Descriptions of Spectra of Algebras of Bounded-Type Block-Symmetric Analytic Functions

Abstract

This paper is devoted to the study of the algebra of bounded-type block-symmetric analytic functions on the Banach space ${\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ In particular, it presents a description of the spectrum of this algebra in terms of exceptional characters ${\varphi }_{\alpha }$ and characters that can be associated with exponential-type functions of several variables with plane zeros. Due to this representation, it is proven that every element of the spectrum is a convolution of an exceptional character with a point evaluation functional.

1. Introduction

Let X be a complex Banach space. We say that a function $f:X\to \mathbb{C}$ is S-symmetric if for every $T\in S$ we have $f\circ T=f,$ where S is a group or semigroup of isometric operators on $X.$ Algebras of S-symmetric analytic functions on infinite-dimensional spaces have been extensively investigated in recent years using methods of nonlinear functional analysis. An important task in the investigation of these algebras is the description of their spectra, that is, the set of nonzero continuous linear multiplicative functionals (or characters). In the case where $S=S_{\mathbb{N}}$ is the group of all permutations of the natural numbers $\mathbb{N},$ we say that an S-symmetric function is symmetric.

The algebra of bounded-type symmetric analytic functions is defined as the completion of the algebra of symmetric continuous polynomials on X with respect to the topology of uniform convergence on bounded subsets. The investigation of symmetric polynomials in infinite-dimensional settings was pioneered by Nemirovski and Semenov in [1], where they introduced algebraic bases for algebras of symmetric real-valued polynomials on the Banach spaces ${\mathcal{l}}_p$ and $L_p[0,1],$ for $1\le p<\infty .$ This line of research was further developed in [2], where the construction of algebraic bases was extended to separable-sequence Banach spaces with symmetric bases, as well as to separable rearrangement-invariant Banach function spaces. When $X={\mathcal{l}}_p$, $1\le p<\infty ,$ polynomials $F_k\left(x\right)={\displaystyle \sum _{n=\lceil p\rceil }^{\infty }}{x}_n^k,k\in \mathbb{N},$ form an algebraic basis of algebra for symmetric polynomials [2], where $\lceil p\rceil$ is the ceiling of p (the smallest integer not less than p). Later on, numerous papers with investigations of symmetric structures and mappings on infinite-dimensional Banach spaces were published (see, e.g., [3,4,5,6,7] and the references therein). For classical symmetric function theory and its application in algebra, combinatorics, and classical invariant theory, we refer the reader to [8,9,10].

The study of the spectra of the algebras ${\mathcal{H}}_b\left(X\right)$, consisting of all entire bounded-type (analytic) functions on Banach spaces $X,$ was initiated by Aron, Cole, and Gamelin in [11]. They discovered that the spectrum of ${\mathcal{H}}_b\left(X\right)$ can possess a complex structure, which, in particular, includes extended point-evaluation functionals corresponding to elements of the second dual space $X^{**}$ (see also [12,13,14,15]). The study of the algebra of bounded-type symmetric analytic functions on the Banach space ${\mathcal{l}}_1$ and its spectrum is presented in [16,17]. In particular, in [17], a representation of the spectrum of the algebra of symmetric analytic functions on ${\mathcal{l}}_1$, as a multiplicative semigroup of exponential-type entire functions of a complex variable, was found. Using this representation, in [16], a complete description of the spectrum of this algebra is given.

It is well-known that every commutative uniform topological algebra can be represented as an algebra of continuous functions on its spectrum (the set of complex homomorphisms). This fact is convenient for understanding the structure of a given topological algebra, and it is useful for constructing operators that extend elements of the algebra, represented as functions, to larger domains. For example, since the second dual space $X^{**}$ to a Banach space X is in the spectrum of the algebra of bounded-type analytic functions on $X$, ${\mathcal{H}}_b\left(X\right),$ the Aron–Berner extension operator [18,19] is a well-defined homomorphism of algebras ${\mathcal{H}}_b\left(X\right)$ and ${\mathcal{H}}_b\left(X^{**}\right)$ (see [11] for details).

Algebras of bounded-type symmetric analytic functions on spaces of Lebesgue-measurable functions and their spectra on $L_{\infty }[0,1]$ and $L_p[0,1]$ have been studied in [20,21,22]. The topological algebra of entire bounded-type functions, generated by a countable set of homogeneous polynomials on a complex Banach space, and other generalizations of symmetric analytic functions were considered in [23,24,25].

This paper explores alternative ways in which the symmetric group $S_{\mathbb{N}}$ of all permutations of natural numbers can be represented in a Banach space. Specifically, we examine the case where the space X is constructed as a direct sum of infinitely many identical “blocks”, each being a linear subspace isomorphic to the others. In this context, $S_{\mathbb{N}}$ acts by permuting these blocks, which naturally leads to the study of block-symmetric analytic functions. Algebras of block-symmetric continuous polynomials on Cartesian products, bounded-type analytic functions, and their spectra were considered in particular in [26,27]. Combinatorial properties of block-symmetric polynomials were studied in [28].

Symmetric functions with respect to various groups (or semigroups) of operators are important in informatics, particularly in neural networks. If data and/or spaces of parameters have some symmetric structures, all functions of data and parameters will be symmetric [29,30,31]. For example, if data is organized as an unordered sequence of numbers, then every function of data must be symmetric with respect to permutations of these numbers. Another more typical situation is that if data is organized as an unordered sequence of vectors, then we have symmetry about permutations of vectors, and each function of the data should be block-symmetric. Such a situation is common in language models, where it is often assumed that the meaning of a phrase does not depend on the order of words, while each word, of course, has a fixed order of letters.

Algebras of symmetric analytic functions also appear in quantum statistical physics for descriptions of grand canonical partition functions of quantum ideal gases (see [32,33,34]). As it was observed in [34], block-symmetric analytic functions can be applied for descriptions of grand canonical partition functions of quantum gases with entangled particles.

Using the main idea in [17], in the present work, we construct a special representation for the algebraic spectrum of bounded-type block-symmetric analytic functions on ${\mathcal{l}}_1$ by entire functions of several complex variables that are exponential and have plane zeros. Next, we apply the main result of [16] to find a complete description of the spectrum of this algebra. Additionally, several results obtained here generalize findings from [35]. In Section 2.1, we present the main important issues related to entire functions of several exponential variables with plane zeros. Section 2.2 introduces the key concepts of block-symmetric polynomial algebras and analytic functions defined on Cartesian products of Banach spaces. In Section 3, we represent the main result concerning the algebraic spectrum of bounded-type block-symmetric analytic functions on ${\mathcal{l}}_1.$

We refer the reader to [36,37] for details on polynomials and analytic functions on Banach spaces.

2. Preliminaries

2.1. Entire Exponential Functions with Plane Zeros

Let $f\left(z\right)$ be an entire function of n variables:

$$f\left(z\right)=\sum _{k_i\ge 0}{b}_{k_1\dots k_n}{z}_1^{k_1}\dots z_n^{k_n}$$

and $\nu =({\nu }_1,\dots ,{\nu }_n)\in {\mathbb{C}}^n,{\nu }_i>0.$ We say that $f\left(z\right)$ is an exponential-type function $\nu$ if for every $\epsilon >0$ there exists $A_{\epsilon }>0$ such that

$$\left|f\left(z\right)\right|\le A_{\epsilon }exp\sum _{j=1}^n({\nu }_j+\epsilon )\left|z_j\right|$$

(see [38]). We assume that $f\left(z\right)$ has plane zeros; that is, the set of zeros is $Z_f={\displaystyle \bigcup _{k=1}^{\infty }}{U}_k,$ where each $U_k=\{z:\langle z,a_k\rangle =|a_k{|}^2\}$ is a hyperplane, and

$$|a_k|=\sqrt{|a_k^1{|}^2+\cdots +{|a_k^n|}^2}.$$

Here $\langle z,\omega \rangle =z^1{\overline{\omega }}^1+\cdots +z^n{\overline{\omega }}^n$ is the Hermitian inner product in ${\mathbb{C}}^n,$ and $a_k=(a_k^1,\dots ,a_k^n)$ are the minima of perpendiculars dropped from the origin onto zero hyperplanes $U_k.$ The study of entire exponential functions with plane zeros has been conducted in the works [39,40,41,42].

Let $n\left(r\right)$ denote the number of points $a_k$ inside a ball of radius $r.$ We also define the order of growth of $f\left(z\right)$ by $\rho =\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{lnln{M}_f\left(r\right)}{lnr}},$ where $M_f\left(r\right)=\underset{\left|z\right|=r}{max}\left|f\left(z\right)\right|,$ and the order of the counting function by ${\rho }_1=\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{lnn\left(r\right)}{lnr}}.$ Assuming that ${\rho }_1<\infty ,$ we define

$$\pi \left(z\right)=\prod _{k=1}^{\infty }G(\langle z,a_k|a_k{|}^{-2}\rangle ,p),$$

(1)

where $G(u,p)=(1-u)exp(u+{\displaystyle \frac{u^2}{2}}+\cdots +{\displaystyle \frac{u^p}{p}})$ is the first Weierstrass factor, and $p\in \mathbb{N}$ is chosen such that

$$\sum _{k=1}^{\infty }|a_k{|}^{-p}=\infty ,and\sum _{k=1}^{\infty }{\left|a_k\right|}^{-p-1}<\infty .$$

(2)

We need the following results, which were established in [41].

Proposition 1. 

${\rho }_1=inf\{\lambda \in {\mathbb{R}}^{+}:{\displaystyle \sum _{k=1}^{\infty }}|a_k{|}^{-\lambda }<\infty \}.$

Theorem 1. 

For a canonical product of the form (1), ${\rho }_1=\rho .$

An analogue of the one-dimensional Hadamard theorem (see [43], p. 27) also holds:

Theorem 2. 

Let $f\left(z\right)$ be an entire function with plane zeros of finite order and assume that $f\left(0\right)=0.$ Then we can use the representation

$$f\left(z\right)=exp\left(P\right(z\left)\right)\prod _{k=1}^{\infty }G(\langle z,a_k|a_k{|}^{-2}\rangle ,p),$$

(3)

where p is chosen according to (2), and $P\left(z\right)$ is a polynomial of degree ρ, at most. In particular, if f is exponential (i.e., $p=1$), f can be represented as

$$f\left(z\right)=e^{{\lambda }_1{z}^1+\cdots +{\lambda }_n{z}^n}\prod _{k=1}^{\infty }\left(1-\left({\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{z}^1+\cdots +{\displaystyle \frac{{\overline{a}}_k^n}{|a_k{|}^2}}{z}^n\right)\right)e^{{\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{z}^1+\cdots +{\displaystyle \frac{{\overline{a}}_k^n}{|a_k{|}^2}}{z}^n}$$

(4)

for some complex numbers ${\lambda }_1,\dots ,{\lambda }_n.$

Let us define

$${\sigma }_f=\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{ln{M}_f\left(r\right)}{r^{\rho }}},{\Delta }_f=\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{n\left(r\right)}{r^{\rho }}}.$$

Furthermore, we say that the sequence $\left\{a_k\right\}$ with convergence exponent ${\rho }_1=p$ (where ${\rho }_1$ is from Proposition 1 and p from condition (2)) satisfies condition A (respectively, condition $A_0$) if for every $(s_1,\dots ,s_n)\in {\mathbb{Z}}_{+}^n$ such that $s_1+\cdots +s_n=p$, the following quantity is finite (respectively, zero):

$$D_s=\overline{\underset{r\to \infty }{lim}}\left|\sum _{|a_k|<r}{\displaystyle \frac{{\left(a_k^1\right)}^{s_1}\dots {\left(a_k^n\right)}^{s_n}}{|a_k{|}^{2p}}}\right|.$$

The following theorem holds (see [41]):

Theorem 3. 

For a canonical product $\pi \left(z\right)$ of the form (1) with $\rho =p,$ we have

$${\sigma }_{\pi }<\infty \iff \left\{\begin{array}{c}{\Delta }_{\pi }<\infty \hfill \\ \mathit{condition}A\mathrm{is}\mathit{satisfied},\hfill \end{array}\right.$$

$${\sigma }_{\pi }=0\iff \left\{\begin{array}{c}{\Delta }_{\pi }=0\hfill \\ \mathit{condition}{A}_0\mathrm{is}\mathit{satisfied}.\hfill \end{array}\right.$$

2.2. Block-Symmetric Polynomials and Bounded-Type Analytic Functions

Let us denote by ${\mathcal{l}}_p\left({\mathbb{C}}^s\right)={\mathcal{l}}_p(\mathbb{N},{\mathbb{C}}^s)$, $1\le p<\infty ,$ the space of sequences

$$x=(x_1,x_2,\dots ,x_j,\dots ),$$

(5)

where each $x_j=(x_j^{\left(1\right)},\dots ,x_j^{\left(s\right)})\in {\mathbb{C}}^s$ for $j\in \mathbb{N},$ such that the series ${\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \sum _{r=1}^s}{\left|x_j^{\left(r\right)}\right|}^p$ converges.

The space ${\mathcal{l}}_p\left({\mathbb{C}}^s\right),$ equipped with the norm

$$∥x∥={\left(\sum _{j=1}^{\infty }\sum _{r=1}^s{\left|x_j^{\left(r\right)}\right|}^p\right)}^{1/p},$$

is a Banach space. A polynomial P on ${\mathcal{l}}_p\left({\mathbb{C}}^s\right)$ is called block-symmetric (or vector-symmetric) if

$$P(x_1,x_2,\dots ,x_m,\dots )=P(x_{\sigma \left(1\right)},x_{\sigma \left(2\right)},\dots ,x_{\sigma \left(m\right)},\dots )$$

for any permutation $\sigma \in S_{\mathbb{N}},$ where $S_{\mathbb{N}}$ is the group of permutations on the set of natural numbers $\mathbb{N},$ and $x_m\in {\mathbb{C}}^s.$ We denote by ${\mathcal{P}}_{vs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ the algebra of block-symmetric polynomials on ${\mathcal{l}}_p\left({\mathbb{C}}^s\right).$

Let $\mathbf{k}=(k_1,k_2,\dots ,k_s)$ be a multi-index variable with non-negative integers $k_1,k_2,\dots ,k_s,$ and we use the standard notations $\left|\mathbf{k}\right|=k_1+k_2+\cdots +k_s$, $\mathbf{k}!=k_1!k_2!\dots k_s!.$

It is known (see e.g., [26]) that the polynomials

$$H^{\mathbf{k}}\left(x\right)=H^{k_1,k_2,\dots ,k_s}\left(x\right)=\sum _{j=1}^{\infty }\prod _{\begin{array}{c}r=1\\ \left|\mathbf{k}\right|\ge \lceil p\rceil \end{array}}^s{\left(x_j^{\left(r\right)}\right)}^{k_r}$$

(6)

form an algebraic basis of the algebra ${\mathcal{P}}_{vs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$, $1\le p<\infty ,$ where $x=(x_1,\dots ,x_m,\dots )\in {\mathcal{l}}_p\left({\mathbb{C}}^s\right)$, $x_j=(x_j^{\left(1\right)},\dots ,x_j^{\left(s\right)})\in {\mathbb{C}}^s$ and $\lceil p\rceil$ is the ceiling of $p.$

In the case of ${\mathcal{l}}_1\left({\mathbb{C}}^s\right),$ there exists another algebraic basis of the algebra of block-symmetric polynomials (see [10,28,35]):

$$\begin{array}{c}{R}^{\mathbf{k}}\left(x\right)=R^{k_1,k_2,\dots ,k_s}\left(x\right)={\displaystyle \sum _{\begin{array}{c}{i}_1^j<\cdots <i_{k_j}^j,\\ 1\le j\le s\end{array}}^{\infty }}{\displaystyle \prod _{j=1}^s}{x}_{i_1^j}^{\left(j\right)}\dots x_{i_{k_j}^j}^{\left(j\right)}.\hfill \end{array}$$

(7)

The connection between the basis of power block-symmetric polynomials and the basis of elementary block-symmetric polynomials is given by an analogue of the Newton formulas and Waring–Girard formulas (see [44,45]). For given multi-indexes $\mathbf{k}$ and $\mathbf{q}$, we denote $\mathbf{k}-\mathbf{q}=(k_1-q_1,k_2-q_2,\dots ,k_s-q_s).$ In addition, we write $\mathbf{k}\ge \mathbf{q}$ whenever $k_1\ge q_1$, $k_2\ge q_2,$…, $k_s\ge q_s.$ In [45], the following generalization of Newton’s formula is proven:

$$\left|\mathbf{k}\right|R^{\mathbf{k}}=\sum _{j=1}^{\left|\mathbf{k}\right|}{(-1)}^{j-1}\sum _{\begin{array}{c}\left|\mathbf{q}\right|=j\\ \mathbf{k}\ge \mathbf{q}\end{array}}{\displaystyle \frac{\left|\mathbf{q}\right|!}{\mathbf{q}!}}{H}^{\mathbf{q}}{R}^{\mathbf{k}-\mathbf{q}}.$$

(8)

We denote by ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ the algebra of all bounded-type block-symmetric analytic functions on ${\mathcal{l}}_p\left({\mathbb{C}}^s\right).$ In other words, ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ is the completion of ${\mathcal{P}}_{vs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ in the algebra of all bounded-type analytic functions, ${\mathcal{H}}_b\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ (in the sense of the topology of uniform convergence on bounded sets). We also denote by ${\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ the spectrum of ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),$ that is, the set of all nonzero continuous complex-valued homomorphisms on ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right).$ Clearly, every $x\in {\mathcal{l}}_p\left({\mathbb{C}}^s\right)$ defines a point-evaluation continuous complex homomorphism ${\delta }_x:{\delta }_x\left(f\right)=f\left(x\right)$, $f\in {\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right).$ In [27,35], the algebraic operations “$•$” and “$\diamond$” were introduced for elements $x=(x_1,\dots ,x_n,\dots )$, $y=(y_1,\dots ,y_n,\dots )\in {\mathcal{l}}_p\left({\mathbb{C}}^s\right)$ as follows:

• The intertwining operation is given by $x•y=(x_1,y_1,\dots ,x_n,y_n,\dots )$ and the corresponding intertwining operator is defined by ${\mathcal{T}}_y\left(f\right)\left(x\right)=f(x•y).$
• The multiplicative shift of x and $y,$ denoted by $x\diamond y,$ is defined as a vector consisting of the elements $x_i{y}_j=(x_i^{\left(1\right)}{y}_j^{\left(1\right)},\dots ,x_i^{\left(s\right)}{y}_j^{\left(s\right)})$, $i,j\in \mathbb{N},$ listed in an arbitrary fixed order. The associated multiplicative operator is defined by $M_y\left(f\right)\left(x\right)=f(x\diamond y).$

These operations were subsequently extended to symmetric and multiplicative convolutions on the spectrum of ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ as follows:

• For $f\in {\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ and $\theta \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),$ the symmetric convolution is defined as $(\theta \star f)\left(x\right)=\theta \left[{\mathcal{T}}_x\left(f\right)\right].$
  For $\varphi ,\theta \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),$ one defines $(\varphi \star \theta )\left(f\right)=\varphi (\theta \star f)=\varphi (y\mapsto \theta \left({\mathcal{T}}_{y}f\right)).$
• For $f\in {\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ and $\theta \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),$ the multiplicative convolution is defined as $(\theta ◊f)\left(x\right)=\theta \left[M_x\left(f\right)\right].$
  For $\varphi ,\theta \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),$ one defines $(\varphi ◊\theta )\left(f\right)=\varphi (\theta ◊f).$

From [27,35], it follows that $({\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),\star ,◊)$ is a commutative semiring.

2.3. Summary of Preliminaries

Summarizing the preliminary results, we would like to highlight the most important facts for us.

• Any entire exponential function with plane zeros, $f:{\mathbb{C}}^n\to \mathbb{C}$, can be represented by Equation (4).
• The order of growth ${\sigma }_{\pi }$ of the canonical product $\pi \left(z\right)$ in (1) is finite if and only if ${\Delta }_{\pi }$ is finite and

  $$\overline{\underset{r\to \infty }{lim}}\left|\sum _{|a_k|<r}{\displaystyle \frac{{\left(a_k^1\right)}^{s_1}\dots {\left(a_k^n\right)}^{s_n}}{|a_k{|}^{2p}}}\right|<\infty .$$
• We have two algebraic bases of block-symmetric polynomials on ${\mathcal{l}}_1\left({\mathbb{C}}^s\right)$ given by (6) and (7), and the Newton-type Formula (8) gives a relation between these bases.
• There is an operation “⋆” on the spectrum ${\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ of ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right)$ such that $({\mathcal{M}}_{bvs}\left({\mathcal{l}}_p\left({\mathbb{C}}^s\right)\right),\star )$ is a commutative semigroup.

3. The Main Result

Throughout this section we consider block-symmetric analytic functions on the space ${\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ Let us define the following formal series that are generating functions of $\phi \left(R^{\mathbf{k}}\right)$ and $\phi \left(H^{\mathbf{k}}\right)$, respectively:

$$\mathcal{R}\left(\phi \right)\left(t\right)=\sum _{\left|\mathbf{k}\right|=0}^{\infty }\prod _{i=1}^s{t}_i^{k_i}\phi \left(R^{\mathbf{k}}\right),$$

$$\mathcal{H}\left(\phi \right)\left(t\right)=\sum _{\left|\mathbf{k}\right|=1}^{\infty }\prod _{i=1}^s{t}_i^{k_i}\phi \left(H^{\mathbf{k}}\right),$$

where $t=(t_1,\dots ,t_s)\in {\mathbb{C}}^s$ and $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right).$ In [35] (Proposition 3), it was proven that $t\mapsto \mathcal{R}\left(\phi \right)\left(t\right)$ is an entire exponential-type function for any fixed $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ and $\mathcal{R}\left(\phi \right)\left(0\right)=1.$ Moreover,

$$\mathcal{R}(\phi \star \theta )=\mathcal{R}\left(\phi \right)·\mathcal{R}\left(\theta \right),$$

$\phi$, $\theta \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$, and if $\phi ={\delta }_x$ for some $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right),$ then

$$\begin{array}{cc}\hfill \mathcal{R}\left({\delta }_x\right)\left(t\right)=\sum _{\left|\mathbf{k}\right|=0}^{\infty }\prod _{i=1}^s{t}_i^{k_i}{R}^{\mathbf{k}}\left(x\right)& =\sum _{\left|\mathbf{k}\right|=0}^{\infty }\prod _{i=1}^s{t}_i^{k_i}\sum _{\begin{array}{c}{i}_1^j<\cdots <i_{k_j}^j,\\ 1\le j\le s\end{array}}^{\infty }\prod _{j=1}^s{x}_{i_1^j}^{\left(j\right)}\dots x_{i_{k_j}^j}^{\left(j\right)}\hfill \\ \hfill & =\prod _{i=1}^{\infty }(1+x_i^{\left(1\right)}{t}_1+\cdots +x_i^{\left(s\right)}{t}_s).\hfill \end{array}$$

(9)

Note that if $x=0,$ then $\mathcal{R}\left({\delta }_0\right)\left(t\right)\equiv 1$ and so $\phi \star {\delta }_0=\phi$ for every $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right).$ In other words, ${\delta }_0$ is the neutral element in the commutative semigroup $\left({\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right),\star \right).$

Let us denote

$${\mathbf{n}}_i=(0,\dots ,\underset{i}{\underbrace{n}},\dots ,0),{\mathbf{n}}_{i,j}=(0,\dots ,\underset{i}{\underbrace{n}},0,\dots ,\underset{j}{\underbrace{n}},\dots ,0),n\in \mathbb{N}.$$

In [35], a family of characters was constructed (so-called exceptional characters) in ${\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$, which are not point-evaluation functionals. Let us consider this example in more detail. Let $\alpha =({\alpha }_1,\dots ,{\alpha }_s)$ be a nonzero vector in ${\mathbb{C}}^s$ and we consider the sequence ${\mathfrak{e}}_n\left(\alpha \right)=(\underset{n}{\underbrace{0,\dots ,\alpha }},0,\dots )$ of vectors in ${\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ For every $n\in \mathbb{N}$ we put

$$v_n\left(\alpha \right)={\displaystyle \frac{1}{n}}({\mathfrak{e}}_1\left(\alpha \right)+\cdots +{\mathfrak{e}}_n\left(\alpha \right))\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right).$$

By direct calculation, we can see that ${\delta }_{v_n}\left(H^{{\mathbf{1}}_i}\right)={\alpha }_i$, $i=1,\dots ,s,$ and

$$\underset{n\to \infty }{lim}{\delta }_{v_n}\left(H^{\mathbf{k}}\right)=0\mathrm{for} \mathrm{all}\left|\mathbf{k}\right|>1.$$

Since $\parallel v_n\parallel =|{\alpha }_1|+\cdots +|{\alpha }_n|=\parallel \alpha \parallel ,$ the sequence $\left\{{\delta }_{v_n}\right\}$ has an accumulation point ${\varphi }_{\alpha }$ in the spectrum of $A_{uvs}\left(r{B}_{{\mathcal{l}}_1\left({\mathbb{C}}^s\right)}\right),$ where $A_{uvs}\left(r{B}_{{\mathcal{l}}_1\left({\mathbb{C}}^s\right)}\right)$ is the Banach algebra of all uniformly continuous block-symmetric analytic functions on the ball $r{B}_{{\mathcal{l}}_1\left({\mathbb{C}}^s\right)}$ of radius $r\ge \parallel \alpha \parallel .$ Indeed, all $v_n\in r{B}_{{\mathcal{l}}_1\left({\mathbb{C}}^s\right)}$ for $r\ge \parallel \alpha \parallel$ and so $\left\{{\delta }_{v_n}\right\}$ is a subset in the spectrum of $A_{uvs}\left(r{B}_{{\mathcal{l}}_1\left({\mathbb{C}}^s\right)}\right).$ Also, the spectrum of a Banach algebra is compact (with respect to the Gelfand topology), and every sequence in a compact set has an accumulation point. Hence, the accumulation point ${\varphi }_{\alpha }$ is in ${\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ and has the following property:

$${\varphi }_{\alpha }\left(H^{\mathbf{k}}\right)=\left\{\begin{array}{cc}{\alpha }_i,i=1,\dots ,s\hfill & if \mathbf{k}={\mathbf{1}}_i,\hfill \\ 0\hfill & if \left|\mathbf{k}\right|>1.\hfill \end{array}\right.$$

(10)

Formally, the accumulation point does not need to be unique. However, property (10) uniquely defines the character ${\varphi }_{\alpha }$ because the polynomials $H^{\mathbf{k}}$ form an algebraic basis in the algebra of block-symmetric polynomials on ${\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ In [35], it was shown that

$$\mathcal{R}\left({\varphi }_{\left(\alpha \right)}\right)\left(t\right)=e^{{\alpha }_1{t}_1+\cdots +{\alpha }_s{t}_s},$$

(11)

and

$$\mathcal{H}\left({\varphi }_{\left(\alpha \right)}\right)\left(t\right)={\alpha }_1+\cdots +{\alpha }_s.$$

Moreover, in [35] (Theorem 4), it was shown that if $\phi ={\varphi }_{({\alpha }_1,\dots ,{\alpha }_s)}\star {\delta }_x,$ then $\mathcal{R}\left(\phi \right)\left(t\right)$ is an entire exponential-type function with plane zeros.

According to Theorem 2, Equation (4) is an exponential-type function $\mathcal{R}\left(\phi \right)\left(t\right)$ with plane zeros and admits the following representation:

$$\mathcal{R}\left(\phi \right)\left(t\right)=e^{{\lambda }_1{t}_1+\cdots +{\lambda }_s{t}_s}\prod _{k=1}^{\infty }\left(1-\left({\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{t}_1+\cdots +{\displaystyle \frac{{\overline{a}}_k^s}{|a_k{|}^2}}{t}_s\right)\right)e^{{\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{t}_1+\cdots +{\displaystyle \frac{{\overline{a}}_k^s}{|a_k{|}^2}}{t}_s},$$

(12)

where $a_k$ are the points defining the orthogonal bases to the hyperplanes $U_k,$ which are in the zero sets of $\mathcal{R}\left(\phi \right)\left(t\right).$ If ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{|a_k|}}<\infty ,$ we obtain

$$\mathcal{R}\left(\phi \right)\left(t\right)=e^{{\lambda }_1{t}_1+\cdots +{\lambda }_s{t}_s}\prod _{k=1}^{\infty }\left(1-\left({\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{t}_1+\cdots +{\displaystyle \frac{{\overline{a}}_k^s}{|a_k{|}^2}}{t}_s\right)\right).$$

(13)

Combining (9) and (11), we can see that if $\phi ={\varphi }_{({\lambda }_1,\dots ,{\lambda }_s)}\star {\delta }_x$, $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right),$ then $\mathcal{R}\left(\phi \right)\left(t\right)$ is of the form (13) for $a_k^j=-{\displaystyle \frac{{\overline{x}}_k^{\left(j\right)}}{|x_k{|}^2}}$ if $x_k\ne 0.$ If $x_k=0,$ we can set $a_k^j=0$, $k\in \mathbb{N}$, $j=1,\dots ,s.$ Indeed, from here we have $x_k^{\left(j\right)}=-{\displaystyle \frac{{\overline{a}}_k^j}{|a_k{|}^2}}$, $j=1,\dots ,s$ and

$$\sum _{k=1}^{\infty }{\displaystyle \frac{1}{|a_k|}}\le \sum _{k=1}^{\infty }\sum _{j=1}^s|x_k^{\left(j\right)}|=\parallel x\parallel <\infty .$$

Let us consider the following numerical example for the calculation of some particular character $\phi .$

Example 1. 

Let $s=2$; ${\lambda }_1=1$ and ${\lambda }_2=3$; and

$$x=\left(\left({\displaystyle \genfrac{}{}{0pt}{}{1}{-2}}\right),\left({\displaystyle \genfrac{}{}{0pt}{}{2}{3}}\right),\left({\displaystyle \genfrac{}{}{0pt}{}{0}{0}}\right),\left({\displaystyle \genfrac{}{}{0pt}{}{0}{0}}\right),\dots \right).$$

To find the character $\phi ={\varphi }_{\lambda }\star {\delta }_x$, we need to compute its evaluation at basis polynomials $H^{\mathbf{k}}=H^{i,j}$ for $i+j\ge 1.$ Thus, for $i+j=1,$

$$\phi \left(H^{1,0}\right)={\varphi }_{\lambda }\left(H^{1,0}\right)+{\delta }_x\left(H^{1,0}\right)=1+H^{1,0}\left(x\right)=1+1+2=4,$$

$$\phi \left(H^{0,1}\right)={\varphi }_{\lambda }\left(H^{0,1}\right)+{\delta }_x\left(H^{0,1}\right)=3+H^{0,1}\left(x\right)=3-2+3=4.$$

If $i+j\ge 2,$ such that $i\ne 0$ and $j\ne 0,$

$$\phi \left(H^{i,j}\right)={\varphi }_{\lambda }\left(H^{i,j}\right)+{\delta }_x\left(H^{i,j}\right)=0+H^{i,j}\left(x\right)=1^i·{(-2)}^j+2^i·3^j={(-2)}^j+2^i·3^j.$$

If $i\ge 2$ and $j\ge 2,$

$$\phi \left(H^{i,0}\right)=1+2^i,and\phi \left(H^{0,j}\right)={(-2)}^j+3^j.$$

Also,

$$\mathcal{R}\left(\phi \right)(t_1,t_2)=e^{t_1+3t_2}(1+t_1-2t_2)(1+2t_1+3t_2).$$

Note that in the general case, entire exponential functions with plane zeros can be of the form

$$f\left(t\right)=\prod _{k=1}^{\infty }\left(1-\left({\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{t}_1+\cdots +{\displaystyle \frac{{\overline{a}}_k^s}{|a_k{|}^2}}{t}_s\right)\right)e^{{\displaystyle \frac{{\overline{a}}_k^1}{|a_k{|}^2}}{t}_1+\cdots +{\displaystyle \frac{{\overline{a}}_k^s}{|a_k{|}^2}}{t}_s}$$

(14)

with ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{|a_k|}}=\infty ,$ so that, according to Theorem 1 and Proposition 1, applying this to the case $\rho =1,$ the series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{|a_k{|}^{1+d}}}$ converges for any $d>0.$ The main purpose of this paper is to show that every $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ is such that the representation $\mathcal{R}\left(\phi \right)\left(t\right)$ is of the form (13).

In order to apply Theorem 3 to (14), we observe that for the case $p=1,$

$${\sigma }_f=\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{ln{M}_f\left(r\right)}{r}},\mathrm{and}{\Delta }_f=\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{n\left(r\right)}{r}}.$$

Also, condition A (respectively, condition $A_0$) means that

$$D_i=\overline{\underset{r\to \infty }{lim}}\left|\sum _{|a_k|<r}{\displaystyle \frac{a_k^i}{|a_k{|}^2}}\right|=\overline{\underset{r\to \infty }{lim}}\left|\sum _{|x_k^{\left(i\right)}|>1/r}{x}_k^{\left(i\right)}\right|<\infty ,i=1,\dots ,s$$

(respectively, $D_i=0$). Thus, according to Theorem 3, a function f has a finite type ${\sigma }_f$ (equal to zero) if and only if all ${\Delta }_{\pi }$ and $D_i$, $i=1,\dots ,s$, are finite (are equal to zero). Hence, a function $f\left(t\right)$ of the form (14) is exponential if and only if the series ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{1}{|a_k{|}^{1+d}}}$ converges for all $d>0$ and both ${\Delta }_{\pi }$ and all $D_i$ are finite. Applying this fact to exponential-type functions $\mathcal{R}\left(\phi \right)(·),$ we have the following corollary.

Corollary 1. 

Let $x_n\in {\mathbb{C}}^s$, $n\in \mathbb{N},$ but the sequence $\left(x_n\right)\notin {\mathcal{l}}_p\left({\mathbb{C}}^s\right)$ for some $p>1.$ Then there does not exist $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ such that

$$\phi \left(H^{\mathbf{k}}\right)=\sum _{j=1}^{\infty }\prod _{\begin{array}{c}r=1\\ \left|\mathbf{k}\right|\ge 1\end{array}}^s{\left(x_j^{\left(r\right)}\right)}^{k_r}\forall \left|\mathbf{k}\right|\ge 1.$$

Let us consider a sequence $x=(x_1,\dots ,x_n,\dots )$ of complex vectors in ${\mathcal{l}}_{1+d}\left({\mathbb{C}}^s\right)$ for every $d>0,$ where each $x_j=(x_j^{\left(1\right)},\dots ,x_j^{\left(s\right)})\in {\mathbb{C}}^s.$ Assume that this sequence satisfies the following conditions:

$$\overline{\underset{r\to \infty }{lim}}{\displaystyle \frac{n\left(r\right)}{r}}<\infty ,\overline{\underset{r\to \infty }{lim}}\left|\sum _{|x_n^{\left(i\right)}|>1/r}{x}_n^i\right|<\infty \forall i=1,\dots ,s,$$

(15)

where $n\left(r\right)$ is the number of points $x_n$ inside a ball of radius r and $\lambda =({\lambda }_1,\dots ,{\lambda }_s)\in {\mathbb{C}}^s.$ We denote by ${\delta }_{(x,\lambda )}$ the homomorphism of the algebra of block-symmetric polynomials ${\mathcal{P}}_{vs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ defined by

$${\delta }_{(x,\lambda )}\left(H^{{\mathbf{1}}_i}\right)={\lambda }_i,i=1,\dots ,s,$$

$${\delta }_{(x,\lambda )}\left(H^{\mathbf{k}}\right)=\sum _{j=1}^{\infty }\prod _{\begin{array}{c}r=1\\ \left|\mathbf{k}\right|\ge 1\end{array}}^s{\left(x_j^{\left(r\right)}\right)}^{k_r},\left|\mathbf{k}\right|>1.$$

Theorem 4. 

Let $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right).$ Then the restriction of φ to ${\mathcal{P}}_{vs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ is of the form ${\delta }_{(x,\lambda )}$ for some $\lambda \in {\mathbb{C}}^s$, $x\in {\mathcal{l}}_{1+d}\left({\mathbb{C}}^s\right)$ for every $d>0$ and x satisfying (15).

Proof. 

Let us consider an entire exponential-type function with plane zeros of the form (12) and a sequence $x=(x_1,\dots ,x_k,\dots ),x_k=(x_k^{\left(1\right)},\dots ,x_k^{\left(s\right)})$, $x_k^{\left(i\right)}=-{\displaystyle \frac{{\overline{a}}_k^i}{|a_k{|}^2}}.$

If $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right),$ then, according to (13), we obtain $\phi ={\varphi }_{\lambda }\star {\delta }_x={\delta }_{(x,\alpha )},$ where ${\alpha }_i={\lambda }_i+{\sum }_{k=1}^{\infty }{x}_k^{\left(i\right)}.$ Now, let $x\notin {\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ Then

$$\begin{array}{cc}\hfill \mathcal{R}\left(\phi \right)\left(t\right)& =e^{{\lambda }_1{t}_1+\cdots +{\lambda }_s{t}_s}\prod _{k=1}^{\infty }\left(1+x_k^{\left(1\right)}{t}_1+\cdots +x_k^{\left(s\right)}{t}_s\right)e^{-x_k^{\left(1\right)}{t}_1-\cdots -x_k^{\left(s\right)}{t}_s}\hfill \\ \hfill & =e^{\left({\lambda }_1-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(1\right)}\right)t_1+\cdots +\left({\lambda }_s-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(s\right)}\right)t_s}\prod _{k=1}^{\infty }\left(1+x_k^{\left(1\right)}{t}_1+\cdots +x_k^{\left(s\right)}{t}_s\right)\hfill \\ \hfill & =e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right),\hfill \end{array}$$

(16)

and on the other hand,

$$\mathcal{R}\left(\phi \right)\left(t\right)=\sum _{\left|\mathbf{k}\right|=0}^{\infty }\prod _{i=1}^s{t}_i^{k_i}\phi \left(R^{\mathbf{k}}\right).$$

(17)

To represent (16), we compute the coefficients of the Taylor series expansion of the right part of (16):

$$\begin{array}{cc}\hfill {\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)}_{t_i}^{\prime }& ={\left(e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)}_{t_i}^{\prime }\hfill \\ \hfill & =\left({\lambda }_i-\sum _{k=1}^{\infty }{x}_k^{\left(i\right)}\right)e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\hfill \\ \hfill & +e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_m^{\left(r\right)}{t}_r\right)\right)\hfill \end{array}$$

for all $i=1,\dots ,s.$ Then

$${\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)}_{t_i}^{\prime }{|}_{t=0}={\lambda }_i,i=1,\dots ,s.$$

On the other hand, the coefficient at $t_i$ in (17) is $\phi \left(R^{{\mathbf{1}}_i}\right).$ Hence, $\phi \left(R^{{\mathbf{1}}_i}\right)=\phi \left(H^{{\mathbf{1}}_i}\right)={\lambda }_i$, $i=1,\dots ,s.$ In the same way, for the second derivatives of the right part of (16), we have

$$\begin{array}{cc}\hfill {\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)}_{{\left(t_i\right)}^2}^{″}& ={\left(\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)}_{t_i}^{\prime }\hfill \\ \hfill & +{\left(e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)\right)}_{t_i}^{\prime }\hfill \\ \hfill & ={\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)}^2{e}^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\hfill \\ \hfill & +\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)\hfill \\ \hfill & +\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)\hfill \\ \hfill & +e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\left({\displaystyle \sum _{k\ne l}^{\infty }}{x}_k^{\left(i\right)}{x}_l^{\left(i\right)}\prod _{m\ne k\ne l}^{\infty }\left(1+\sum _{r=1}^s{x}_m^{\left(r\right)}{t}_r\right)\right).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)}_{{\left(t_i\right)}^2}^{″}{|}_{t=0}& ={\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)}^2+\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)+\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)\hfill \\ \hfill & +\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)+\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)+{\displaystyle \sum _{k\ne l}^{\infty }}{x}_k^{\left(i\right)}{x}_l^{\left(i\right)}={\left({\lambda }_i\right)}^2-{\left({\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)}^2\hfill \\ \hfill & +{\displaystyle \sum _{k\ne l}^{\infty }}{x}_k^{\left(i\right)}{x}_l^{\left(i\right)}={\left({\lambda }_i\right)}^2-{\displaystyle \sum _{k=1}^{\infty }}{\left(x_k^{\left(i\right)}\right)}^2-{\displaystyle \sum _{k\ne l}^{\infty }}{x}_k^{\left(i\right)}{x}_l^{\left(i\right)}+{\displaystyle \sum _{k\ne l}^{\infty }}{x}_k^{\left(i\right)}{x}_l^{\left(i\right)}\hfill \\ \hfill & ={\left({\lambda }_i\right)}^2-H^{{\mathbf{2}}_i}\left(x\right).\hfill \end{array}$$

From (17), $\phi \left(R^{{\mathbf{2}}_i}\right)={\displaystyle \frac{{\left({\lambda }_i\right)}^2-H^{{\mathbf{2}}_i}\left(x\right)}{2}}.$ On the other hand, from the Newton-type Formula (8), we have $R^{{\mathbf{2}}_i}\left(x\right)={\displaystyle \frac{{\left(H^{{\mathbf{1}}_i}\left(x\right)\right)}^2-H^{{\mathbf{2}}_i}\left(x\right)}{2}}.$ Applying the character $\phi ,$ to the last equality, we obtain $\phi \left(R^{{\mathbf{2}}_i}\right)={\displaystyle \frac{{\left(\phi \left(H^{{\mathbf{1}}_i}\right)\right)}^2-\phi \left(H^{{\mathbf{2}}_i}\right)}{2}}.$ So, $\phi \left(H^{{\mathbf{2}}_i}\right)=H^{{\mathbf{2}}_i}\left(x\right)$ $\forall i=1,\dots ,s.$ Similarly, the mixed second derivatives are

$$\begin{array}{cc}\hfill {\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)}_{t_i{t}_j}^{″}& =\left[\left(\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right.\right.\hfill \\ \hfill & +{\left.\left.{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\right]}_{t_j}^{\prime }\hfill \\ \hfill & =\left[\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right){\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(j\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \sum _{k\ne m}^{\infty }}{x}_k^{\left(i\right)}{x}_m^{\left(j\right)}\prod _{l\ne m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_l^{\left(r\right)}{t}_r\right)\right]e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\hfill \\ \hfill & +\left[\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right.\hfill \\ \hfill & +\left.{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\prod _{m\ne k}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right]\left({\lambda }_j-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(j\right)}\right)e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)}_{t_i{t}_j}^{″}{|}_{t=0}& =\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right){\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(j\right)}+{\displaystyle \sum _{k\ne m}^{\infty }}{x}_k^{\left(i\right)}{x}_m^{\left(j\right)}\hfill \\ \hfill & +\left(\left({\lambda }_i-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)+{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}\right)\left({\lambda }_j-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(j\right)}\right)\hfill \\ \hfill & ={\lambda }_i{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(j\right)}-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}{x}_k^{\left(j\right)}-{\displaystyle \sum _{k\ne m}^{\infty }}{x}_k^{\left(i\right)}{x}_m^{\left(j\right)}+{\displaystyle \sum _{k\ne m}^{\infty }}{x}_k^{\left(i\right)}{x}_m^{\left(j\right)}\hfill \\ \hfill & +{\lambda }_i{\lambda }_j-{\lambda }_i{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(j\right)}={\lambda }_i{\lambda }_j-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(i\right)}{x}_k^{\left(j\right)}={\lambda }_i{\lambda }_j-H^{{\mathbf{1}}_{i,j}}\left(x\right).\hfill \end{array}$$

From (17), $\phi \left(R^{{\mathbf{1}}_{i,j}}\right)={\lambda }_i{\lambda }_j-H^{{\mathbf{1}}_{i,j}}\left(x\right).$ On the other hand, from the analogue of Newton’s Formula (8), we have that $R^{{\mathbf{1}}_{i,j}}\left(x\right)=H^{{\mathbf{1}}_i}\left(x\right)H^{{\mathbf{1}}_j}\left(x\right)-H^{{\mathbf{1}}_{i,j}}\left(x\right).$ Applying the character $\phi$ to the last equality, we obtain $\phi \left(R^{{\mathbf{1}}_{i,j}}\right)=\phi \left(H^{{\mathbf{1}}_i}\right)\phi \left(H^{{\mathbf{1}}_j}\right)-\phi \left(H^{{\mathbf{1}}_{i,j}}\right)={\lambda }_i{\lambda }_j-\phi \left(H^{{\mathbf{1}}_{i,j}}\right).$ So,

$$\phi \left(H^{{\mathbf{1}}_{i,j}}\right)=H^{{\mathbf{1}}_{i,j}}\left(x\right).$$

Therefore, we proved that if $\left|\mathbf{k}\right|=1,$ then $\phi \left(H^{\mathbf{k}}\right)={\varphi }_{\lambda }\left(H^{\mathbf{k}}\right)$, and if $\left|\mathbf{k}\right|=2,$ then $\phi \left(H^{\mathbf{k}}\right)=H^{\mathbf{k}}\left(x\right).$ To finish the proof of the theorem, we need to show that $\phi \left(H^{\mathbf{k}}\right)=H^{\mathbf{k}}\left(x\right)$ for every $\mathbf{k}$ such that $\left|\mathbf{k}\right|>2.$ Let us prove it using the method of mathematical induction. Assume that for all $\left|\mathbf{k}\right|<\left|\mathbf{n}\right|$, we have $\phi \left(H^{\mathbf{k}}\right)={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \prod _{\begin{array}{c}r=1\\ \left|\mathbf{k}\right|\ge 1\end{array}}^s}{\left(x_j^{\left(r\right)}\right)}^{k_r}.$ Now we prove that $\phi \left(H^{\mathbf{k}}\right)={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \prod _{\begin{array}{c}r=1\\ \left|\mathbf{k}\right|\ge 1\end{array}}^s}{\left(x_j^{\left(r\right)}\right)}^{k_r}$ is true for $\left|\mathbf{k}\right|=\left|\mathbf{n}\right|.$

Let us rewrite the function $\mathcal{R}\left(\phi \right)\left(t\right)$ as

$$\begin{array}{cc}\hfill \mathcal{R}\left(\phi \right)\left(t\right)& =exp\left(ln\left(e^{{\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r}\prod _{k=1}^{\infty }\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right)\right)\hfill \\ \hfill & =exp\left({\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r+{\displaystyle \sum _{k=1}^{\infty }}ln\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right)\right).\hfill \end{array}$$

(18)

Let us denote

$$L\left(t\right)=L(t_1,\dots ,t_s)={\displaystyle \sum _{r=1}^s}\left({\lambda }_r-{\displaystyle \sum _{k=1}^{\infty }}{x}_k^{\left(r\right)}\right)t_r+{\displaystyle \sum _{k=1}^{\infty }}ln\left(1+\sum _{r=1}^s{x}_k^{\left(r\right)}{t}_r\right).$$

Then, from direct computations,

$${\partial }_t^{\mathbf{m}}{\left(L\left(t\right)\right)|}_{t=0}=\left\{\begin{array}{cc}{\lambda }_i,\hfill & \mathbf{m}={\mathbf{1}}_i;\hfill \\ {(-1)}^{\left|\mathbf{m}\right|-1}\left(\right|\mathbf{m}|-1)!H^{\mathbf{m}}\left(x\right),\hfill & \left|\mathbf{m}\right|\ge 2,\hfill \end{array}\right.$$

(19)

where $\mathbf{m}=(m_1,\dots ,m_s).$

For given multi-indexes $\mathbf{m}=(m_1,\dots ,m_s)$ and $\mathbf{n}=(n_1,\dots ,n_s)$ we denote $\mathbf{n}-\mathbf{m}=(n_1-m_1,\dots ,n_s-m_s)$, $\mathbf{n}-{\mathbf{1}}_i=(n_1,\dots ,n_i-1,\dots ,n_s)$, $\mathbf{m}\le \mathbf{n}$⇔$m_i\le n_i$ for all $i=1,\dots ,s$, and

$$\left(\begin{array}{c}\mathbf{n}\\ \mathbf{m}\end{array}\right)=\prod _{r=1}^s\left(\begin{array}{c}{n}_r\\ m_r\end{array}\right)=\prod _{r=1}^s{\displaystyle \frac{n_r!}{(n_r-m_r)!m_r!}}.$$

Applying the multi-variable form of the Leibniz formula (see [46], p. 10) for the computation of derivatives of the product of two functions ${\left(L\left(t\right)\right)}_{t_i}^{\prime }{e}^{L\left(t\right)},$ we obtain

$$\begin{array}{cc}\hfill {\partial }_t^{\mathbf{n}}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}& ={\partial }_t^{\mathbf{n}-{\mathbf{1}}_i}\left({\left(L\left(t\right)\right)}_{t_i}^{\prime }{e}^{L\left(t\right)}\right){|}_{t=0}\hfill \\ \hfill & =\sum _{\mathbf{m}\le \mathbf{n}-{\mathbf{1}}_i}\left(\begin{array}{c}\mathbf{n}-{\mathbf{1}}_i\\ \mathbf{m}\end{array}\right){\partial }_t^{\mathbf{m}}\left({\left(L\left(t\right)\right)}_{t_i}^{\prime }\right){{|}_{t=0}{\partial }_t^{\mathbf{n}-{\mathbf{1}}_i-\mathbf{m}}\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}\hfill \\ \hfill & ={\lambda }_i{\partial }_t^{\mathbf{n}-{\mathbf{1}}_i}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}+\hfill \\ \hfill & +\sum _{0<\mathbf{m}\le \mathbf{n}-{\mathbf{1}}_i}\left(\begin{array}{c}\mathbf{n}-{\mathbf{1}}_i\\ \mathbf{m}\end{array}\right){\partial }_t^{\mathbf{m}}\left({\left(L\left(t\right)\right)}_{t_i}^{\prime }\right){{|}_{t=0}{\partial }_t^{\mathbf{n}-{\mathbf{1}}_i-\mathbf{m}}\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}\hfill \\ \hfill & ={\lambda }_i{\partial }_t^{\mathbf{n}-{\mathbf{1}}_i}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}\hfill \\ \hfill & +\sum _{0<\mathbf{m}\le \mathbf{n}-{\mathbf{1}}_i}{\displaystyle \frac{n_1!}{(n_1-m_1)!m_1!}}\dots {\displaystyle \frac{(n_i-1)!}{(n_i-(m_i+1))!m_i!}}\dots {\displaystyle \frac{n_s!}{(n_s-m_s)!m_s!}}\hfill \\ \hfill & \times {\partial }_t^{\mathbf{m}+{\mathbf{1}}_i}\left(L\left(t\right)\right){{|}_{t=0}{\partial }_t^{\mathbf{n}-(\mathbf{m}+{\mathbf{1}}_i)}\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}\hfill \end{array}$$

for all $\left|\mathbf{n}\right|\ge 1$ and ${\partial }_t^{\mathbf{0}}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}=1.$

Let us re-denote the vector $\mathbf{m}+{\mathbf{1}}_i=(m_1,\dots ,m_i+1,\dots ,m_s)$ by $\mathbf{q}=(q_1,\dots ,q_i,\dots ,q_s).$ Then we obtain the following formula:

$$\begin{array}{cc}\hfill {\partial }_t^{\mathbf{n}}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}& ={\lambda }_i{\partial }_t^{\mathbf{n}-{\mathbf{1}}_i}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}\hfill \\ \hfill & +\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{\displaystyle \frac{n_1!}{(n_1-q_1)!q_1!}}\dots {\displaystyle \frac{(n_i-1)!}{(n_i-q_i)!(q_i-1)!}}\dots {\displaystyle \frac{n_s!}{(n_s-q_s)!q_s!}}\hfill \\ \hfill & \times {\partial }_t^{\mathbf{q}}\left(L\left(t\right)\right){{|}_{t=0}{\partial }_t^{\mathbf{n}-\mathbf{q}}\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}.\hfill \end{array}$$

(20)

From the Taylor series expansion, we have

$$\phi \left(R^{\mathbf{n}-\mathbf{q}}\right)={\displaystyle \frac{{\partial }_t^{\mathbf{n}-\mathbf{q}}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}}{(\mathbf{n}-\mathbf{q})!}}.$$

Substituting $\phi \left(R^{\mathbf{n}-\mathbf{q}}\right)$ from the last equation and ${\partial }_t^{\mathbf{m}}{\left(L\left(t\right)\right)|}_{t=0}$ from (19) into Formula (20), we obtain

$$\begin{array}{cc}\hfill \phi \left(R^{\mathbf{n}}\right)& ={\displaystyle \frac{1}{\mathbf{n}!}}{\partial }_t^{\mathbf{n}}{\left(\mathcal{R}\left(\phi \right)\left(t\right)\right)|}_{t=0}\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathbf{n}!}}({\lambda }_i(\mathbf{n}-{\mathbf{1}}_i)!\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{\displaystyle \frac{n_1!}{(n_1-q_1)!q_1!}}\dots {\displaystyle \frac{(n_i-1)!}{(n_i-q_i)!(q_i-1)!}}\dots \hfill \\ \hfill & \times {\displaystyle \frac{n_s!}{(n_s-q_s)!q_s!}}{(-1)}^{\left|\mathbf{q}\right|-1}\left(\right|\mathbf{q}|-1)!H^{\mathbf{q}}\left(x\right)(\mathbf{n}-\mathbf{q})!\phi \left(R^{\mathbf{n}-\mathbf{q}}\right))={\displaystyle \frac{1}{n_i}}{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathbf{n}!}}\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{\displaystyle \frac{\mathbf{n}!}{(\mathbf{n}-\mathbf{q})!\mathbf{q}!}}{\displaystyle \frac{q_i}{n_i}}{(-1)}^{\left|\mathbf{q}\right|-1}\left(\right|\mathbf{q}|-1)!H^{\mathbf{q}}\left(x\right)(\mathbf{n}-\mathbf{q})!\phi \left(R^{\mathbf{n}-\mathbf{q}}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{n_i}}{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+{\displaystyle \frac{1}{n_i}}\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{(-1)}^{\left|\mathbf{q}\right|-1}{\displaystyle \frac{\left(\right|\mathbf{q}|-1)!}{\mathbf{q}!}}{q}_i{H}^{\mathbf{q}}\left(x\right)\phi \left(R^{\mathbf{n}-\mathbf{q}}\right).\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill n_i\phi \left(R^{\mathbf{n}}\right)={\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{(-1)}^{\left|\mathbf{q}\right|-1}{\displaystyle \frac{\left(\right|\mathbf{q}|-1)!}{\mathbf{q}!}}{q}_i{H}^{\mathbf{q}}\left(x\right)\phi \left(R^{\mathbf{n}-\mathbf{q}}\right).\end{array}$$

(21)

Summing Equation (21) over all $i=1,\dots ,s,$ we obtain

$$\begin{array}{cc}\hfill \sum _{i=1}^s{n}_i\phi \left(R^{\mathbf{n}}\right)& =\sum _{i=1}^s{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{i=1}^s\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{(-1)}^{\left|\mathbf{q}\right|-1}{\displaystyle \frac{\left(\right|\mathbf{q}|-1)!}{\mathbf{q}!}}{q}_i{H}^{\mathbf{q}}\left(x\right)\phi \left(R^{\mathbf{n}-\mathbf{q}}\right)\hfill \\ \hfill & =\sum _{i=1}^s{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{(-1)}^{\left|\mathbf{q}\right|-1}{\displaystyle \frac{\left(\right|\mathbf{q}|-1)!}{\mathbf{q}!}}\left|\mathbf{q}\right|H^{\mathbf{q}}\left(x\right)\phi \left(R^{\mathbf{n}-\mathbf{q}}\right)\hfill \\ \hfill & =\sum _{i=1}^s{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|\ge 2\end{array}}{(-1)}^{\left|\mathbf{q}\right|-1}{\displaystyle \frac{\left|\mathbf{q}\right|!}{\mathbf{q}!}}{H}^{\mathbf{q}}\left(x\right)\phi \left(R^{\mathbf{n}-\mathbf{q}}\right).\hfill \end{array}$$

(22)

Remember that we already have $\phi \left(H^{{\mathbf{1}}_i}\right)={\lambda }_i,i=1,\dots ,s$ and, according to the induction hypothesis, $\phi \left(H^{\mathbf{q}}\right)={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \prod _{\begin{array}{c}r=1\\ \left|\mathbf{q}\right|\ge 1\end{array}}^s}{\left(x_j^{\left(r\right)}\right)}^{q_r}=H^{\mathbf{q}}\left(x\right)$ for all $1<\left|\mathbf{q}\right|<\left|\mathbf{n}\right|.$ Thus, from (22), we obtain

$$\begin{array}{cc}\hfill \left|\mathbf{n}\right|\phi \left(R^{\mathbf{n}}\right)& =\sum _{i=1}^s{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{j=2}^{\left|\mathbf{n}\right|}{(-1)}^{j-1}\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}\\ \left|\mathbf{q}\right|=j\end{array}}{\displaystyle \frac{\left|\mathbf{q}\right|!}{\mathbf{q}!}}{H}^{\mathbf{q}}\left(x\right)\phi \left(R^{\mathbf{n}-\mathbf{q}}\right)\hfill \\ \hfill & =\sum _{i=1}^s{\lambda }_i\phi \left(R^{\mathbf{n}-{\mathbf{1}}_i}\right)+\sum _{j=2}^{\left|\mathbf{n}\right|-1}{(-1)}^{j-1}\sum _{\begin{array}{c}\mathbf{q}\le \mathbf{n}-1\\ \left|\mathbf{q}\right|=j\end{array}}{\displaystyle \frac{\left|\mathbf{q}\right|!}{\mathbf{q}!}}\phi \left(H^{\mathbf{q}}\right)\phi \left(R^{\mathbf{n}-1-\mathbf{q}}\right)\hfill \\ \hfill & +{(-1)}^{\left|\mathbf{n}\right|-1}{\displaystyle \frac{\left|\mathbf{n}\right|!}{\mathbf{n}!}}{H}^{\mathbf{n}}\left(x\right).\hfill \end{array}$$

(23)

Applying the character $\phi$ to the Newton-type Formula (8) and equating it to equality (23), we obtain $\phi \left(H^{\mathbf{n}}\right)=H^{\mathbf{n}}\left(x\right).$ Thus, the identity $\phi \left(H^{\mathbf{n}}\right)=H^{\mathbf{n}}\left(x\right)$ holds for all $\left|\mathbf{n}\right|\ge 2$ and so the theorem is proved. □

Proposition 2. 

Let $f\in {\mathcal{H}}_{bs}\left({\mathcal{l}}_1\right).$ For every fixed $t=(t_1,\dots ,t_s)\in {\mathbb{C}}^s$, the function

$$g_t\left(x\right)=f(t_1{x}^{\left(1\right)}+\cdots +t_s{x}^{\left(s\right)})$$

is in ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right),$ and the operator $f\mapsto g_t$ is a continuous homomorphism from ${\mathcal{H}}_{bs}\left({\mathcal{l}}_1\right)$ to ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right).$

Proof. 

The function $g_t$ is a composition of the entire bounded-type function f and the linear map $x\mapsto t_1{x}^{\left(1\right)}+\cdots +t_s{x}^{\left(s\right)}$, which is also entire and bounded. Thus, if $f\in {\mathcal{H}}_b\left({\mathcal{l}}_1\right),$ the composition operator $f\mapsto g_t$ is a continuous homomorphism from $H_b\left({\mathcal{l}}_1\right)$ to $H_b\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ (see, e.g., [11]). In particular, $g_t$ is an entire bounded-type function on ${\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ From the symmetry of f, it follows that $g_t$ is block-symmetric. Hence, if $f\in {\mathcal{H}}_{bs}\left({\mathcal{l}}_1\right)$, $g_t\in {\mathcal{H}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ for every fixed vector $t,$ and the operator $f\mapsto g_t$ is a continuous homomorphism from ${\mathcal{H}}_{bs}\left({\mathcal{l}}_1\right)$ to ${\mathcal{H}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right).$ □

Theorem 5. 

Let $\phi \in {\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right).$ Then the restriction of φ to ${\mathcal{P}}_{vs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ is of the form ${\varphi }_{\alpha }\star {\delta }_x$ for some $\alpha =({\alpha }_1,\dots ,{\alpha }_s)\in {\mathbb{C}}^s$ and $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right).$

Proof. 

According to Theorem 4, $\phi ={\delta }_{(x,\lambda )}$ for some $x\in {\mathcal{l}}_{1+d}\left({\mathbb{C}}^s\right)$, $d>0,$ and $\lambda \in {\mathbb{C}}^s.$ Let us suppose that $x=(x^{\left(1\right)},\dots ,x^{\left(s\right)})\notin {\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ Then there is $1\le j\le s$ such that $x^{\left(j\right)}\notin {\mathcal{l}}_1.$ Consider the following character $\theta$ on ${\mathcal{H}}_{bs}\left({\mathcal{l}}_1\right)$:

$$\theta \left(f\right)={\delta }_{(x,\lambda )}\left(g_t\right),\mathrm{for}t=(\underset{j}{\underbrace{0,\dots ,0,1}},0,\dots ,0).$$

Since the operator $f\mapsto g_t$ is a continuous homomorphism according to Proposition 2 and ${\delta }_{(x,\lambda )}$ is continuous, their composition, $\theta$, is a continuous linear multiplicative functional. On the other hand, the restriction of $\theta$ to the basis polynomials $F_n$ is $\theta \left(F_1\right)={\lambda }_j$ and $\theta \left(F_n\right)=F_n\left(x\right)$ if $n>1.$ But according to Theorem 2.12 in [16], this is impossible because we assumed that $x\notin {\mathcal{l}}_1.$

Now, let $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right)$. We set

$${\alpha }_i={\lambda }_i-\sum _{n=1}^{\infty }{x}_n^{\left(i\right)},1\le i\le s.$$

Then

$${\varphi }_{\alpha }\star {\delta }_x\left(H^{{\mathbf{1}}_i}\right)={\varphi }_{\alpha }\left(H^{{\mathbf{1}}_i}\right)+H^{{\mathbf{1}}_i}\left(x\right)={\alpha }_i+\sum _{n=1}^{\infty }{x}_n^{\left(i\right)}={\lambda }_i={\delta }_{(x,\lambda )}\left(H^{{\mathbf{1}}_i}\right).$$

For every $\mathbf{k}$ such that $\left|\mathbf{k}\right|>1,$

$${\varphi }_{\alpha }\star {\delta }_x\left(H^{\mathbf{k}}\right)={\varphi }_{\alpha }\left(H^{\mathbf{k}}\right)+{\delta }_x\left(H^{\mathbf{k}}\right)=0+H^{\mathbf{k}}\left(x\right)={\delta }_{(x,\lambda )}\left(H^{\mathbf{k}}\right).$$

Thus, ${\varphi }_{\alpha }\star {\delta }_x={\delta }_{(x,\lambda )}=\phi .$ □

The following corollary gives an illustration of the main results.

Corollary 2. 

The set of exceptional characters ${\varphi }_{\lambda }$ is a commutative group in $({\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right),\star ).$ If $x\ne 0$, $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right),$ then there is no opposite element to ${\delta }_x$ in $({\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right),\star ).$

Proof. 

As we observed above, ${\delta }_0$ is a neutral element in $({\mathcal{M}}_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right),\star ).$ Clearly,

$${\varphi }_{\lambda }\star {\varphi }_{-\lambda }\left(H^{\mathbf{k}}\right)={\varphi }_{\lambda }\left(H^{\mathbf{k}}\right)+{\varphi }_{-\lambda }\left(H^{\mathbf{k}}\right)=0={\delta }_0\left(H^{\mathbf{k}}\right)$$

for every multi-index $\mathbf{k}.$ Thus, ${\varphi }_{-\lambda }$ is opposite to ${\varphi }_{\lambda }$ and so the set of exceptional characters forms a commutative group.

Let $x\ne 0$ and $\psi$ be a character such that ${\delta }_x\star \psi ={\delta }_0.$ Then, according to Theorem 5, $\psi ={\varphi }_{\alpha }\star {\delta }_y$ for some $\alpha \in {\mathbb{C}}^s$ and $y\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ Hence,

$$\mathcal{R}({\delta }_x\star \psi )\left(t\right)=\mathcal{R}\left({\delta }_x\right)\left(t\right)\mathcal{R}({\varphi }_{\alpha }\star {\delta }_y)\left(t\right)=\mathcal{R}\left({\delta }_0\right)\left(t\right)=1.$$

Both functions $\mathcal{R}\left({\delta }_x\right)$ are entire functions of $t\in {\mathbb{C}}^s$ and $\mathcal{R}\left({\delta }_x\right)$ is of the form (9). So $\mathcal{R}\left({\delta }_x\right)$ has nontrivial zeros, while the right part is equal to 1 and has no zeros. This is a contradiction. Thus, there is no opposite element $\psi$ to ${\delta }_x.$ □

4. Conclusions

This study examined the algebra of bounded-type block-symmetric analytic functions on the space ${\mathcal{l}}_1\left({\mathbb{C}}^s\right),$ providing a detailed description of its spectrum through the exceptional characters ${\varphi }_{\lambda }$, $\lambda =({\lambda }_1,\dots ,{\lambda }_s)\in {\mathbb{C}}^s$ and characters given by exponential-type functions on ${\mathbb{C}}^s$ with plane zeros. Such a representation allowed us to prove that any character $\phi$ of the algebra of bounded-type block-symmetric analytic functions on ${\mathcal{l}}_1\left({\mathbb{C}}^s\right)$ is of the form $\phi ={\varphi }_{{\lambda }_1,\dots ,{\lambda }_s}\star {\delta }_x$ for some $\lambda \in {\mathbb{C}}^s$ and $x\in {\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ These findings contribute to a deeper understanding of the spectral structure of such algebras and establish a foundation for further exploration of their analytic and algebraic properties.

Therefore, using the Papush theory of entire functions with plane zeros, we obtained a representation of the spectrum of bounded-type block-symmetric analytic functions on ${\mathcal{l}}_1\left({\mathbb{C}}^s\right)$ as a special subset of entire exponential-type functions with plane zeros on ${\mathbb{C}}^s.$ From this representation and the known result for the case $s=1,$ we deduce the general form $\phi ={\varphi }_{\lambda }\star {\delta }_x$ for elements of the spectrum. The

Table 1. Comparison of the known results and the new results in the paper.

Known Results	New Results in the Paper
The representation of the spectrum of $H_{bs}\left({\mathcal{l}}_1\right)$ by special entire exponential functions ([17])	The representation of the spectrum of $H_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right)$ by special entire exponential functions with plane zeros (Theorem 4)
The general form of characters of $H_{bs}\left({\mathcal{l}}_1\right),$ $\phi ={\varphi }_{\lambda }\star {\delta }_x$ ([16])	The general form of characters of $H_{bvs}\left({\mathcal{l}}_1\left({\mathbb{C}}^s\right)\right),$ $\phi ={\varphi }_{\lambda }\star {\delta }_x$ (Theorem 5)

presents a comparison of the known results and the new results from this paper.

Further investigations will focus on a complete description of the spectrum of block-symmetric analytic functions of bounded type on the space ${\mathcal{l}}_1\left({\mathbb{C}}^s\right)$ and on the space ${\mathcal{l}}_p\left({\mathbb{C}}^s\right),$ where $1<p<\infty .$