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\title[Infinite order Euler operators]{Infinite order Euler operators
in spaces of holomorphic functions and Stirling numbers}% Указываем название статьи

\author{O.A. Ivanova, S.N. Melikhov}%Указываем авторов

\address{Olga Alexandrovna Ivanova,
\newline\hphantom{iii} Southern Federal University,
\newline\hphantom{iii} Institute of Mathematics, Mechanics
\newline\hphantom{iii} and Computer Sciences named after I.I. Vorovich
\newline\hphantom{iii} Milchakova str.  8a, % Адрес (улица, дом, строение и т.п.)
\newline\hphantom{iii} 344090, Rostov--on--Don, Russia}%  Адрес (почтовый индекс, город, страна)
\email{ivolga@sfedu.ru}% Ваш электронный адрес для переписки

\address{Sergej Nikolaevich Melikhov,
\newline\hphantom{iii} Southern Federal University,
\newline\hphantom{iii} Institute of Mathematics, Mechanics
\newline\hphantom{iii} and Computer Sciences named after I.I. Vorovich
\newline\hphantom{iii} Milchakova str.  8a, % Адрес (улица, дом, строение и т.п.)
\newline\hphantom{iii} 344090, Rostov--on--Don, Russia
\smallskip
\newline\hphantom{iii} Southern Mathematical Institute
 \newline\hphantom{iii} Vladikavkaz Scientific Center of RAS
\newline\hphantom{iii} Vatutin str. 53, % Адрес (улица, дом, строение и т.п.)
\newline\hphantom{iii} 362025, Vladikavkaz, Russia}%  Адрес (почтовый индекс, город, страна)
\email{snmelihov@yandex.ru, snmelihov@sfedu.ru}% Ваш электронный адрес для переписки


\thanks{\sc O.A. Ivanova, S.N. Melikhov, %  Ф.И.О. авторов на английском языке
Infinite order Euler operators in spaces of holomorphic functions and Stirling numbers}% название статьи на английском языке
\thanks{\copyright \ Ivanova O.A., Melikhov S.N. \ 2026}
\thanks{\rm The research is supported by Russian Science Foundation, project no. 25-21-00062, https://rscf.ru/project/25-21-00062/}
\thanks{\it Submitted December 04, 2025.}

\maketitle {\small
\begin{quote}
\noindent{\bf Abstract.}  We study infinite order Euler operators in the space $H(\Omega)$ of all functions holomorphic
on an open set $\Omega$ in $\mathds C^N$, with the topology of uniform convergence on compact sets in $\Omega$.
In terms of their characteristic functions,  we prove necessary and sufficient conditions
for the applicability of these operators to $H(\Omega)$. The special case $\Omega=(\mathds C\backslash\{0\})^N$ is considered.  We study
the relationship between the two representations for the Euler operator, in which the Stirling numbers of the first and second kinds play a significant role.
 It is expressed by means of  associated functions, one of which is the sum of the Newton interpolation series.
The obtained results imply that each entire function of exponential type $0$ on $\mathds C^N$ can be expanded into a
multidimensional Newton interpolation series. A multidimensional version of the Wigert~---~Leau theorem is proved. We show   that in the space $H(\mathds C^N)$ of all integer functions in $\mathds C^N$, each Hadamard type operator in
$H(\mathds C^N)$ is an Euler operator, that is,  each continuous linear operator in $H(\mathds C^N),$ for which each monomial is an eigenvector.

\medskip
\noindent{\bf Keywords:} holomorphic function, infinite order  Euler operator, Stirling numbers of the first and second kind.

\medskip
\noindent{\bf Mathematics Subject Classification:} {46E10, 47B99, 11B73}

\end{quote}
}

\vskip30pt

\section{Introduction}

In this paper, we study infinite order Euler operators  acting in the space $H(\Omega)$ of all functions holomorphic on an open set $\Omega$ in $\mathds C^N$,
with the topology of uniform convergence on compact sets in $\Omega$, which can be
represented as
\begin{equation} \label{REPR1}
\mathcal E_a(f)(t):=\sum\limits_{\beta\in\mathds N_0^N} a_\beta t^\beta f^{(\beta)}(t)
\end{equation}
or
 \begin{equation} \label{REPR2}
 \mathcal E_{\theta, a}(f)(t):=\sum\limits_{\beta\in\mathds N_0^N} a_\beta\theta^\beta(f)(t),
 \end{equation}
where
\begin{align*}
\theta^\beta:=\theta_1^{\beta_1}\cdots\theta_N^{\beta_N},\qquad  \theta_j(f)(t):=t_j \frac{\partial f}{\partial t_j}(t),\qquad a_\beta\in\mathds C,\qquad \beta\in\mathds N_0^N,\qquad \mathds N_0:=\mathds N\cup\{0\}.
\end{align*}
These operators in $H(\Omega)$ were studied in a lot of papers. The problem of well--definedness and surjectivity of Euler operators was mostly addressed.
In  \cite{KORSMJ69}, \cite{KOR69}, \cite{LIN}, \cite{DAVIS} (for $N=1$) and in \cite{ISHIM1}, \cite{ISHIM2} (for $N\geq 1$), operators of  form \eqref{REPR1} were studied. In \cite{ZNAM}, \cite{TRYBULA}, Euler operators in $H(\Omega)$
with the representation \eqref{REPR2} for $N=1$ were investigated.
Infinite order Euler operators
in spaces of real analytic functions of one and several variables obeying representation \eqref{REPR2}
were studied in  \cite{DOMLANG1}--\cite{DOMLANG3}.

The main goals of this paper are to study the convergence of series defining Euler operators and the relationship between their representations.
The problem on the action of $\mathcal E_a$ and $\mathcal E_{\theta,a}$
in $H(\Omega)$ is resolved in Section~2.
We follow the definition of a summable and absolutely summable family in a locally convex space from \cite[Ch. 1, Sect. 1.3]{PIETCH}. For a countable set of indices
$\Lambda$, instead of the term ``the family $\{x_\beta:\,\beta\in\Lambda\}$ is absolutely summable'', we use the term ``the series $\sum\limits_{\beta\in\Lambda} x_\beta$ converges absolutely''.  Following Yu.F.~Korobeynik, we call the operators defined by identities \eqref{REPR1} and \eqref{REPR2} applicable to $H(\Omega)$ if the series in their
definition absolutely converges at each point $t\in\Omega$ for each function $f\in H(\Omega)$.
A sufficient condition for both \eqref{REPR1} and \eqref{REPR2} to be applicable to $H(\Omega)$ for all open sets $\Omega$ in $\mathds C^N$ is the identity
$$\lim\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}=0.$$ For a wide class of open sets $\Omega\subset\mathds C^N$, this identity is necessary. A necessary and sufficient condition for the applicability of both types of operators to $H(\mathds C^N)$ is the relation $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$
The mentioned conditions ensure the absolute convergence of  series \eqref{REPR1} and \eqref{REPR2} in $H(\Omega)$, and hence the continuity of $\mathcal E_a$ and $\mathcal E_{\theta,a}$ in $H(\Omega)$. We note  that
the aforementioned results on applicability for $N=1$ were known; only the question of whether the condition
$$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty$$
is necessary for the applicability of $\mathcal E_{\theta, a}$ to $H(\mathds C)$ remained open; this condition was formulated in   \cite[Prop. 6.2]{TRYBULA}. We consider the special case $\Omega=(\mathds C\backslash\{0\})^N$, which was not investigated in this direction even for $N=1$. For this domain $\Omega$, the criterion for applicability to $H(\Omega)$ of the operator $\mathcal E_a$
is the condition $$\limsup\limits_{|\beta|\to\infty}(\beta !|a_\beta|)^{\frac{1}{|\beta|}}<1,$$
and $\mathcal E_{\theta,a}$ is the relation $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$
When proving necessary conditions on the coefficients $a_\beta$, $\beta\in\mathds N_0^N$, sets that are K\"othe duals to special spaces of (multi)sequences are used. Information about them is given, for example, in \cite[Sect. 2]{KOR83}.

 Section~3 is devoted to the  relations between representations
  \eqref{REPR1} and \eqref{REPR2}. It seems that this problem was not systematically studied for infinite order  Euler operators. We study this problem by using Stirling numbers of the first and second kind. When the order of $\mathcal E_a$ and $\mathcal E_{\theta,a}$ is finite, by means of these numbers, each of these operators can be expressed in terms of the other. This follows easily from the well--known one--dimensional identities relating the operators $f\mapsto t^\beta f^{(\beta)}(t)$ and $\theta^\beta$.
If the orders of $\mathcal E_a$ and $\mathcal E_{\theta,a}$ are infinite, then the connection between the representations can be established due to the availability of the needed upper bounds for the Stirling numbers of the second kind
and for the moduluses of the Stirling numbers of the first kind, which ensure the absolute convergence of the corresponding series
and the validity of the needed identities. It is described by means of functions associated with $a$, one of which is the sum of the  Newton interpolation series.
We also show that each entire function of exponential type $0$ in $\mathds C^N$ can be expanded into the corresponding multidimensional Newton series. This allows us to prove a certain multidimensional version of the Wigert~---~Leau theorem in terms of the defining compact set of an analytic functional in Section~4.
In Section~5 we establish that each operator of Hadamard type in $H(\mathds C^N)$ is an Euler operator of form \eqref{REPR1} (and \eqref{REPR2}).

\vskip20pt

\section{Conditions for coefficients}

\subsection{Applicability of  $\mathcal E_a$}
We first  study the action of the operator $\mathcal E_a$ in $H(\Omega)$.
For $L\in\mathds N$, $c\in \mathds C^L$, $c\ne 0$, $d\in\mathds C$, we introduce the  hyperplane in $\mathds C^L$
$$
T(c,d):=\{t\in\mathds C^L: \,\langle c, t\rangle= d\}.
$$
Here $\langle z,w\rangle:=\sum\limits_{k=1}^L z_kw_k$, $z,w\in\mathds C^L$.
The proof of the necessary condition for the applicability of the Euler operator involves a  certain local condition of linear convexity.
It is formulated in terms of the existence of a complex hyperplane passing through the boundary point of the set and not intersecting it.
Further, $\mathds C^*:=\mathds C\backslash\{0\}$. For a set $Q$ in $\mathds C^L$, we denote by $\partial\,Q$ the boundary of $Q$ in $\mathds C^L$;
\begin{equation*}
t^\beta:=t_1^{\beta_1}\cdots t_L^{\beta_L},\qquad |\beta|:=\beta_1+\cdots+\beta_L, \qquad  \beta !:=\beta_1!\cdots\beta_L!,\qquad t\in\mathds C^L,\qquad \beta\in\mathds N_0^L.
\end{equation*}
\begin{theorem} \label{PRIM1}
\begin{enumerate}
\item[(i)] If $$\lim\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}=0, $$ then
for each nonempty open set
$\Omega$ in $\mathds C^N$ and each function $f\in H(\Omega)$
 series \eqref{REPR1} converges absolutely in $H(\Omega)$.

 \vskip5pt

\item[(ii)] Let
\begin{equation*}
\Omega=\Omega_1\times\cdots\times\Omega_M,\qquad 1\leq M\leq N,\qquad  N_m\in\mathds N,\qquad \sum\limits_{m=1}^M N_m=N,
\end{equation*}
and $\Omega_m\ne\mathds C^{N_m}$ be a a non--empty open set in $\mathds C^{N_m}$ with  following property in $\mathds C^{N_m}$:
there exist $c^{(m)}\in (\mathds C^*)^{N_m}$, $d_m\in\mathds C$, points $$w^{(m)}\in T(c^{(m)}, d_m)\cap(\mathds C^*)^{N_m}\cap\partial \Omega_m$$ such that $$T(c^{(m)}, d_m)\cap\Omega_m=\emptyset.$$ If the operator $\mathcal E_a$ is applicable to $H(\Omega)$, then
$$\lim\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}=0.$$
\end{enumerate}
\end{theorem}

\begin{proof} Statement (i) can proved in the standard way by means of estimates obtained by using the Cauchy formula.


 (ii): We write the point $t\in\mathds C^N$ as $t=(t^{(1)},\ldots, t^{(M)})$, where $t^{(m)}\in \mathds C^{N_m}$, \linebreak $1\leq m\leq M$.
There exists a sequence $t(n)\in\Omega\cap(\mathds C^*)^N$, $n\in\mathds N$, converging to the point $w=(w^{(1)},\ldots,w^{(M)})$ in $\mathds C^N$.
Since the function $$  f(t)=\prod\limits_{m=1}^M\frac{1}{d_m-\langle c^{(m)},t^{(m)}\rangle}$$ is holomorphic in $\Omega$ and
$$
f^{(\beta)}(t)=\prod\limits_{m=1}^M\frac{|\beta^{(m)}|! (c^{(m)})^{\beta^{(m)}}}{(d_m-\langle c^{(m)},t^{(m)}\rangle)^{|\beta^{(m)}|+1}},  \qquad \beta\in\mathds N_0^N,
$$
for each $n\in\mathds N$ there exists a constant $A_n>0$ obeying
\begin{equation} \label{ONE}
|a_\beta|\left(\prod\limits_{m=1}^M|\beta^{(m)}|!\right)|(t(n))^\beta||c^\beta|
\leq A_n\prod\limits_{m=1}^M|d_m-\langle c^{(m)},(t(n))^{(m)}\rangle|^{|\beta^{(m)}|+1}, \qquad \beta\in\mathds N_0^N.
\end{equation}
There exits  $\delta>0$ such
$|(t(n))^\beta||c^\beta|\geq\delta^{|\beta|}$ for all $n\in\mathds N$, $\beta\in\mathds N_0^N$. By  \eqref{ONE} and the estimate
$\beta!\leq\prod\limits_{m=1}^M|\beta^{(m)}| !$ we find
$$
\beta!|a_\beta|\leq\frac{A_n}{\delta^{|\beta|}}\prod\limits_{m=1}^M|d_m-\langle c^{(m)},(t(n))^{(m)}\rangle|^{|\beta^{(m)}|+1}, \qquad n\in\mathds N,
$$
and
\begin{align*}
(\beta!|a_\beta|)^{\frac{1}{|\beta|}}\leq&\frac{(A_n)^{\frac{1}{|\beta|}}}{\delta}\prod\limits_{m=1}^M|d_m-\langle c^{(m)},(t(n))^{(m)}\rangle|^{\frac{|\beta^{(m)}|+1}{|\beta|}}
\\
\leq&\frac{(A_n)^{\frac{1}{|\beta|}}}{\delta}\left(\prod\limits_{m=1}^M|d_m-\langle c^{(m)},(t(n))^{(m)}\rangle|\right)^{\frac{1}{|\beta|}}
\\
&\cdot\sum\limits_{m=1}^M|d_m-\langle c^{(m)},(t(n))^{(m)}\rangle|, \quad n\in\mathds N, \beta\in\mathds N_0^N.
\end{align*}
This yields that for all  $n\in\mathds N$
$$
\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}\leq\frac{1}{\delta}\sum\limits_{m=1}^M|d_m-\langle c^{(m)},(t(n))^{(m)}\rangle|.
$$
This is why the limit $\lim\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}$ exists and it is zero. The proof is complete.
\end{proof}

\begin{remark}
The proven theorem is known for $N=1$  \cite{KORSMJ69}. In this case, the additional condition in (ii) is satisfied by
any nonempty open set $\Omega$ in $\mathds C$ distinct from $\mathds C$ and $\mathds C^*$.

For $N\geq 2$, the set $\Omega\subset\mathds C^N$ possesses the property assumed in
statement (ii) of Theorem \ref{PRIM1}, for example, in the following situations:
if $\Omega=\Omega_1\times\cdots\times\Omega_N$, where $\Omega_j$, $1\leq j\leq N$, is a nonempty open set in $\mathds C$ distinct from $\mathds C$ and $\mathds C^*$;
if $\Omega$ is a bounded convex domain in $\mathds C^N$, each point of its boundary is smooth, i.e., through each point in $\partial \Omega$ there passes a unique
(real) support hyperplane to $\Omega$ \cite[Def. 10.7]{LEIHT}.

Let us justify the last example. Let $H_\Omega(t):=\sup\limits_{z\in\Omega}\RE \langle z, t\rangle$, $t\in\mathds C^N$, be the support function of $\Omega$,
$S:=\Big\{z\in\mathds C^N: \,\sum\limits_{j=1}^N |z_j|^2=1\Big\}$ be the unit sphere in $\mathds C^N$, $\omega:\partial\Omega\to S$ be the mapping that assigns
$z\in\partial\Omega$ to a point $\omega(z)\in S$ such that the real hyperplane $$W(z)=\{t\in\mathds C^N:\RE \langle \omega(z), t \rangle=H_\Omega(\omega(z))\}$$
is  a support hyperplane to $\Omega$ at $z$. By \cite[Thm. 10.8]{LEIHT}, the mapping $\omega:\partial\Omega\to S$ is continuous (when $\partial\Omega$ and $S$
are equipped with the topologies induced by the Euclidean topology in $\mathds C^N$). Since $\omega$ is bijective and $\partial\Omega$ is compact, $\omega$ is a homeomorphism of $\partial\Omega$ onto $S$.
Fix $z\in(\partial\Omega)\cap(\mathds C^*)^N$. There exists a neighborhood $U$ of $z$ in $\mathds C^N$ such that $(\partial\Omega)\cap U\subset (\mathds C^*)^N$. The set $V=\omega((\partial\Omega)\cap U)$ is a neighborhood of $\omega(z)$ in $S$, and
there exists a point $c\in V\cap(\mathds C^*)^N$. The complex hyperplane $T(w(c), \langle w(c), c \rangle )$ passes through $c$, is contained in $W(c)$, and therefore does not intersect $\Omega$.
\end{remark}

We proceed to the case  $\Omega=\mathds C^N$. We denote by {\bf 1} a point in $\mathds C^N$ (and a multi--index in $\mathds N_0^N$),  all coordinates of which are equal to $1.$


\begin{theorem} \label{PRIM2}
If the operator $\mathcal E_a$ is applicable to $H(\mathds C^N)$, then
$$\limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}<+\infty.$$
If the last condition is satisfied, then   series \eqref{REPR1} converges absolutely in $H(\mathds C^N)$ for each function $f\in H(\mathds C^N)$.
\end{theorem}

\begin{proof}
Let us prove that  the condition
$$
\limsup\limits_{|\beta|\to\infty} \left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}<+\infty
$$
is necessary for the applicability of $\mathcal E_a$. We introduce the space of sequences
$$
\Lambda_\infty=\bigg\{c\in\mathds C^{\mathds N_0^N}:\,\forall m\in\mathds N\,\,\sum\limits_{\beta\in\mathds N_0^N}|c_\beta|\frac{m^{|\beta|}}{\beta!}<+\infty\bigg\}.
$$
As it is known, for $d\in\mathds C^{\mathds N_0^N}$, the series $\sum\limits_{\beta\in\mathds N_0^N} |d_\beta c_\beta|$ converges for all $c\in\Lambda_\infty$
if and only if there exists $m\in\mathds N,$ for which $$ \sup\limits_{\beta\in\mathds N_0^N}\frac{|d_\beta|\beta !}{m^{|\beta|}}<+\infty,$$
see, for example, \cite[Sect.~2]{KOR83}.

For each $c\in\Lambda_\infty$, the function
$$ f_c(t)=\sum\limits_{\beta\in\mathds N_0} \frac{c_\beta}{\beta!} (t-{\bf 1})^\beta$$ is entire in $\mathds C^N$. Therefore, for each $c\in\Lambda_\infty$, the series
$\displaystyle \sum\limits_{\beta\in\mathds N_0^N}|a_\beta||f_c^{(\beta)}({\bf 1})|$ converges, i.e. $\displaystyle \sum\limits_{\beta\in\mathds N_0^N}|a_\beta||c_\beta|<+\infty$. Therefore,
there exists $m\in\mathds N$ for which $$ \sup\limits_{\beta\in\mathds N_0^N}\frac{\beta!|a_\beta|}{m^{|\beta|}}<+\infty,$$ and
$$ \limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}<+\infty.
$$

And in this case, the fact that the condition $$ \limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}<+\infty$$ ensures
the absolute convergence of   series \eqref{REPR1} in $H(\mathds C^N)$ is shown in the standard way by using the Cauchy formula. The proof is complete.
\end{proof}

\begin{remark}
The applicability (in another sense) to $H(\Omega)$ of infinite order differential operators
$$\mathcal L(f)(t)=\sum\limits_{\beta\in\mathds N_0^N}\varphi_\beta(t) f^{(\beta)}(t),$$
where $\varphi_\beta$, $\beta\in\mathds N_0^N$ are functions defined on a compact subset of the holomorphy domain  $\Omega$ in $\mathds C^N$, was studied in  \cite{BRMOR}.
\end{remark}

\subsection{Applicability  $\mathcal E_{\theta,a}$}
Let us study the applicability of the operator $\mathcal E_{\theta,a}$. We let  $f_\alpha(t):=t^\alpha:=t_1^{\alpha_1}\cdots t_N^{\alpha_N}$, \, $\alpha\in\mathds Z^N$.
The following statement on the action of the operator $\theta^\beta$ on the functions $f_\alpha$ is obvious.

\begin{lemma} \label{ENTIRE}
For all $\alpha\in\mathds Z^N$, $\beta\in\mathds N_0^N$ in the domain  of $f_\alpha$ the identity holds
$\theta^\beta(f_\alpha)=\alpha^\beta f_\alpha$.
\end{lemma}


We introduce the dilation operator $M_v(f)(t):=f(vt)$, $vt:=(v_j t_j)_{j=1}^N$, $v, t\in\mathds C^N$.

\begin{theorem} \label{THETA}
\begin{enumerate}
\item[(i)] If $\lim\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}=0$,   then    for each non--empty open set $\Omega$ in $\mathds C^N$ and each  function $f\in H(\Omega)$  series \eqref{REPR2} converges absolutely in $H(\Omega)$.

    \vskip5pt

\item[(ii)]  Let $\Omega=\Omega_1\times\cdots\times\Omega_N$, where $\Omega_j$, $1\leq j\leq N$, is an open set in $\mathds C$, distinct from $\mathds C$
and $\mathds C^*$.
If the operator $\mathcal E_{\theta, a}$ is applicable to $H(\Omega)$, then
$$\lim\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}=0.$$

\vskip5pt

\item[(iii)] If the operator $\mathcal E_{\theta, a}$ is applicable to  $H(\mathds C^N)$, then $$\limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}} <+\infty.$$
\end{enumerate}

\vskip5pt

\noindent
If the latter condition is satisfied, then for each function $f\in H(\mathds C^N)$   series \eqref{REPR2} converges absolutely in $H(\mathds C^N)$.
\end{theorem}

\begin{proof}
Statement (i) was essentially proved in \cite{DL2016}; see the proof of implication (c)$\Rightarrow$(d) in Theorem 2.2 in \cite{DL2016}, which goes back to \cite[Lm. 11.2]{HILLE} and  uses
the Cauchy integral formula and upper bounds for the modulus of  function $\theta^\beta(g_w)$,
where   $$g_w(t):=\frac{1}{(w_1-t_1)\cdots(w_N-t_N)}.$$

The proof of (ii) reduces to the single  variable case considered in \cite{TRYBULA}.
There exist $v_j\in\mathds C^*\cap(\partial\Omega_j)$, $1\leq j\leq N$.
Since $\theta^\beta M_v(f)=M_v\theta^\beta(f)$ for all $\beta\in\mathds N_0^N$, $f\in H\left(\Omega\right)$,
the operator $\mathcal E_{\theta,a}$ is applicable to $\displaystyle H\Big(\frac{1}{v}\Omega\Big)$, where
where
$$\frac{1}{v}\Omega:=\left\{\left(\frac{t_1}{v_1},\ldots,\frac{t_N}{v_N}\right) :\, t\in\Omega\right\}.$$
The function $$ g(t)=\frac{1}{(1-t_1)\cdots(1-t_N)}$$
is holomorphic in $\displaystyle\frac{1}{v}\Omega$ and
$$
\theta^\beta(g)(t)=\frac{A_{\beta_1}(t_1)}{(1-t_1)^{\beta_1+1}}\cdots\frac{A_{\beta_N}(t_N)}{(1-t_N)^{\beta_N+1}}, \qquad \beta\in\mathds N_0^N,
$$
where $A_n$, $n\in\mathds N_0$, is an Euler polynomial of degree $n$. By the proof of implication (3)$\Rightarrow$(5) in Theorem 6.3 in \cite{TRYBULA} and due to the fact that the polynomials $A_n$ have non--positive real roots we conclude
that there exists a sequence $\displaystyle w^{(k)}\in\frac{1}{v}\Omega$, $k\in \mathds N$, such that $w^{(k)}\to {\bf 1}$ and the inequalities
$$
|\theta^\beta(g)(w^{(k)})|\geq k^{|\beta|+N}2^N\beta!, \quad \beta\in\mathds N_0^N, \quad k\in\mathds N,
$$
hold.
This implies $$\lim\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}=0.$$

(iii): The condition $$
\limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}} <+\infty
$$
ensures the absolute convergence of  series \eqref{REPR2} in $H(\mathds C^N)$ for each function $f\in H(\mathds C^N)$ due to Lemma~\ref{ENTIRE}
and estimates implied by the Cauchy integral formula.

Suppose that  series \eqref{REPR2} converges absolutely for all $f\in H(\mathds C^N)$, $t\in\mathds C^N$. For each sequence
$c\in\Lambda_\infty$, the function $$ g_c(t):=\sum\limits_{\alpha\in\mathds N_0^N} \frac{|c_\alpha|}{\alpha !} t^\alpha$$ is entire in $\mathds C^N$. This is  why
$$
+\infty>\sum\limits_{\beta\in\mathds N_0^N} |a_\beta||\theta^\beta(g_c)({\bf 1})|
\geq \sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\frac{|c_\beta|}{\beta !}\theta^\beta(f_\beta)({\bf 1})=
\sum\limits_{\beta\in\mathds N_0^N} |a_\beta|\frac{|c_\beta|}{\beta !}\beta^\beta.
$$
This yields $$\limsup\limits_{|\beta|\to\infty}(\beta ! |a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$  The proof is complete.
 \end{proof}

 \begin{remark}
\begin{enumerate}
	\item[(i)] The condition
$$
\limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}} <+\infty
$$
is equivalent to the fact that the function $$ a(t)=\sum\limits_{\beta\in\mathds N_0^N} a_\beta t^\beta$$ is an entire function of exponential type in $\mathds C^N$,
and the condition $$\lim\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}=0$$ is equivalent to the fact that $a$ is an entire function of exponential type $0$ in $\mathds C^N$ \cite{GOLDBERG}, \cite[Ch. 3, Sect. 1]{RONKIN}, \cite[Lm. 2.1]{DL2016}.

\vskip5pt

\item[(ii)] If   $\limsup\limits_{|\beta|\to\infty}\left(\beta!|a_\beta|\right)^{\frac{1}{|\beta|}}<+\infty$, then the identities hold
     $$\mathcal E_a(f_\beta)=\lambda_\beta f_\beta, \quad \text{where} \quad \lambda_\beta=\beta!\sum\limits_{0\leq\gamma\leq\beta}\frac{a_\gamma}{(\beta-\gamma)!}, \quad \beta\in\mathds N_0^N,$$
 and
$$\mathcal E_{\theta,a}(f_\beta)=a(\beta) f_\beta, \quad \beta\in\mathds N_0^N.$$
\end{enumerate}
 \end{remark}

In what follows, $\EXP (\mathds C^N)$ (respectively, $\EXP (\{0\})$) is the space of all entire functions of exponential type (respectively, of exponential type $0$)
in $\mathds C^N$.


\vskip20pt

\subsection{Case $\Omega=(\mathds C^*)^N$}
We consider the case $\Omega=(\mathds C^*)^N$. The set $\Omega$ does not satisfy the additional assumptions made in the proof of the necessary applicability conditions in Theorems~\ref{PRIM1},~\ref{THETA}.
We will use multidimensional decreasing factorials
 $$
(t)_\beta:=(t_1)_{\beta_1}\cdots(t_N)_{\beta_N}, \quad t\in\mathds C^N,\quad \beta\in\mathds N_0^N,
$$
where
$$
(z)_n:=z(z-1)\cdots(z-n+1), \quad n\in\mathds N,\quad (z)_0:=1, \quad z\in\mathds C.
$$

 \begin{lemma} \label{ZERO}
 For eachg compact set $Q$ in $(\mathds C^*)^N$, each function $f\in H((\mathds C^*)^N)$,
each $R>1$, there exists a constant $C>0$ such that
 $$
 |t^\beta||f^{(\beta)}(t)|\leq C R^{|\beta|}, \quad t\in Q, \quad \beta\in\mathds N_0^N.
 $$
 \end{lemma}

 \begin{proof}
 For $R>1$, we let $\displaystyle\varepsilon:=1- \frac{1}{R}$. The function $f\in H((\mathds C^*)^N)$ can be expanded into the Laurent series
 $$
 f(t)=\sum\limits_{\nu\in\mathds Z^N} d_\nu t^\nu,\quad t\in(\mathds C^*)^N.
 $$
For all $\beta\in\mathds N_0^N$, $t\in(\mathds C^*)^N$
 $$
t^\beta f^{(\beta)}(t)=\sum\limits_{\nu\in\mathds Z^N} d_\nu (\nu)_\beta t^\nu,
 $$
and hence
 $$
|t^\beta|| f^{(\beta)}(t)|\leq \sum\limits_{\nu\in\mathds Z^N} |d_\nu||(\nu)_\beta||t^\nu|.
$$
We employ the Cauchy inequalities for the coefficients of the Laurent series
$$
|d_\nu|\leq\frac{M(f;r_1,\ldots,r_N)}{(r_1)^{\nu_1}\cdots(r_N)^{\nu_N}}, \quad\nu\in\mathds Z^N,
$$
 for all  $r_j>0$, $1\leq j\leq N$, where $M(f;r_1,\ldots,r_N):=\max\{|f(z)|:\, |z_j|=r_j, 1\leq j\leq N\}$.
 There exist $\tau, T>0$ such that $\tau\leq|z_j|\leq T$, $1\leq j\leq N$, for each $z\in Q$. For $\nu\in\mathds Z^N$, we define the numbers $r_j$, $1\leq j\leq N$.
If $\nu_j\leq -1$, then $r_j:=\varepsilon\tau$; if $\nu_j\geq 0$, then we let $\displaystyle r_j:=\frac{T}{\varepsilon}$.
Then, for all $t\in Q$, $\nu\in\mathds Z^N$, $1\leq j\leq N$, the inequality
$\displaystyle\left(\frac{|t_j|}{r_j}\right)^{\nu_j}\leq\varepsilon^{|\nu_j|}$ holds. Therefore, for all $t\in Q$, $\beta\in\mathds N_0^N$,
for constants
\begin{equation*}
B:=\sup\{ M(f; \rho_1,\ldots,\rho_N):\, \varepsilon\tau\leq \rho_j\leq T/\varepsilon, 1\leq j\leq N\},\qquad C:= 2^{N+1}B R^N,
\end{equation*}
we have
$$
|t^\beta||f^{(\beta)}(t)|\leq B 2^{N+1}\sum\limits_{\mu\in\mathds N_0^N}\varepsilon^{|\mu|} C_{\mu+\beta}^\mu=
\frac{B 2^{N+1}}{(1-\varepsilon)^N}\frac{\beta !}{(1-\varepsilon)^{|\beta|}}=C\beta! R^{|\beta|}.
$$
The proof is complete.
\end{proof}

We introduce the space of sequences
$$
\Lambda_1:=\left\{d\in\mathds C^{\mathds N_0^N}:\,\forall n\in\mathds N\,\, \sup\limits_{\beta\in\mathds N_0^N}\frac{|d_\beta|}{e^{|\beta|/n}}<+\infty\right\}.
$$

It coincides with the set of all sequences $d\in\mathds C^{\mathds N_0^N}$ obeying
 $$\limsup\limits_{|\beta|\to\infty}|d_\beta|^{\frac{1}{|\beta|}}\leq 1.$$

We extend Vostretsov lemma \cite{VOSTR} to the multi--dimensional case.

\begin{lemma} \label{LEMMAVOSTR}
For each $d\in\Lambda_1$ there exists a function $g\in\EXP (\{0\})$ such that $g(\beta)\geq |d_\beta|$ for all $\beta\in\mathds N_0^N$.
\end{lemma}

\begin{proof}
Let $c_\beta:=\max\{1; |d_\beta|\}$, $\beta\in\mathds N_0^N$. Then $\lim\limits_{|\beta|\to\infty} (c_\beta)^{\frac{1}{|\beta|}}=1$.
We  define the majorant sequences
$$
x_{j,n}:=\max\{c_\gamma:\, 0\leq\gamma_k\leq n,\, 1\leq k\leq N, k\ne j; \gamma_j=n\},\quad n\in\mathds N_0, \quad 1\leq j\leq N.
$$
The identities  $x_{j,n}\geq 1$, $1\leq j\leq N$, $n\in\mathds N_0$,  and
$c_\beta\leq x_{1,\beta_1}\cdots x_{N,\beta_N},$ $\beta\in\mathds N_0^N,$ hold. Indeed, for
$\beta_k=\max\{\beta_j\,: \, 1\leq j\leq N\}$ we have
$$
c_\beta\leq x_{k,\beta_k}\leq x_{1,\beta_1}\cdots x_{N,\beta_N}.
$$
The relations
\begin{align*}
(x_{j,n})^{\frac{1}{n}}&=\max\left\{\left(c_\gamma\right)^{\frac{1}{n}}:\, 0\leq\gamma_k\leq n,\, 1\leq k\leq N,  k\ne j; \gamma_j=n\right\}
\\
&\leq\max\left\{\left(c_\gamma\right)^{\frac{N}{|\gamma|}}: \,0\leq\gamma_k\leq n,\, 1\leq k\leq N, k\ne j; \gamma_j=n\right\},
\end{align*}
$1\leq j\leq N$, $n\in\mathds N$, yield
$$
\lim\limits_{n\to\infty}(x_{j,n})^{\frac{1}{n}}=1,\qquad 1\leq j\leq N.
$$
By the lemma in \cite{VOSTR}, there exist entire functions $h_j$, $1\leq j\leq N$, in $\mathds C$ of zero exponential type with non--negative Taylor
coefficients such that
$$
h_j(n)\geq x_{j,n}, \quad n\in\mathds N.
$$
For entire  functions $g_j(z)=x_{j,0}+ z h_j(z)$, $1\leq j\leq N$, in $\mathds C$ of exponential type zero  the inequalities
$h_j(n)\geq x_{j,n}$ hold for all $n\in\mathds N_0$.
The function $g(t):=g_1(t_1)\cdots g_N(t_N)$, $t\in\mathds C^N$, is an entire function of exponential type zero in $\mathds C^N$, and
$g(\beta)\geq c_\beta\geq |d_\beta|$ for each $\beta\in\mathds N_0^N$.
The proof is complete.
\end{proof}

 \begin{theorem} \label{LAST}
\begin{enumerate}
	\item[(i)] If $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1,$$ then series \eqref{REPR1} converges absolutely in $H((\mathds C^*)^N)$.
If the operator $\mathcal E_a$ is applicable to $H((\mathds C^*)^N)$, then $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1.$$

\vskip5pt

\item[(ii)] The condition $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty$$
    implies
  the absolute convergence of series \eqref{REPR2} in $H((\mathds C^*)^N)$.
If the operator $\mathcal E_{\theta,a}$ is applicable to $H((\mathds C^*)^N)$, then   $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$
\end{enumerate}
 \end{theorem}

 \begin{proof}
 (i): Suppose that $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1.$$
 By Lemma \ref{ZERO}, for each compact set $Q$ in $\mathds C^N$,
the series $$\sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\sup\limits_{t\in Q}(|t^\beta||f^{(\beta)}(t)|)$$ converges.


 Let the operator $\mathcal E_a$ be applicable to $H((\mathds C^*)^N)$. Fix $d\in\Lambda_1$.
By Lemma \ref{LEMMAVOSTR}, there exists a function $$g\in\EXP (\{0\}),\qquad  g(t)=\sum\limits_{\gamma\in\mathds N_0^N}\frac{c_\gamma}{\gamma !}t^\gamma,$$
for which $g(\beta)\geq |d_\beta|$ for all $\beta\in\mathds N_0^N$ and $c_\gamma\geq 0$,
$\gamma\in\mathds N_0^N$.
The function $$  f(t)=\sum\limits_{\gamma\in\mathds N_0^N}\frac{c_\gamma}{t^{\gamma+{\bf 1}}}$$ is holomorphic in $(\mathds C^*)^N$, and for
each $t\in(\mathds C^*)^N$ the series converges
 \begin{align*}
 \sum\limits_{\beta\in\mathds N_0^N}|a_\beta|t^\beta|\left|f^{(\beta)}(t)\right|
&=\sum\limits_{\beta\in\mathds N_0^N}|a_\beta||t^\beta|\left|\sum\limits_{\gamma\in\mathds N_0^N}\frac{(-1)^{|\beta|}c_\gamma(\gamma+\beta)!}{\gamma! t^{\gamma+\beta+{\bf 1}}}\right|
 \\
  &=\sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\left|\sum\limits_{\gamma\in\mathds N_0^N}\frac{c_\gamma(\gamma+\beta)!}{\gamma! t^{\gamma+{\bf 1}}}\right|.
  \end{align*}
  Hence, the latter series converges for  $t={\bf 1}$, that is, the series converges
  $$
   \sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\beta!\sum\limits_{\gamma\in\mathds N^N}\frac{c_\gamma(\gamma+\beta)!}{\gamma! \beta!}\geq
    \sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\beta!\sum\limits_{\gamma\in\mathds N_0^N}\frac{c_\gamma}{\gamma!}\beta^\gamma\geq \sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\beta ! |d_\beta|.
   $$
Thus, $$\sum\limits_{\beta\in\mathds N_0^N}|a_\beta|\beta! |d_\beta|<+\infty$$ for each sequence $d\in\Lambda_1$. This implies
$$\limsup\limits_{|\beta|\to\infty}(\beta! |a_\beta|)^{\frac{1}{|\beta|}}<1.$$

  (ii): Suppose that $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$ We let $$  |a|(t):=\sum\limits_{\beta\in\mathds N_0^N}|a_\beta|t^\beta,\qquad t\in\mathds C^N.$$
  We take a function $f\in H((\mathds C^*)^N)$  and expand it into the Laurent series in $(\mathds C^*)^N$
  $$  f(t)=\sum\limits_{\nu\in\mathds Z^N} d_\nu t^\nu.$$ We fix a compact set $Q$  in  $(\mathds C^*)^N$. There exists $\rho>0$ such that $|t^\nu|\leq\rho^{|\nu_1|+
\cdots+|\nu_N|}$ for all $t\in Q$, $\nu\in\mathds Z^N$.
Since $a\in \EXP (\mathds C^N)$, we also have $|a|\in \EXP (\mathds C^N)$, and
there exist constants $C, R>0$ such that
  $$
  |a|(|t_1|,\ldots,|t_N|)\leq
  C e^{R(|t_1|+\cdots+|t_N|)},\quad t\in\mathds C^N.
  $$
  Hence,
  \begin{align*}
  \sum\limits_{\beta\in\mathds N_0^N} |a_\beta|\sup\limits_{t\in Q}(|\theta^\beta(f)(t)|)&\leq\sum\limits_{\beta\in\mathds N_0^N} |a_\beta|\sum\limits_{\nu\in\mathds Z^N} |d_\nu||\nu^\beta|
  \rho^{|\nu_1|+ \cdots+|\nu_N|}
  \\
 &=\sum\limits_{\nu\in\mathds Z^N} |d_\nu|\rho^{|\nu_1|+ \cdots+|\nu_N|} |a|(|\nu_1|,\ldots,|\nu_N|)\\
&\leq  C\sum\limits_{\nu\in\mathds Z^N} |d_\nu|\left(\rho e^R\right)^{|\nu_1|+\cdots+|\nu_N|}<+\infty.
\end{align*}

Now suppose that the operator $\mathcal E_{\theta,a}$ is applicable to $H((\mathds C^*)^N)$.
Then, for each entire function $f$ in $\mathds C^N$, the series
$$\sum\limits_{\beta\in\mathds N_0^N}|a_\beta||\theta^\beta(f)({\bf 1})| $$ converges.
By the proof of Assertion (iii) in Theorem~\ref{THETA}, this implies
 $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$
The proof is complete.
 \end{proof}

Theorems~\ref{PRIM1}--\ref{LAST} and  Banach~---~Steinhaus theorem
imply

 \begin{corollary}
\begin{enumerate}
	\item[(i)] If  $$\lim\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}=0,$$ then
the operators $\mathcal E_a$, $\mathcal E_{\theta,a}$ are linear and continuous in $H(\Omega)$ for each nonempty open
set $\Omega$ in $\mathds C^N$.

\vskip5pt

 \item[(ii)] If
 $$
     \limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<+\infty,
     $$
 then the operators $\mathcal E_a$, $\mathcal E_{\theta,a}$ are linear and continuous in $H(\mathds C^N)$, while $\mathcal E_{\theta,a}$ also is  linear and continuous in $H((\mathds C^*)^N)$.

\vskip5pt

 \item[(iii)] If $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1,$$ then the operator $\mathcal E_a$ is linear and continuous in  $H((\mathds C^*)^N)$.
\end{enumerate}
 \end{corollary}

\vskip20pt

\section{Relations between representations for Euler operators}

For $\beta, \gamma\in\mathds N_0^N$, $\gamma\leq\beta$, we let
$$
s(\beta,\gamma):=\prod\limits_{j=1}^N s(\beta_j,\gamma_j), \qquad S(\beta,\gamma):=\prod\limits_{j=1}^N S(\beta_j,\gamma_j),
$$
where $s(n,k)$, respectively, $S(n,k)$, for $k, n\in\mathds N_0$, $k\leq n$, is the Stirling number of the first, respectively, second kind.
The notation $\gamma\leq\beta$ means that $\gamma_j\leq\beta_j$, $1\leq j\leq N$.
These numbers are defined in various equivalent ways. One definition uses decreasing factorials;
Stirling numbers are defined by the identities
 \cite[Ch. 2, Sect. 7]{RIORDAN}
$$
(z)_n=\sum\limits_{k=0}^n s(n,k)z^k,\qquad z^n=\sum\limits_{k=0}^n S(n,k) (z)_k,\quad n\in\mathds N_0, \quad z\in\mathds C.
$$
For all  $t\in\mathds C^N$, $\beta\in\mathds N_0^N$
$$
(t)_\beta=\sum\limits_{0\leq\gamma\leq\beta} s(\beta,\gamma)t^\gamma, \qquad t^\beta=\sum\limits_{0\leq\gamma\leq\beta} S(\beta,\gamma)(t)_\gamma.
$$

The relations in the following statement are implied by  known similar equalities for one variable \cite[Ch. 2, Probl. 18]{RIORDAN}.

\begin{lemma} \label{OPERATORY}
For each non--empty open set $\Omega$ in $\mathds C^N$, all $\beta\in\mathds N_0^N$, $f\in H(\Omega)$, $t\in\Omega$
$$
\theta^\beta(f)(t)=\sum\limits_{0\leq\gamma\leq\beta} S(\beta,\gamma) t^\gamma f^{(\gamma)}(t), \qquad
t^\beta f^{(\beta)}(t)=\sum\limits_{0\leq \gamma\leq\beta} s(\beta,\gamma) \theta^\gamma(f)(t).
$$

\end{lemma}

 Let
 $$a\in \EXP (\mathds C^N),\qquad a(t)=\sum\limits_{\beta\in\mathds N_0^N} a_\beta t^\beta, \qquad t\in\mathds C^N.$$
 To analyze the relationship between infinite order Euler operators, we need series
\begin{align} \label{NEWTON}
&\widetilde a(t):=\sum\limits_{\beta\in\mathds N_0^N} a_\beta(t)_\beta,\\
&\widehat a(t):=\sum\limits_{\gamma\in\mathds N_0^N} \left(\sum\limits_{\beta\geq\gamma} S(\beta,\gamma) a_\beta\right) t^\gamma, \quad t\in\mathds C^N. \nonumber
\end{align}
We are going to show that the series $$\sum\limits_{\beta\geq\gamma} S(\beta,\gamma)a_\beta,\qquad \gamma\in\mathds N_0^N,$$ are absolutely convergent;
we let $$\widehat a_\gamma:=\sum\limits_{\beta\geq\gamma} S(\beta,\gamma)a_\beta.$$
Series \eqref{NEWTON} (formal in the general case) is a multidimensional version of Newton  interpolation series; for holomorphic functions of one variable, it was studied, for example, in  \cite{KELDYSH}, \cite{GELF}, \cite[Sect. 1.10]{LEON}.


We shall  use some properties of Stirling numbers, in particular, the estimates for $S(\beta,\gamma)$ and $|s(\beta,\gamma)|$,


\begin{lemma} \label{ABSCHAETZUNG}
\begin{enumerate}
	\item[(i)] For all $\beta,\gamma\in\mathds N_0^N$, $\gamma\leq\beta$,
$$
\sum\limits_{\gamma\leq\nu\leq\beta} s(\beta,\nu)S(\nu,\gamma)=\sum\limits_{\gamma\leq\nu\leq\beta} S(\beta,\nu)s(\nu,\gamma)=\delta_{\gamma,\beta},
$$
$\delta_{\gamma,\beta}$ is the Kronecker delta.

\vskip5pt

\item[(ii)] $\displaystyle S(\beta,\gamma)\leq C_\beta^\gamma \gamma^{\beta-\gamma}$, \quad $\beta, \gamma\in\mathds N_0^N$, \quad $\gamma\leq\beta$.

\vskip5pt

\item[(iii)] $\displaystyle |s(\beta,\gamma)|\leq C_{\beta}^{\gamma}\frac{\beta !}{\gamma !},$ \quad $\beta, \gamma\in\mathds N_0^N,$ \quad $\gamma\leq\beta$.

\vskip5pt

\item[(iv)] For all $t\in\mathds C^N$, $\gamma\in\mathds N_0^N$ the identity holds
$$
\sum\limits_{\beta\geq\gamma} S(\beta,\gamma)\frac{t^\beta}{\beta!}=\frac{(e^{t_1} - 1)^{\gamma_1}\cdots(e^{t_N}-1)^{\gamma_N}}{\gamma!}.
$$
\end{enumerate}
\end{lemma}

\begin{proof}
The identities in (i) are implied by the same identities for $N=1$ \cite[Ch. 2, Sect. 7, (39)]{RIORDAN}.

(ii): By \cite[Probl. 7]{COMTET} the identities hold
$$
S(\beta,\gamma)\leq C_{\beta- {\bf 1}}^{\gamma-{\bf 1}}\gamma^{\beta-\gamma}, \quad \beta, \gamma\in\mathds N^N, \gamma\leq\beta.
$$
This implies inequalities in  (ii) for all $\beta, \gamma\in\mathds N_0^N$, $\gamma\leq\beta$.

(iii): Let $n\in\mathds N$. Since
$$
z(z+1)\cdots(z+n-1)=\sum\limits_{k=1}^n|s(n,k)|z^k, \quad z\in\mathds C,
$$
we have
$$
|s(n,k)|\leq C_{n-1}^{k-1}\frac{(n-1)!}{(k-1)!}\leq C_n^k\frac{n!}{k!}, \quad 1\leq k\leq n.
$$
The above can be also obtained from the identity
$$
\sum\limits_{j=k}^n |s(n,j)| S(j,k)=C_{n-1}^{k-1}\frac{n!}{k!}
$$
\cite[Ch. 2, Probl. 16~(d)]{RIORDAN} and the identity $S(k,k)=1$. The above inequalities imply the estimate in  (iii) for all $\beta, \gamma\in\mathds N_0^N$, $\gamma\leq\beta$.

(iv): For $N=1$ this relation was given, for instance, in  \cite[Ch. 2, Probl. 14]{RIORDAN}.
We note that the inequalities in  (ii) ensure the absolute convergence of the series in  (iv) for each $t\in\mathds C^N$. The proof is complete.
\end{proof}

\begin{lemma} \label{BEZIEHUNG}
\begin{enumerate}
	\item[(i)] If  $a\in \EXP (\mathds C^N)$, then $\widehat a\in \EXP (\mathds C^N)$.

\vskip5pt

\item[(ii)] If $$\limsup\limits_{|\beta|\to \infty}(\beta !|a_\beta|)^{\frac{1}{|\beta|}}<1,$$ then $\widetilde a\in\EXP (\mathds C^N)$ and $a=\widehat{\widetilde a}$.

\vskip5pt

\item[(iii)]  If $$\limsup\limits_{|\beta|\to \infty}(\beta !|a_\beta|)^{\frac{1}{|\beta|}}<\log 2,$$ then
$$\limsup\limits_{|\beta|\to \infty}(\beta !|\widehat a_\beta|)^{\frac{1}{|\beta|}}< 1$$ and
$a=\widehat{\widetilde a}=\widetilde{\widehat a}$.

\vskip5pt

\item[(iv)] If $a\in \EXP (\{0\})$, then $\widetilde a, \widehat a\in \EXP (\{0\})$ and $a=\widehat{\widetilde a}=\widetilde{\widehat a}$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i): Let $$a\in\EXP (\mathds C^N),\qquad\displaystyle a(t)=\sum\limits_{\beta\in\mathds N_0^N}a_\beta t^\beta,\qquad \sigma:=\limsup\limits_{|\beta|\to\infty}(\beta !|a_\beta|)^{\frac{1}{|\beta|}}<+\infty.$$

We fix $\varepsilon >0$. There exists $A>0$ such that $$  |a_\beta|\leq A\frac{(\sigma+\varepsilon)^{|\beta|}}{\beta !},\qquad \beta\in\mathds N_0^N.$$
By Lemma \ref{ABSCHAETZUNG} for
 $\gamma\in\mathds N_0^N$
$$
\sum\limits_{\beta\geq\gamma}S(\beta,\gamma)|a_\beta|\leq
A\sum\limits_{\beta\geq\gamma}S(\beta,\gamma)\frac{(\sigma+\varepsilon)^{|\beta|}}{\beta !}=A\frac{(e^{\sigma+\varepsilon}-1)^{|\gamma|}}{\gamma!}.
$$
Hence, $$\limsup\limits_{|\gamma|\to\infty}(\gamma ! |\widehat a_\gamma|)^{\frac{1}{|\gamma|}}\leq e^\sigma - 1,\qquad  \widehat a_\gamma=\sum\limits_{\beta\geq\gamma}S(\beta,\gamma)a_\beta,$$ and an entire function $$\widehat a(t)=\sum\limits_{\gamma\in\mathds N_0^N}\widehat a_\gamma t^\gamma,\qquad t\in\mathds C^N,$$ of exponential type
 is well--defined in $\mathds C^N$.

(ii): We choose $\delta<1$ such that $$\limsup\limits_{|\beta|\to \infty}(\beta !|a_\beta|)^{\frac{1}{|\beta|}}<\delta.$$
There exists a constant $B>0$ such that $\displaystyle |a_\beta|\leq B\frac{\delta^{|\beta|}}{\beta!}$ for each $\beta\in\mathds N_0^N$.
In view of Lemma~\ref{ABSCHAETZUNG}  we obtain
\begin{align*}
\sum\limits_{\beta\geq\gamma} |s(\beta,\gamma)||a_\beta|&\leq B \sum\limits_{\beta\geq\gamma}C_{\beta}^{\gamma}\frac{\beta!}{\gamma!}\frac{\delta^{|\beta|}}{\beta!}
= B \sum\limits_{\beta\in\mathds N_0^N}C_{\beta+\gamma}^{\gamma}\frac{\delta^{|\beta|+|\gamma|}}{\gamma!}
\\
&\leq  B \frac{\delta^{|\gamma|}}{\gamma!}\sum\limits_{\beta\in\mathds N_0^N}\frac{(\beta+\gamma)!}{\beta!\gamma!}\delta^{|\beta|}=
B\frac{\delta^{|\gamma|}}{\gamma!(1-\delta)^{|\gamma|+N}}.
\end{align*}
Therefore,
$$\sum\limits_{\beta\in\mathds N_0^N} |a_\beta|\sum\limits_{0\leq\gamma\leq\beta}|s(\beta,\gamma)||t^\gamma|<+\infty$$
for each $t\in\mathds C^N$ and once $$\widetilde a(t)=\sum\limits_{\gamma\in\mathds N_0^N}\widetilde a_\gamma t^\gamma,\quad t\in\mathds C^N,$$ we have
$$
\limsup\limits_{|\gamma|\to\infty}(\gamma ! |\widetilde{a}_\gamma|)^{\frac{1}{|\gamma|}}\leq \frac{\delta}{1-\delta}.
$$
At the same time, $$\widetilde a_\gamma=\sum\limits_{\beta\in\mathds N_0^N} s(\beta,\gamma)a_\beta,\qquad \gamma\in\mathds N_0^N.$$ Thus,  $\widetilde a\in\EXP (\mathds C^N)$.
By (i), an entire function $\widehat{\widetilde a}$ of exponential type is well--defined.
For $\beta\in\mathds N_0^N$, by Lemma~\ref{ABSCHAETZUNG},
\begin{align*}
\sum\limits_{\nu\geq\beta} S(\nu,\beta)\widetilde a_\nu&=\sum\limits_{\nu\geq\beta} S(\nu,\beta)\sum\limits_{\gamma\geq\nu} s(\gamma,\nu) a_\gamma
\\
&=\sum\limits_{\gamma\geq\beta} a_\gamma\sum\limits_{\beta\leq\nu\leq\gamma}s(\gamma,\nu)S(\nu,\beta)=a_\beta.
\end{align*}
Hence, $a=\widehat{\widetilde a}$.

Statement (iii) follows from (i) and (ii).

(iv): The belonging $\widetilde a, \widehat a\in \EXP (\{0\})$ is implied by the estimates obtained in the proof of (i) and (ii). The identities hold due to (iii). The proof is complete.
\end{proof}

We interpret statement (iii) in Lemma~\ref{BEZIEHUNG} in terms of the representation of an entire function by its Newton series.
For an entire function $a$ in $\mathds C^N$ we introduce the numbers
$$
\Delta^\gamma(a):=\sum\limits_{0\leq\nu\leq\gamma}(-1)^{\gamma-\nu} C_\gamma^{\gamma-\nu} a(\nu), \quad \gamma\in\mathds N_0^N.
$$
For $N=1$, they coincide with the finite difference of order $\gamma$ of the function $a$ at the point $0$.
Let $$\limsup\limits_{|\beta|\to\infty}(\beta !|a_\beta|)^{\frac{1}{|\beta|}}<\log 2.$$ By Lemma~\ref{BEZIEHUNG}, the function $\widehat a$ is well--defined.
We are going to show that $$\frac{1}{\gamma !}\Delta^\gamma(a)=\widehat a_\gamma,\qquad \gamma\in\mathds N_0^N,$$ where $\widehat a_\gamma$ are the Taylor coefficients of
$\widehat a$. To prove this, we use the ``direct'' identities for $S(\beta,\gamma)$, $0\leq\gamma\leq\beta$,
$$
S(\beta,\gamma)=\frac{1}{\gamma!}\sum\limits_{0\leq\nu\leq\gamma}(-1)^{\gamma-\nu} C_\gamma^{\gamma-\nu}\nu^\beta,
$$
which are implied by the corresponding relations for  $N=1$ \cite[Ch. 2, Sect. 7, Ident.  (38)]{RIORDAN}). For $\gamma\in\mathds N_0^N$
\begin{align*}
\frac{\Delta^\gamma(a)}{\gamma !}&=\frac{1}{\gamma !}
\sum\limits_{0\leq\nu\leq\gamma}(-1)^{\gamma-\nu} C_\gamma^{\gamma-\nu} \sum\limits_{\beta\in\mathds N_0^N} a_\beta\nu^\beta
\\
&=\sum\limits_{\beta\in\mathds N_0^N} a_\beta\frac{1}{\gamma !}
\sum\limits_{0\leq\nu\leq\gamma}(-1)^{\gamma-\nu} C_\gamma^{\gamma-\nu} \nu^\beta=\sum\limits_{\beta\geq\gamma} a_\beta S(\beta,\gamma).
\end{align*}
The latter identity holds since
$$
\sum\limits_{0\leq\nu\leq\gamma}(-1)^{\gamma-\nu} C_\gamma^{\gamma-\nu} \nu^\beta=0
$$
for
 $\gamma\nleqslant\beta$. This is, for instance, since $g^{(m)}(0)=0$ for $m,n \in\mathds N_0$, $m<n$, where
$$ g(z)=(e^z-1)^n=\sum\limits_{k=0}^n(-1)^{n-k} C_n^{n-k}e^{kz},\qquad z\in\mathds C.$$
Thus, the identity $a=\widetilde{\widehat a}$ in Lemma~\ref{BEZIEHUNG}~(iii) means that each entire function $a$ obeying
\begin{equation*}
\limsup\limits_{|\beta|\to\infty}(\beta ! |a_\beta|)^{\frac{1}{|\beta|}}<\log 2
\end{equation*}
can be expanded into the  Newton series.
For $N=1$, this is known, see, for example, \cite[Thm. 5.11]{LEON}. In \cite{LEON}, the proof was made by using integral representations related with the function
associated with $a$  in Borel sense.


\begin{remark}
\begin{enumerate}
	\item[(i)] In statement (i) of Lemma \ref{BEZIEHUNG}, in the condition $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1,$$ the number $1$ cannot be replaced by a larger number.
Indeed, for $\displaystyle a_\beta:=\frac{1}{\beta!}$, $\beta\in\mathds N_0^N$ (and then $a(t)=e^{\langle 1,t\rangle}$), for
$t_j=-1$, $1\leq j\leq N$, the series in the definition of   function $\widetilde a(t)$ is not convergent for any permutation of indices from $\mathds N_0^N$.

In the condition $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<\log 2$$ of statement (iii) in Lemma~\ref{BEZIEHUNG}, the number $\log 2$ also cannot
be replaced by a larger one. For example, if
$a(t)=2^{\langle {\bf 1}, t\rangle}$, then $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}=\log 2,\qquad \widehat a(t)=e^{\langle {\bf 1},t\rangle},$$ and the function $\widetilde{\widehat a}$ is ill--defined.

 \vskip5pt

\item[(ii)] The quantity $$\sigma=\limsup\limits_{|\beta|\to \infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}$$ can be characterized in terms of the set
$$
L_1:=\left\{x\in [0,+\infty)^N:\, \sum\limits_{j=1}^N x_j\leq 1\right\}.
$$
For $a\in\EXP (\mathds C^N)$ the number $\sigma$ coincides with  the $L_1$--type of function  $a$ \cite[Ch. 3, Sect. 1]{RONKIN}.
\end{enumerate}
\end{remark}

The following statement is implied the solvability of a more general interpolation problem (see, for example, \cite[Thm. 1]{BERTAY}; for $N=1$, its solvability was proved in
\cite[Thm. 2]{LEONINT}). We present an independent, simple proof that employs the specificity of the node set in this situation. It is made by the method applied in the proof of its one--dimensional analog in \cite[Thm. 1.2.1]{BIBER}.



\begin{lemma} \label{INTERP}
For each sequence $g_\beta\in\mathds C$, $\beta\in\mathds N_0^N$ such that
$$\limsup\limits_{|\beta|\to\infty}(|g_\beta|)^{\frac{1}{|\beta|}}<+\infty,$$ there exists a function $f\in \EXP (\mathds C^N)$ such that
$f(\beta)=g_\beta$ for each $\beta\in\mathds N_0^N$.
\end{lemma}

\begin{proof}
The function $g(t)=\sum\limits_{\beta\in\mathds N_0^N} g_\beta t^\beta$ is holomorphic on the closed polydisk $\{t\in\mathds C^N\,|\,|t_j|\leq\varepsilon, 1\leq j\leq N\}$
for some $\varepsilon>0$. By the Cauchy integral formula we have
$$
g_\beta=\frac{1}{(2\pi i)^N}\int\limits_{|t_1|=\varepsilon}\cdots\int\limits_{|t_N|=\varepsilon}\frac{g(t)}{t^{\beta+{\bf 1}}}dt_N\cdots dt_1, \quad \beta\in\mathds N_0^N.
$$
We make the change $\tau_j=\log t_j$, $1\leq j\leq N$ ($\log  z$ denotes any continuous univalent branch of the logarithm).
We obtain
$$
g_\beta=\frac{1}{(2\pi i)^N}\int\limits_{\Gamma_1}\cdots\int\limits_{\Gamma_N}g\left(e^{\tau_1},\ldots, e^{\tau_N}\right) e^{-\langle\beta,\tau\rangle} d\tau_N\cdots d\tau_1,
$$
where $\Gamma_j$ is the vertical segment in  $\mathds C$. The function
$$
f(z):=\frac{1}{(2\pi i)^N}\int\limits_{\Gamma_1}\cdots\int\limits_{\Gamma_N} g\left(e^{\tau_1},\ldots, e^{\tau_N}\right) e^{-\langle z,\tau\rangle} d\tau_N\cdots d\tau_1
$$is an entire function of exponential type in $\mathds C^N$ and $f(\beta)=g_\beta$ for all $\beta\in\mathds N_0^N$. The proof is complete.
\end{proof}

\begin{theorem} \label{MAIN}
\begin{enumerate}
	\item[(i)] For each function $a\in \EXP (\{0\})$, each non--empty open set $\Omega$ in $\mathds C^N$ in $H(\Omega)$, the identities $\mathcal E_a=\mathcal E_{\theta, \widetilde a}$ and $\mathcal E_{\theta,a}=\mathcal E_{\widehat a}$ hold.


\vskip5pt

\item[(ii)] For each function $a\in{\rm Exp}(\mathds C^N)$ in $H(\mathds C^N)$ the identity $\mathcal E_{\theta,a}=\mathcal E_{\widehat a}$ holds.

\vskip5pt

\item[(iii)] For each function $a\in{\rm Exp}(\mathds C^N)$ there exists $c\in \EXP (\mathds C^N)$ such that
$\mathcal E_a=\mathcal E_{\theta,c}$ in $H(\mathds C^N)$.

\vskip5pt

\item[(iv)] If  $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1,$$ then  $\mathcal E_a=\mathcal  E_{\theta,\widetilde a}$ in $H(\mathds C^N)$.

\vskip5pt

\item[(v)] If
$$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1,$$ then $\mathcal E_a=\mathcal E_{\theta,\widetilde a}$ in $H((\mathds C^*)^N)$. If $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<\log 2,$$ then $\mathcal E_{\theta,a}=\mathcal E_{\widehat a}$ in $H((\mathds C^*)^N)$.
\end{enumerate}

\end{theorem}

\begin{proof}
(i): By the proofs of Lemma~\ref{BEZIEHUNG} and Theorems~\ref{PRIM1} and~\ref{THETA} for all   $f\in H(\Omega)$, $t\in\Omega$ we have
\begin{align} \label{PEREST1}
&\sum\limits_{\gamma\in\mathds N_0^N}|\theta^\gamma(f)(t)|\sum\limits_{\beta\geq\gamma}|s(\beta,\gamma)||a_\beta|<+\infty,
\\ \label{PEREST2}
&\sum\limits_{\gamma\in\mathds N_0^N}|t^\gamma||f^{(\gamma)}(t)|\sum\limits_{\gamma\geq\gamma}S(\beta,\gamma)|a_\beta|<+\infty.
\end{align}
Due to  \eqref{PEREST1} and Lemma~\ref{OPERATORY} for all  $f\in H(\Omega)$, $t\in\Omega$,
$$
\mathcal E_{\theta,\widetilde a}(f)(t)=\sum\limits_{\gamma\in\mathds N_0^N}\theta^\gamma(f)(t)\sum\limits_{\beta\geq\gamma} s(\beta,\gamma)a_\beta=
\sum\limits_{\beta\in\mathds N_0^N} a_\beta\sum\limits_{0\leq\gamma\leq\beta}s(\beta,\gamma)\theta^\gamma(f)(t)=\mathcal E_a(f)(t).
$$
By  (\ref{PEREST2}) and Lemma~\ref{OPERATORY}, for all $f\in H(\Omega)$, $t\in\Omega$,
$$
\mathcal E_{\widehat a}(f)(t)=\sum\limits_{\gamma\in\mathds N_0^N}t^\gamma f^{(\gamma)}(t)\sum\limits_{\beta\geq\gamma}S(\beta,\gamma)a_\beta=
\sum\limits_{\beta\in\mathds N_0^N} a_\beta\sum\limits_{0\leq\gamma\leq\beta}S(\beta,\gamma)\theta^\gamma(f)(t)=\mathcal E_{\theta,a}(f)(t).
$$

(ii): In this case, for all $f\in H(\mathds C^N)$, $t\in\mathds C^N$ the relation \eqref{PEREST2} is valid, and hence,
$\mathcal E_{\widehat a}(f)(t)=\mathcal E_{\theta,a}(f)(t)$.

(iii): The identities  hold $${\mathcal E}_a(f_\beta)=\lambda_\beta f_\beta,\qquad
 \lambda_\beta=\beta! \sum\limits_{0\leq\gamma\leq\beta}\frac{a_\gamma}{(\beta-\gamma)!},\qquad \beta\in\mathds N_0^N.$$
 Let us estimate  $|\lambda_\beta|$ from above. Since
$$
|\lambda_\beta|\leq \beta!\sum\limits_{0\leq\gamma\leq\beta}\frac{|a_\gamma|}{(\beta-\gamma)!}
$$
and there exist $C,\sigma>0$ such that $\displaystyle|a_\nu|\leq C\frac{\sigma^{|\nu|}}{\nu !}$ for all $\nu\in\mathds N_0^N$,
we have
$$
|\lambda_\beta|\leq C\sum\limits_{0\leq\gamma\leq\beta}\frac{\beta !\sigma^{|\gamma|}}{\gamma!(\beta-\gamma)!}
=C(1+\sigma)^{|\beta|}, \quad \beta\in\mathds N_0^N.
$$
By Lemma \ref{INTERP}, there exists a function $c\in \EXP (\mathds C^N)$ such that $c(\beta)=\lambda_\beta$ for all $\beta\in\mathds N_0^N$.
Since $\mathcal E_{\theta,c}(f_\beta)=c(\beta)f_\beta=\mathcal E_a(f_\beta)$ for each $\beta\in\mathds N_0^N$,
the set of all polynomials is dense in $H(\mathds C^N)$ and the operators $\mathcal E_{\theta,c}$ and $\mathcal E_a$ are linear and continuous in $H(\mathds C^N)$ ,
then $\mathcal E_{\theta,c}=\mathcal E_a$ on the entire space $H(\mathds C^N)$.


(iv): By (ii) and Lemma~\ref{BEZIEHUNG} $\mathcal E_{\theta,\widetilde a}=\mathcal E_{\widehat{\widetilde a}}=\mathcal E_a$ in $H(\mathds C^N)$.

(v): If $\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<1$, by Theorem~\ref{LAST} and the proof of Lemma~\ref{BEZIEHUNG}~(ii) for all  $f\in H((\mathds C^*)^N)$,
$t\in(\mathds C^*)^N$ relation  \eqref{PEREST1} holds. This is why  $\mathcal E_a=\mathcal E_{\theta,\widetilde a}$ in $H((\mathds C^*)^N)$.

If $$\limsup\limits_{|\beta|\to\infty}(\beta!|a_\beta|)^{\frac{1}{|\beta|}}<\log 2,$$ then by Theorem~\ref{LAST} and the proof of Assertion~(i) in  Lemma~\ref{BEZIEHUNG}, for all  $f\in H((\mathds C^*)^N)$,
$t\in(\mathds C^*)^N$ relation  \eqref{PEREST2} holds. This is why $\mathcal E_{\theta,a}=\mathcal E_{\widehat a}$ in $H((\mathds C^*)^N)$.
The proof is complete.
\end{proof}

\vskip20pt

\section{Multi--dimensional verions of  Wigert~---~Leau theorem}

We present one result that can be considered a multi--dimensional version of the Wigert~---~Leau theorem \cite[Thm. 1.3.2]{BIBER}, the proof of which uses  Lemma~\ref{BEZIEHUNG}.
To justify this terminology, we note that a fact for functions of one variable $t$  essentially contained in \cite[Sect. 2]{KOR69}.
The function $f(t)$ is holomorphic in a neighborhood of $0$ and holomorphically extends to $\overline{\mathds C}\setminus\{1\}$ if and only if
the function $\displaystyle\frac{1}{t}f\left(\frac{1}{t}\right)$ is holomorphic in a neighborhood of infinity and holomorphically extends to $\overline{\mathds C}\backslash\{1\}$.
Thus, if $$  g(t)=\sum\limits_{n=0}^\infty \frac{g^{(n)}(0)}{n!}t^n$$ is a nonzero entire function of exponential type in $\mathds C$, then
the function $$g_0(t)=\sum\limits_{n=0}^\infty g^{(n)}(0) t^n$$ is entire in $\displaystyle\frac{1}{1-t}$ if and only if the conjugate diagram of $g$ coincides
with $\{1\}$. Therefore, the Wigert~---~Leau theorem can be reformulated as follows: the conjugate diagram of an entire function $g$ of exponential type coincides with
$\{1\}$ if and only if there exists an entire function $a$ of exponential type 0 such that $a(n)=g^{(n)}(0)$ for all $n\in\mathds N_0$.
Therefore, the theorem below can be considered a multidimensional analogue of the Wigert~---~Leau theorem.
We will use information  on analytic functionals from \cite[Ch. 4, Sect. 4.5]{KHERM}.
Below, $H(\mathds C^N)'$ is the topological dual space of $H(\mathds C^N)$;
$\mathcal F: H(\mathds C^N)'\to \EXP (\mathds C^N)$ is the Laplace transform
$$
\mathcal F(\varphi)(t):=\varphi_z(e^{\langle z,t\rangle}),\quad\varphi\in H(\mathds C^N)',\quad t\in\mathds C^N.
$$

\begin{theorem} For each non-zero functional $\varphi\in H(\mathds C^N)'$ the following conditions are equivalent:

\vskip5pt

\begin{enumerate}
\item[(i)] The defining compact set of $\varphi$ coincides with   $\{{\bf 1}\}$.

\vskip5pt

\item[(ii)] There exists a function $a\in\EXP (\{0\})$ such that  $a(\beta)=(\mathcal F(\varphi))^{(\beta)}(0)$ for each $\beta\in\mathds N_0^N$.
\end{enumerate}
\end{theorem}

\begin{proof} (i)$\Rightarrow$(ii): We introduce the function $a(t)=\mathcal F(\varphi)(t)e^{-\langle 1, t\rangle}$, $t\in\mathds C^N$.
Since $a\in\EXP (\{0\})$, then by Lemma~\ref{BEZIEHUNG} the function $\widetilde a\in \EXP (\{0\})$ is well--defined, and $$  \widetilde a(\beta)=\beta! \sum\limits_{0\leq\gamma\leq\beta}
\frac{a_\gamma}{(\beta-\gamma)!}$$ for all $\beta\in\mathds N_0^N$.
Since $\mathcal F(\varphi)(t)=a(t) e^{\langle 1,t\rangle}$, $t\in\mathds C^N$, we obtain
$(\mathcal F(\varphi))^{(\beta)}(0)=\widetilde a(\beta)$, $\beta\in\mathds N_0^N$.

(ii)$\Rightarrow$(i):
Let $a=\mathcal F(\psi)$ for a functional $\psi\in H(\mathds C^N)'$, the defining compact set of which coincides
with
 $\{ 0\}$. For each $t\in\mathds C^N$
\begin{align*}
\mathcal F(\varphi)(t)&=\sum\limits_{\beta\in\mathds N_0^N}\frac{(\mathcal F(\varphi))^{(\beta)}(0)}{\beta !} t^\beta=\sum\limits_{\beta\in\mathds N_0^N}\frac{a(\beta)}{\beta !} t^\beta
=\sum\limits_{\beta\in\mathds N_0^N}\frac{\psi_u(e^{\langle\beta, u\rangle})}{\beta !}t^\beta
\\
&=\psi_u\left(\sum\limits_{\beta\in\mathds N_0^N}\frac{e^{\langle\beta, u\rangle}}{\beta !}t^\beta\right)=\psi_u\left(\sum\limits_{\beta\in\mathds N_0^N}\frac{(e^{u_1})^{\beta_1}\cdots(e^{u_N})^{\beta_N} t^\beta}{\beta !}\right)
=\psi_u\left(e^{e^{u_1}t_1+\cdots e^{u_N} t_N}\right).
\end{align*}
We fix $\varepsilon>0$ and choose $\delta>0$ such that
$|e^w-1|<\varepsilon$ for all $w\in\mathds C^N$ satisfying the inequality $|w|<\delta$. Since the defining compact set $\psi$
coincides with $\{0\}$, there exists $C>0$ such that for each $t\in\mathds C^N$
\begin{align*}
|\mathcal F(\psi)(t)|&\leq C\sup\limits_{|u_j|\leq\delta, 1\leq j\leq N}| e^{e^{u_1}t_1+\cdots +e^{u_N} t_N}|
\\
&\leq C \sup\limits_{|u_j|\leq\delta, 1\leq j\leq N}e^{\RE (e^{u_1}t_1+\cdots e^{u_N}t_N)}
\\
&=C\Bigg(\sup\limits_{|u_j|\leq\delta, 1\leq j\leq N} e^{\RE ((e^{u_1}-1)t_1)+\cdots+\RE ((e^{u_N}-1)t_N)}\Bigg) e^{\RE t_1+\cdots+\RE t_N}
\\
&\leq Ce^{\varepsilon|t_1|+\cdots+\varepsilon|t_N|}e^{\RE \langle 1, t\rangle}.
\end{align*}
Since $\RE t_1+\cdots+\RE t_N=\RE \langle 1, t\rangle$, this implies that defining compact set  of $\varphi$ coincides with  $\{{\bf 1}\}$. The proof is complete.
\end{proof}

\vskip20pt

\section{Each Hadamard operator on  space of \\ all entire functions is   Euler operator}

Below, $\mathcal L_h(H(\mathds C^N))$ is the space of all Hadamard type (Hadamard) operators in $H(\mathds C^N)$, i.e., of continuous linear operators $A$ in $H(\mathds C^N)$,  for which
each monomial $f_\alpha$, $\alpha\in\mathds N_0^N$, is an eigenvector.
By \cite[Thm. 2]{MATZAM21}, the mapping $\varphi\mapsto A_\varphi$, where
$$
A_\varphi(f)(t):=\varphi_z(f(tz)), \qquad t\in\mathds C^N,
$$
is
bijective  from $H(\mathds C^N)'$ into $\mathcal L_h(H(\mathds C^N))$. Let
$\mathcal L_{\mathcal E}(H(\mathds C^N))$ be the set of all operators $\mathcal E_a$, $a\in\EXP (\mathds C^N)$. Each operator $\mathcal E_a$ is Hadamard;
it coincides with $\displaystyle A_{\varphi_a}$, where
$$
\varphi_a=\sum\limits_{\beta\in\mathds N_0^N} a_\beta\delta_{1,\beta}, \qquad \delta_{1,\beta}(f)=f^{(\beta)}({\bf 1}).
$$
Let us show that each Hadamard operator in $H(\mathds C^N)$ is an Euler operator of form \eqref{REPR1}, and hence also  \eqref{REPR2}.


\begin{theorem}
The identity holds  $\mathcal L_h(H(\mathds C^N))=\mathcal L_{\mathcal E}(H(\mathds C^N))$.
\end{theorem}

\begin{proof}
For all  $\varphi\in H(\mathds C^N)'$, $f\in H(\mathds C^N)$
$$
\varphi(f)=\varphi_z\Bigg(\sum\limits_{\beta\in\mathds N_0^N}\frac{f^{(\beta)}({\bf 1})}{\beta !}(z-{\bf 1})^\beta\Bigg)=\sum\limits_{\beta\in\mathds N_0^N}f^{(\beta)}({\bf 1})\frac{1}{\beta !}\varphi_z((z-{\bf 1})^\beta).
$$
There exist $C, R>0$ such that for each $\beta\in\mathds N_0^N$
$$
|\varphi_z((z-{\bf 1})^\beta)|\leq C\sup\limits_{|z_j-1|\leq R, 1\leq j\leq N}|(z-{\bf 1})^\beta|= C R^{|\beta|}.
$$
Hence, if $c_\beta:=\varphi_z((z-{\bf 1})^\beta)$, then $$\limsup\limits_{|\beta|\to\infty}|c_\beta|^{\frac{1}{|\beta|}}\leq R$$
and
 $$ c(t)=\sum\limits_{\beta\in\mathds N_0^N}\frac{c_\beta}{\beta !}t^\beta,\qquad  t\in\mathds C^N,$$ is an entire function of exponential type in  $\mathds C^N$. Therefore, $\varphi=\varphi_c$ and $A_\varphi=\mathcal E_c$.
Thus,  $\mathcal L_h(H(\mathds C^N))=\mathcal L_{\mathcal E}(H(\mathds C^N))$.
The proof is complete.
\end{proof}

\begin{remark}
Let
\begin{align*}
&\Omega:=\Big\{z\in\mathds C^N: \,\sum\limits_{j=1}^N |z_j|^2<R^2\Big\},\qquad R\in (0, +\infty),\qquad \lambda\in\mathds C^N,
\\
& |\lambda_j|\leq 1,\qquad 1\leq j\leq N,\qquad \lambda\ne{\bf  1}.
\end{align*}
The dilation operator $M_\lambda(f)(t)=f(\lambda_1 t_1,\ldots,\lambda_N t_N)$ is Hadamard in $H(\Omega)$,
but is not an Euler operator of  form \eqref{REPR1} and \eqref{REPR2} for $a\in\EXP (\{0\})$. Indeed,
suppose that there exists a function $a\in \EXP (\{0\})$ such that $M_\lambda=\mathcal E_a$ in $H(\Omega)$. Then, for $f(t)=e^{\langle {\bf 1}, t\rangle}$, for all $t\in\mathds C^N$, the identities $a(t)f(t)=\mathcal E_a(f)(t)=f(\lambda t)$ hold, and hence, $a(t)=e^{\langle \lambda-{\bf 1}), t\rangle}$.
We obtain a contradiction  since the function $e^{\langle \lambda- {\bf 1}, t \rangle}$ does not belong to $\EXP (\{0\})$.
\end{remark}


\vskip30pt

\begin{thebibliography}{39}

\bibitem{BRMOR}
G.G. Braichev, V.V. Morzhakov. \textit{Applicability of partial differential operators of infinite order} // Math. Notes {\bf 24}:6,  910--913  (1978). \hrr{https://doi.org/10.1007/BF01140017}


\bibitem{BIBER}
L. Bieberbach. {\it Analytic Continuation.} Nauka, Moscow (1967). (in Russian).

\bibitem{VOSTR}
B.A.  Vostretsov. \textit{On the existence of boundary values and on the integral representation of functions analytic in the unit circle} //  Dokl. Akad. Nauk SSSR. {\bf 65}:1, 7--8 (1949). (in Russian).

\bibitem{GELF}
A.O. Gelfond. {\it Calculus of Finite Differences.} GIMFML, Moscow (1959). (in Russian).

\bibitem{GOLDBERG}
A.A. Goldberg. {\it Elementary remarks on formulas for determining order and type of entire functions of many variables} // Dokl. Akad. Nauk Arm. SSR. {\bf 29}, 145--152 (1959). (in Russian).

\bibitem{ZNAM}
S.V. Znamenskii. \textit{The solvability of differential equations of infinite order in spaces of holomorphic functions, a theorem of Leont'ev, and a formula of Vostrecov} //  Siberian Math. J. \textbf{18}:6, 925--935 (1977). \hrr{https://doi.org/10.1007/BF00969231}


\bibitem{KELDYSH}
I.I. Ibraghimov, M.V. Keldych. {\it On interpolation of entire functions} // Matem. Sb. {\bf 20(62)}:2, 283--291 (1947). \hrr{https://www.mathnet.ru/eng/sm6217}

\bibitem{MATZAM21}
O.A. Ivanova, S.N. Melikhov. \textit{
Operators of almost Hadamard type and Hardy~---~Littlewood operator in the space of entire functions of several complex variables} // Math. Notes \textbf{110}:1, 61--71 (2021). \hrr{https://doi.org/10.1134/S0001434621070063}


\bibitem{KORSMJ69}
Yu.F. Korobeinik. \textit{The applicability of differential operators of infinite order} // Siberian Math. J. {\bf 10}:3, 395--404 (1969). \hrr{https://doi.org/10.1007/BF01078329}


\bibitem{KOR83}
Yu. F. Korobeinik.  {\it Translation operators on number sets.} Rostov State Univ. Publ., Rostov--on--Don (1983). (in Russian).

\bibitem{KOR69}
Yu.F. Korobeinik, Yu.M. Donskov. {\it Analytic solutions of the Euler equation of infinite order} // Izv. Vyssh. Uchebn. Zaved, Mat. {\bf 1969}:11, 44--52 (1969). (in Russian).

\noindent\hrr{https://www.mathnet.ru/eng/ivm3594}

\bibitem{LEIHT}
K. Leichtweiss. \textit{Convex Sets.} Nauka, Moscow (1985). (in Russian).


\bibitem{LEONINT}
A. F. Leont'ev. {\it On a question of interpolation in the class of entire functions of finite order} \\  Matem. Sb. {\bf 83}:1, 81--96 (1957). (in Russian). \hrr{https://www.mathnet.ru/eng/sm5015}

\bibitem{LEON}
A.F. Leont'ev. \textit{Entire Functions. Exponential Series.} Nauka, Moscow (1983). (in Russian).


\bibitem{LIN}
S.S. Linchuk.  {\it Diagonal operators in spaces of analytic functions and their applications} // in ``Topical Questions in Theory of Functions''. Rostov State Univ. Publ., Rostov--on--Don,   118--121 (1987). (in Russian).

\bibitem{PIETCH}
A. Pietsch. \textit{Nuclear Locally Convex Spaces.} Akademie--Verlag, Berlin  (1972).

\bibitem{RIORDAN}
J. Riordan. \textit{An Introduction to Combinatorial Analysis. }  John Wiley \& Sons, Inc. New York  (1958).


\bibitem{RONKIN}
L.I. Ronkin. \textit{Introduction to the Theory of Entire Functions of Several Variables}. Nauka, Moscow (1971); \textsl{English translation:} Amer. Math. Soc., Providence, R.I.  (1974).

\bibitem{KHERM}
L. H\"ormander. \textit{An Introduction to Complex Analysis in Several Variables.}  D. van Nostrand Company, Inc. Princeton  (1966).


\bibitem{BERTAY}
C.A.~Berenstein, B.A.~Taylor. {\it On the geometry of interpolating varieties} // in ``S\'eminaire Pierre Lelong~---~Henri Skoda (Analyse) Ann\'ees 1980/81'', P. Lelong, H. Scoda (eds.), Springer--Verlag, Berlin, 1--25 (1982). \hrr{https://doi.org/10.1007/BFb0097041}

\bibitem{COMTET}
L.~Comtet. {\it Advances Combinatories.The Art of Finite and Infinite Expansions.}  D. Reider Pub. Co., Dordrecht (1974).

\bibitem{DAVIS}
H.T.~Davis. {\it The Euler differential equation of infinite order} // Amer. Math. Monthly {\bf 32}:5, 223--233 (1925). \hrr{https://doi.org/10.2307/2299190}

\bibitem{DOMLANG1}
P.~Doma\'nski, M.~Langenbruch. {\it Representation of multipliers on spaces of real analytic functions}~// Analysis, M\"unchen  {\bf 32}:2, 137--162 (2012). \hrr{https://doi.org/10.1524/anly.2012.1150}

\bibitem{DOMLANG2}
P.~Doma\'nski, M.~Langenbruch. {\it Hadamard multipliers on spaces of real analytic functions} // Adv. Math. {\bf 240}, 575--612 (2013).
 \hrr{https://doi.org/10.1016/j.aim.2013.01.015}

\bibitem{DOMLANG2015_INTERP}
P.~Doma\'nski, M.~Langenbruch. {\it Interpolation of holomorphic functions and surjectivity of Taylor coefficient multipliers} // Adv. Math. {\bf 293}, 782--855 (2016).

\noindent
\hrr{https://doi.org/10.1016/j.aim.2013.01.015}

\bibitem{DL2016}
P.~Doma\'nski, M.~Langenbruch. {\it Euler type partial differential operators on real analytic functions}~// J. Math. Anal. Appl. {\bf 443}:2, 652--674 (2016). \hrr{https://doi.org/10.1016/j.jmaa.2016.05.018}

\bibitem{DOMLANG3}
P.~Doma\'nski, M.~Langenbruch, D.~Vogt. {\it Hadamard type operators on spaces of real analytic functions in several variables} // J. Funct. Anal. {\bf 269}:12,  3868--3913 (2015).

\noindent
\hrr{https://doi.org/10.1016/j.jfa.2015.09.011}

\bibitem{HILLE}
E.~Hille. {\it Analytic Function Theory. Vol. II.}  Chelsea Publishing Company, London (1973).

\bibitem{ISHIM1}
R.~Ishimura. {\it Existence locale de solutions holomorphes pour les \'equations diff\'eretielles d'ordre infini} // Ann. Inst. Fourier {\bf 35}:3, 49--57 (1985). \hrr{https://doi.org/10.5802/aif.1018}

\bibitem{ISHIM2}
R.~Ishimura. {\it Sur les \'equations diff\'erentielles d'ordre infini d'Euler}
// Mem. Fac. Sci., Kyushu Univ., Ser. A  {\bf 44}:1, 1--10 (1990). \hrr{https://doi.org/10.2206/kyushumfs.44.1}


\bibitem{TRYBULA}
M.~Trybula. {\it Hadamard multipliers on spaces of holomorphic functions} // Int. Equ. Oper. Theory {\bf 88}:2, 249--268 (2017). \hrr{https://doi.org/10.1007/s00020-017-2369-7}

\end{thebibliography}

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