin{equation*}
   F=\frac{(n-1)!(|\bm|-|\bk|+\ell)!}{(n-1+|\bm|+\ell)!} \frac{\overline\vecz^\bm}{\overline\vecz^\bk} \; \sum_{|\bbeta|=\ell}\frac{\ell!}{\bbeta!} \frac{(\bm+\bbeta)!}{(\bm-\bk+\bbeta)!} \overline \vecz^{\bbeta} \vecw^{\bbeta}\;.
  \end{equation*}
  
  \item[ b)] If $|\bk|+s=|\bm|+\ell$, and $\bk \ge \bm$, then
  \begin{equation*}
   F=\frac{(n-1)!(s+|\bk|-|\bm|)!}{(|\bk|+s+n-1)!} \frac{\vecw^\bk}{\vecw^\bm} \; \sum_{|\balpha|=s}\frac{s!}{\balpha!} \frac{(\bk+\balpha)!}{(\bk-\bm+\balpha)!} \overline \vecz^{\balpha} \vecw^{\balpha}\;.
  \end{equation*}
 \end{enumerate}
\end{lemma}
\begin{proof}
We note that the function $\vartheta_{\vecz}^{\bk,s}(\vecx)$ is harmonic and homogeneous of order $|\bk|+s$, then $J=0$ if  $|\bk|+s\ne|\bm|+\ell$. We suppose now that $|\bk|+s=|\bm|+\ell$, and $\bm \ge \bk$, then
\begin{align*}
F & = \sum_{|\balpha|=s}\sum_{|\bbeta|=\ell}\frac{s!}{\balpha!}\frac{\ell}{\bbeta!} \overline \vecz^{\balpha}\vecw^{\bbeta} \int_{\Sn}  \vecx^{\bk+\balpha} \overline \vecx^{\bm+\bbeta} \ds(\vecx)\\
& = \sum_{|\bbeta|=\ell}\frac{(|\bm|-|\bk|+\ell)!}{(\bm-\bk+\bbeta)!} \frac{\ell!}{\bbeta!} \overline \vecz^{\bm-\bk+\bbeta}\vecw^{\bbeta}\frac{(n-1)!(\bm+\bbeta)!}{(n-1+|\bm|+\ell)!}\\
& = \overline \vecz^{\bm-\bk}\frac{(n-1)!(|\bm|-|\bk|+\ell)!}{(n-1+|\bm|+\ell)!}\sum_{|\bbeta|=\ell}\frac{(\bm+\bbeta)!}{(\bm-\bk+\bbeta)!} \frac{\ell!}{\bbeta!} \overline \vecz^{\bbeta}\vecw^{\bbeta}\;,
\end{align*}
where we use $\displaystyle\int_{\Sn}|\vecx^{\balpha}|^2 \ds(\vecx)=\frac{(n-1)!\balpha!}{(n-1+|\balpha|)!}$ (see Ref \cite{R08} for details). Similarly we get the other formula.
%\textcolor{red}{Functional theory in the Unit Ball of $\mathbb C^n$, p. 16}
\end{proof}




\section{Appendix B. Asymptotic of the Integral kernel of $\mathbf U_{n,-1}$} \label{AppendixB}


In the particular case when $p=-1$, we can obtain the asymptotic expresion of the coherent states $\K$. First, based on the definition of $\K$  (see Eqs. (\ref{coherent states}) and  (\ref{constants coherent states})) let us define the function $g : \mathbb C\to \mathbb C$ by
\begin{displaymath}
g(z)= \sum_{\ell=0}^\infty \frac{\sqrt{a\ell+1}}{\ell !} \;z^\ell\;,\hspace{1cm} a=\frac{1}{n-1}\,.
\end{displaymath}

Note that the coherent states $\mathbf U_{n,-1}$ are equal to the function $g$ evaluated at $(\vecx\cdot \vecz)/\hbar$. In this appendix we obtain the main asymptotic term for the function $g$.
\begin{lemma}\label{asymptotic p=1}  
For $\Re(z)>0$ and $|\Im(z)| \le C \Re(z)$ with $C$ a positive constant and $\Re(z)\to +\infty$, $g$ has the following asymptotic expansion:
\begin{equation}
g(z)=\sqrt a z^{1/2}\mathrm{exp}(z)\left[ 1+\frac{a_1}{z}+\frac{a_2}{z^2}+\mathrm O(z^{-3})\right]\;,
\end{equation}
with $a_1,a_2$ some constants.
\end{lemma}
\begin{proof}
This follows from Lemma 10.1, which appears in Ref. \cite{D-V09}.
\end{proof}


Using Lemma \ref{asymptotic p=1} with $z = \vecx\cdot\vecz/\hbar$ we obtain the following asymptotic expansion:
\begin{prop}\label{asymptotic coherent states p=1}
Let $\vecz \in \mathbb C^n-\{0\}$. Then for $\hbar \to 0$ and $|\Im(\vecx \cdot\vecz)|\le C\Re(\vecx\cdot\vecz)$, with $C$ a positive constant, we have
\begin{equation*}
\K(\vecx,\vecz)=\left[\frac{\vecx\cdot\vecz}{\hbar(n-1)}\right]^{\frac{1}{2}} \mathrm{exp}\left(\frac{\vecx\cdot\vecz}{\hbar}\right)\left[ 1+\frac{a_1}{\vecx\cdot\vecz} \hbar+\mathrm O(\hbar^2)\right]\;.
\end{equation*}
\end{prop}



\newpage
\begin{comment}


We note that is enough to show that Eq. (\ref{asymptotic Berezin transform}) holds for all points $\vecz$ of the form
\begin{equation}\label{zr}
\vecz=(r,0,\ldots,0)\;,\hspace{0.5cm} r>0\;.
\end{equation}
Indeed, let $U\in \mathrm{SO}(n)$ an arbitrary orthogonal transformation on $\mathbb C^n$, from Eq. (\ref{definition Berezin transform}) and Lemma \ref{Orthogonal composition coherent states}
\begin{equation}\label{Invariance Berezin transform rotations}
\B (\Phi \circ U)= (\B\Phi) \circ U\;.
\end{equation}

So, the validity of Eq. (\ref{asymptotic Berezin transform}) for $f$ at $\vecz$ is therefore equivalent to its validity for $f \circ U$ at $U^{-1} \vecz$. Since any given $\vecz$  can be mapped by a suitable $U$ into a point of the form (\ref{zr}), with $r=|\vecz|$, it is indeed enough to prove (\ref{asymptotic Berezin transform}) only for points $\vecz$ of the latter form.

For the rest of the proof of Theorem \ref{theorem asymptotic Berezin transform}, we thus assume that $\vecz$ is of the form (\ref{zr}) with $r>0$. 
\newpage




\appendix 
\section{SO(n) invariant Berezin transform} \label{AppendixA}

\begin{lemma} \label{Orthogonal composition coherent states}
Let $U\in \mathrm{SO}(n)$ an unitary orthogonal transformation on $\mathbb C^n$, then 
\begin{equation}\label{SO(n) invariant Berezin}
\B (\Phi \circ U)= (\B\Phi) \circ U\;.
\end{equation}
\end{lemma}
\begin{proof}
Let $\vecz\in \mathbb C^n$ and $\mathrm T_U:\mathcal O\to \mathcal O$, defined by $\mathrm T_U\Phi(\vecx)=\Phi(U\vecx)$. Since $\ds(\vecx)=\ds(U\vecx)$, is not difficult show that the composition map  $\mathrm T_U$ is unitary on $\mathcal O$. 

From Eqs. (\ref{U inner product coherent states}) and (\ref{expression U}) we obtain that for any $\Phi \in \mathcal O$
\begin{align*}
\langle \Phi,\K(\cdot,U\vecz))\rangle_{_{\Sn}} & =\U \Phi(U\vecz) \\
%& = \U(\mathrm T_U \Phi)(\vecz)\\
%& =\langle \Phi,\mathrm T_{U^{-1}}\K(\cdot,\vecz)\rangle_{_{\Sn}}\;,
& = \U(\Phi \circ U)(\vecz)\\
& =\langle \Phi,\K(\cdot,\vecz)\circ U^{-1}\rangle_{_{\Sn}}\;,
\end{align*}
where in the last equality we use that the operator $\mathrm T_U$ is unitary. 

So $\K(\cdot,U\vecz)=\K(\cdot,\vecz) \circ U^{-1}$ or 
\begin{equation}\label{SO(n) invariant coherent states}
\K(\vecx,\vecz)=\K(U\vecx, U\vecz)\;,\hspace{0.5cm} \mbox{for any } \vecx \in \Sn, \vecz\in \mathbb C^n.
\end{equation}
So for any arbitrary orthogonal transformation on $\mathbb C^n$, from Eqs. (\ref{definition Berezin transform}) and (\ref{SO(n) invariant coherent states}) we obtain the Eq. (\ref{SO(n) invariant Berezin}).
\end{proof}



\begin{prop}\label{prop basis O} The Hilbert space $\mathcal O$ is equal to the direct sum of the space $V_\ell$, this is
\begin{equation}
 \mathcal O= \bigoplus_{\ell=0}^\infty V_\ell\;.
\end{equation} 
Further, the set $\left\{\phi_{\bk}\;|\; \bk\in \mathbb Z_+^n\right\}$ is orthogonal basis of $\mathcal O$, where $\phi_{\bk}:\Sn \to \mathbb C$ is defining by
\begin{equation}\label{phi}
 \phi_{\bk}(\vecx)= \sqrt{\frac{(n)_{|\bk|}}{\bk!}}
 %\sqrt{\frac{\Gamma(|\bk|+n)}{\Gamma(n)\bk!}}
 \;\vecx^\bk\;,
\end{equation}
where, for $b> 0$, the Pochhammer symbol is given by
 \begin{equation*}
  (b)_\ell=\frac{\Gamma(b+\ell)}{\Gamma(b)}\;.
 \end{equation*}
\end{prop}
\begin{proof} Is a consequence of fact that the monomials are dense in $\mathcal O$ and that the spaces $V_\ell$ and $V_{\ell'}$ are orthogonal for $\ell \ne \ell'$ (see Lemma \ref{product in Sn}).

The second assertion too is a direct consequence of Lemma \ref{product in Sn}, since
%a consequence the fact that for each $\ell \in \mathbb Z_+$ the set 
\begin{displaymath}
\left\{\sqrt{\frac{(n)_\ell}{\bk!}}
%\sqrt{\frac{\Gamma(\ell+n)}{\Gamma(n)\bk!}}
\;\vecx^\bk\;|\;\bk \in \mathbb Z_+^n\;,\; |\bk|=\ell\right\}\;, 
\end{displaymath}
is orthogonal basis of $V_\ell$. 
\end{proof}

\end{comment}





















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\bibitem{D-V09}Diaz-Ortiz, E. I. and Villegas-Blas, C., \emph{Semiclassical properties of coherent states for $L^2(S^n)$, $n = 2, 3, 5$}, Qual. Theory Dyn. Syst. 8(2), 279–317 (2009).
%\bibitem{D-V12}Diaz-Ortiz, E. I. and Villegas-Blas, C., \emph{On a Bargmann transform and coherent states for the n-sphere}, J. Math. Phys. 53, 062103 (2012)
\bibitem{E00} Englis, M., \emph{The asymptotics of a Laplace integral on a K\"{a}hler manifold}, Journal Fur Die Reine und angewandte Mathematik, 528, 1-39, 2000.
\bibitem{G-S64} Gel'fand, I. M. y Shilov, G. E., \emph{Generalized functions}, Vol 1, Properties and Operators, Academic Press, New York and London, 1964.
\bibitem{G94} Gradshteyn, I.S. y Ryzhik, I.M.  \emph{Table of integrals, series, and products}, Fifth edition. Academic Pres, United Kingdom (1994), editado por Alan Jeffrey.
%\bibitem{H94} Hall, B. \emph{The Segal-Bargmann coherent states transform for compact Lie Groups}, J. Funct. Anal. 122, 103-151, 1994.
%\bibitem{B-H00} Hall, B. \emph{Holomorphic methods in analysis and mathematical physics}, Contemporary Mathematics, Vol 260, edited by S. P\'erez-Esteva and C. Villegas-Blas , American Mathematical Society, Providence, RI, 2000.
%\bibitem{G-P90} Graffi, S y  Parmeggiani, A. \emph{Quantum evolution and classical flow in complex space}, Commun, Math. Phys, 128, 393-409 1990.
%\bibitem{HM02} Hall, B y Mitchell, J. \emph{Coherent states on the sphere}, J. Math. Phys. 43, 1211-1236, 2002.
%\bibitem{H81} Helgason, S. \emph{Topics in Harmonic Analysis on Homogeneous Spaces}, Progress in Mathematics, Vol. 13, Birkh\"auser, 1981.
\bibitem{H90} H\"ormander, L. \emph{The analysis of Linear Partial Differential Operators}, Vol I. Distribution Theory and Fourier Analysis, 2nd eds. Springer, Berlin, 1990.
%\bibitem{H-O93} Horowski, Maciej y Odzijewicz,  Anatol \emph{Geometry of the Kepler system in coherent states approach}, Ann. Inst. H. Poincaré Phys. Théor. 59, no. 1, 69–89, 1993.
%\bibitem{I86} Ii, K. \emph{On a Bargmann-type transform and a Hilbert space of holomorphic functions}, Tohoku Math. J. 38, 57-69, 1986.
\bibitem{K03} Karp, Dmitrii \emph{Hypergeometric reproducing kernels and analytic continuation from a half-line}, Integral Transforms and Spec. Funct. 14 (2003), no. 6, 485–498.
%\bibitem{KR01} Kowalski, K y Rembielinski, J. \emph{The Bargmann representation for the quantum mechanics on a sphere}, J. Math. Phys. 42, 4138-4147, 2001.
%\bibitem{K82} Kummer, M. \emph{On the regularization of the Kepler problem}, Commun. Math. Phys. 84, 133–152 1982.
%\bibitem{KS65}  Kustaanheimo, P., and Stiefel, E., \emph{Perturbation theory of Kepler motion based on spinor regularization}, J. Reine Angew. Math. 218, 204–219 1965.
\bibitem{L65} Lebedev, N. \emph{Special functions and their applications}, Pretice-Hall, 1965.
%\bibitem{LC56} Levi-Civita, T. \emph{Sur la résolution qualitative du probleme restréint des trois corps. In: Opere Matematiche}, Vol. Secondo. Zanichelli, Bologna, 1956.
%\bibitem{M02} Martinez, Andr\'e. \emph{Introduction to Semiclassical and Microlocal Analysis}, Springer-Verlag, 2002
%\bibitem{M98} Merzbacher, E. \emph{Quantum Mechanics}. John Wiley and Sons, Inc. 1998. Third edition.
%\bibitem{M70} Moser, J. \emph{Regularization of Kepler’s problem and the averaging method on a manifold}, Commun. Pure Appl. Math. 23, 609–636 1970.
\bibitem{P-B-M90}Prudnikov, A.P. Brychkov Yu. A.  and Marichev, O. I. \emph{Integrals and series, Volume 3: More Special Functions}, Gordon and Breach Science Publishers, 1990.
%\bibitem{R79} Rawnsley, J. \emph{A non unitary pairing of polarizations for the Kepler problem}, Trans. Am. Math. Soc., 250, 167-180, 1979.
%\bibitem{RS72} Reed, M y  Simon, B. \emph{Methods of Modern Mathematical Physics, I: Functional Analysis}, Academic Press, New York, London, 1972.
%\bibitem{R57} Rose, M. E. \emph{Elementary Theory of Angular Momentum} Wiley, New York, 1957.
\bibitem{R08} Rudin, Walter \emph{Function theory in the unit ball on $\mathbb C^n$}, Classics Math. Springer-Verlag, Berlin, 2008, reprint of the 1980 edition.
%\bibitem{S26} Schr\"odinger, E. \emph{Naturwissenschaften}, 14, 664, 1926.
%\bibitem{S99} Stenzel, M. \emph{The Segal-Bargmann transform on a symmetric space of compact type}, J. Funct. Anal. 165, 44-58, 1999.
%\bibitem{TE07} Tawareque, S. y  Englis, M. \emph{Berezin-Toeplitz Quantization over Matrix Domains}, Contributions in mathematical physics, 1–36, Hindustan Book Agency, New Delhi, 2007. 
%\bibitem{TW95} Thomas, L., Wassell, S.. \emph{Semiclassical approximation for Schro\"odinger Operators on a 2-sphere at high energy}, J. Math. Phys. 36, 5480-5505,1995.
%\bibitem{TWM} Tsoy-Wo Ma. \emph{Higher Chain Formula proved by Combinatorics}, Electronic Journal of Combinatorics, Vol 16(1): N21, 2009.
%\bibitem{V02} Villegas-Blas, C. \emph{The Bargmann transform and canonical transformations}, J.Math. Phys. 43(5), 2249–2283 (2002)
%\bibitem{V02-1} Villegas-Blas, C. \emph{ The Bargmann transform for $L^2(S^3)$ and regularization of the Kepler problem}, Mathematical results in quantum mechanics (Taxco, 2001), Contemporary Mathematics, vol. 307, 311–318. American Mathematical Society, Providence 2002.
%\bibitem{V06} Villegas Blas, C. \emph{The Bargmann transform and the regularization of the 2, 3, 5-dimensional Kepler problem}, J. Math. Phys. 47(6), 062110 (2006)
%\bibitem{W86} Wada, R. \emph{On the Fourier-Borel transformation of analytic functionals on the complex-sphere}, Todoku Math, J., 38, 417-432, 1986.
%\bibitem{Z-E} Zworski, M. y Evans, L.C. \emph{Lectures on Semiclassical Analysis}, Lectures notes. http://math.berkeley.edu/~zworski/Pr
\end{thebibliography}


\begin{comment}
\textcolor{blue}{El objetivo principal del presente articulo es introducir un calculo simbolico.
Para hacer esto, contruiremos una familia $\{e_\alpha\}$ de funciones en $\mathcal O$ que satisfaga (1), dado que los estados coherentes satisfacen dicha propiedad y en base a la experiencia obtenida en $L^2(S^m)$, $m=2,3,5$ donde $S^m=\{\vecx \in \mathbb R^{m+1}\;|\;x_1^2+\cdots+x_{m+1}^2=1\}$, propondré estados coherentes en $\mathcal O$ como una adecuada serie de potencias del producto interno $\vecx\cdot\vecz/\hbar$ con $\vecx \in\Sn$, $\vecz\in \mathbb C^n$ y $\hbar$ denotando la constante de Plank. Además para asegurar que se satisface (2) se definirá un nuevo espacio de Hilbert  $L^2(\mathbb C^n, \dm)$ cuya medida $\dm$ es obtenida de tal forma que el producto interno de dichos estados coherentes sea el núcleo reproductor del rango del mapeo $f\to \hat f$ definido en (2).
Hay que hacer notar, que dado que  $H=\mathcal O$ y  $L^2(M,\mathrm d\mu)=L^2(\mathbb C^n, \dm)$,  no hay una inclusion de $H$ en $L^2(M,\mathrm d\mu)$, además la familia $e_\alpha$ no es obtenida por el núcleo reproductor.}
\end{comment}
\end{document}

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{B74 Berezin, F. A., Quantization, Math USSR-Izv., 8, 1116-1175 (1974) %
1702.05708.extracted_bib
{D87 Davtyan, L., Mardoyan, G., Pogosyan, G., Sissakian, S., and Ter-Antonyan, V.,Generalized KS transformation: from five-dimensional hydrogen atom to eight-dimensional isotropic oscillator, J. Phys. A 20, 6121–6125 1987.
1702.05708.extracted_bib
{D-V09Diaz-Ortiz, E. I. and Villegas-Blas, C., Semiclassical properties of coherent states for $L^2(S^n)$, $n = 2, 3, 5$, Qual. Theory Dyn. Syst. 8(2), 279–317 (2009). %
1702.05708.extracted_bib
{D-V12Diaz-Ortiz, E. I. and Villegas-Blas, C., On a Bargmann transform and coherent states for the n-sphere, J. Math. Phys. 53, 062103 (2012)
1702.05708.extracted_bib
{E00 Englis, M., The asymptotics of a Laplace integral on a K\"{ahler manifold, Journal Fur Die Reine und angewandte Mathematik, 528, 1-39, 2000.
1702.05708.extracted_bib
{G-S64 Gel'fand, I. M. y Shilov, G. E., Generalized functions, Vol 1, Properties and Operators, Academic Press, New York and London, 1964.
1702.05708.extracted_bib
{G94 Gradshteyn, I.S. y Ryzhik, I.M. Table of integrals, series, and products, Fifth edition. Academic Pres, United Kingdom (1994), editado por Alan Jeffrey. %
1702.05708.extracted_bib
{H94 Hall, B. The Segal-Bargmann coherent states transform for compact Lie Groups, J. Funct. Anal. 122, 103-151, 1994. %
1702.05708.extracted_bib
{B-H00 Hall, B. Holomorphic methods in analysis and mathematical physics, Contemporary Mathematics, Vol 260, edited by S. P\'erez-Esteva and C. Villegas-Blas , American Mathematical Society, Providence, RI, 2000. %
1702.05708.extracted_bib
{G-P90 Graffi, S y Parmeggiani, A. Quantum evolution and classical flow in complex space, Commun, Math. Phys, 128, 393-409 1990. %
1702.05708.extracted_bib
{HM02 Hall, B y Mitchell, J. Coherent states on the sphere, J. Math. Phys. 43, 1211-1236, 2002. %
1702.05708.extracted_bib
{H81 Helgason, S. Topics in Harmonic Analysis on Homogeneous Spaces, Progress in Mathematics, Vol. 13, Birkh\"auser, 1981.
1702.05708.extracted_bib
{H90 H\"ormander, L. The analysis of Linear Partial Differential Operators, Vol I. Distribution Theory and Fourier Analysis, 2nd eds. Springer, Berlin, 1990. %
1702.05708.extracted_bib
{H-O93 Horowski, Maciej y Odzijewicz, Anatol Geometry of the Kepler system in coherent states approach, Ann. Inst. H. Poincaré Phys. Théor. 59, no. 1, 69–89, 1993. %
1702.05708.extracted_bib
{I86 Ii, K. On a Bargmann-type transform and a Hilbert space of holomorphic functions, Tohoku Math. J. 38, 57-69, 1986.
1702.05708.extracted_bib
{K03 Karp, Dmitrii Hypergeometric reproducing kernels and analytic continuation from a half-line, Integral Transforms and Spec. Funct. 14 (2003), no. 6, 485–498. %
1702.05708.extracted_bib
{KR01 Kowalski, K y Rembielinski, J. The Bargmann representation for the quantum mechanics on a sphere, J. Math. Phys. 42, 4138-4147, 2001. %
1702.05708.extracted_bib
{K82 Kummer, M. On the regularization of the Kepler problem, Commun. Math. Phys. 84, 133–152 1982. %
1702.05708.extracted_bib
{KS65 Kustaanheimo, P., and Stiefel, E., Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math. 218, 204–219 1965.
1702.05708.extracted_bib
{L65 Lebedev, N. Special functions and their applications, Pretice-Hall, 1965. %
1702.05708.extracted_bib
{LC56 Levi-Civita, T. Sur la résolution qualitative du probleme restréint des trois corps. In: Opere Matematiche, Vol. Secondo. Zanichelli, Bologna, 1956. %
1702.05708.extracted_bib
{M02 Martinez, Andr\'e. Introduction to Semiclassical and Microlocal Analysis, Springer-Verlag, 2002 %
1702.05708.extracted_bib
{M98 Merzbacher, E. Quantum Mechanics. John Wiley and Sons, Inc. 1998. Third edition. %
1702.05708.extracted_bib
{M70 Moser, J. Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. Math. 23, 609–636 1970.
1702.05708.extracted_bib
{P-B-M90Prudnikov, A.P. Brychkov Yu. A. and Marichev, O. I. Integrals and series, Volume 3: More Special Functions, Gordon and Breach Science Publishers, 1990. %
1702.05708.extracted_bib
{R79 Rawnsley, J. A non unitary pairing of polarizations for the Kepler problem, Trans. Am. Math. Soc., 250, 167-180, 1979. %
1702.05708.extracted_bib
{RS72 Reed, M y Simon, B. Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, London, 1972. %
1702.05708.extracted_bib
{R57 Rose, M. E. Elementary Theory of Angular Momentum Wiley, New York, 1957.
1702.05708.extracted_bib
{R08 Rudin, Walter Function theory in the unit ball on $\mathbb C^n$, Classics Math. Springer-Verlag, Berlin, 2008, reprint of the 1980 edition. %
1702.05708.extracted_bib
{S26 Schr\"odinger, E. Naturwissenschaften, 14, 664, 1926. %
1702.05708.extracted_bib
{S99 Stenzel, M. The Segal-Bargmann transform on a symmetric space of compact type, J. Funct. Anal. 165, 44-58, 1999. %
1702.05708.extracted_bib
{TE07 Tawareque, S. y Englis, M. Berezin-Toeplitz Quantization over Matrix Domains, Contributions in mathematical physics, 1–36, Hindustan Book Agency, New Delhi, 2007. %
1702.05708.extracted_bib
{TW95 Thomas, L., Wassell, S.. Semiclassical approximation for Schro\"odinger Operators on a 2-sphere at high energy, J. Math. Phys. 36, 5480-5505,1995. %
1702.05708.extracted_bib
{TWM Tsoy-Wo Ma. Higher Chain Formula proved by Combinatorics, Electronic Journal of Combinatorics, Vol 16(1): N21, 2009. %
1702.05708.extracted_bib
{V02 Villegas-Blas, C. The Bargmann transform and canonical transformations, J.Math. Phys. 43(5), 2249–2283 (2002) %
1702.05708.extracted_bib
{V02-1 Villegas-Blas, C. The Bargmann transform for $L^2(S^3)$ and regularization of the Kepler problem, Mathematical results in quantum mechanics (Taxco, 2001), Contemporary Mathematics, vol. 307, 311–318. American Mathematical Society, Providence 2002. %
1702.05708.extracted_bib
{V06 Villegas Blas, C. The Bargmann transform and the regularization of the 2, 3, 5-dimensional Kepler problem, J. Math. Phys. 47(6), 062110 (2006) %
1702.05708.extracted_bib
{W86 Wada, R. On the Fourier-Borel transformation of analytic functionals on the complex-sphere, Todoku Math, J., 38, 417-432, 1986. %
1702.05708.extracted_bib
{Z-E Zworski, M. y Evans, L.C. Lectures on Semiclassical Analysis, Lectures notes. http://math.berkeley.edu/~zworski/Pr