# A Biorthogonal Approach to the Infinite Dimensional Fractional Poisson
Measure

Jerome B. Bendong
DMS, MSU-Iligan Institute of Technology,
Tibanga, 9200 Iligan City, Philippines
Email<EMAIL_ADDRESS>Sheila M. Menchavez
DMS, MSU-Iligan Institute of Technology,
Tibanga, 9200 Iligan City, Philippines
Email<EMAIL_ADDRESS>José Luís da Silva
Faculdade de Ciências Exatas e da Engenharia,
CIMA, Universidade da Madeira,
Campus Universitário da Penteada,
9020-105 Funchal, Portugal
Email<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

In this paper we use a biorthogonal approach to the analysis of the infinite
dimensional fractional Poisson measure $\pi_{\sigma}^{\beta}$, $0<\beta\leq
1$, on the dual of Schwartz test function space $\mathcal{D}^{\prime}$. The
Hilbert space $L^{2}(\pi_{\sigma}^{\beta})$ of complex-valued functions is
described in terms of a system of generalized Appell polynomials
$\mathbb{P}^{\sigma,\beta,\alpha}$ associated to the measure
$\pi_{\sigma}^{\beta}$. The kernels $C_{n}^{\sigma,\beta}(\cdot)$,
$n\in\mathbb{N}_{0}$, of the monomials may be expressed in terms of the
Stirling operators of the first and second kind as well as the falling
factorials in infinite dimensions. Associated to the system
$\mathbb{P}^{\sigma,\beta,\alpha}$, there is a generalized dual Appell system
$\mathbb{Q}^{\sigma,\beta,\alpha}$ that is biorthogonal to
$\mathbb{P}^{\sigma,\beta,\alpha}$. The test and generalized function spaces
associated to the measure $\pi_{\sigma}^{\beta}$ are completely characterized
using an integral transform as entire functions.

Keywords: fractional Poisson measure, generalized Appell system, Wick
exponential, test functions, generalized functions, Stirling operators,
$S$-transform.

###### Contents

  1. 1 Introduction
  2. 2 Tensor Powers of Nuclear Spaces
  3. 3 Finite Dimensional Fractional Poisson Measure
  4. 4 Infinite Dimensional Fractional Poisson Measure
    1. 4.1 Fractional Poisson Measure on the Linear Space $\mathcal{D}^{\prime}$
    2. 4.2 Configuration Space
    3. 4.3 Fractional Poisson Measure on $\Gamma$
  5. 5 Generalized Appell System
    1. 5.1 Generalized Appell Polynomials
    2. 5.2 Generalized Dual Appell System
  6. 6 Test and Generalized Function Spaces
  7. 7 Conclusion and Outlook
  8. A Appendix
    1. A.1 Kolmogorov extension theorem on configuration space
    2. A.2 Stirling Operators
    3. A.3 An Alternative Proof of the Theorem 5.7 (Biorthogonal Property)

##  1 Introduction

In this paper we develop a biorthogonal approach to the analysis of the
infinite dimensional fractional Poisson measure (fPm) on the configuration
space $\Gamma$ or over $\mathcal{D}^{\prime}$ (the dual of the Schwartz test
function space $\mathcal{D}$). As a special case of a non-Gaussian measure
(for which this biorthogonal approach was developed in [1, 23, 21]) the fPm
revealed an interesting connection with the Stirling operators and falling
factorials in the context of infinite dimensional analysis introduced recently
in [11].

To describe our results more precisely, let us recall that there are different
ways to introduce a total set of orthogonal polynomials in the Hilbert space
of square integrable functions with respect to (wrt) a probability measure.
For example, applying the Gram-Schmidt method to an independent sequence of
functions or using generating functions. In the case at hand, that is, the fPm
$\pi_{\sigma}^{\beta}$ ($0<\beta\leq 1$, $\sigma$ a non-degenerate and non-
atomic measure in $\mathbb{R}^{d}$), we have chosen the generating function
procedure because the Gram-Schmidt method is not practical. In addition, the
generating function is picked in a way such that at $\beta=1$, we recover the
classical Charlier polynomials, that is, $\pi_{\sigma}^{1}$ coincides with the
standard Poisson measure $\pi_{\sigma}$ on $\Gamma$, see [3] for more details.
In explicit, given the map

$\alpha:\mathcal{D}_{\mathbb{C}}\longrightarrow\mathcal{D}_{\mathbb{C}},\;\varphi\mapsto\alpha(\varphi)(x):=\log(1+\varphi(x)),\quad
x\in\mathbb{R}^{d},$

we define the modified Wick exponential

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w):=\frac{\exp(\langle
w,\alpha(\varphi)\rangle)}{l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))}=\sum_{n=0}^{\infty}\frac{1}{n!}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{\otimes n}\rangle,\quad
w\in\mathcal{D}^{\prime}_{\mathbb{C}},$

where $\varphi$ is properly chosen from a neighborhood of zero in
$\mathcal{D}_{\mathbb{C}}$. The monomials $\langle
C_{n}^{\sigma,\beta}(w),\varphi^{\otimes n}\rangle$, $n\in\mathbb{N}_{0}$
generates a system of polynomials $\mathbb{P}^{\sigma,\beta,\alpha}$ which
forms a total set in the space $L^{2}(\pi_{\sigma}^{\beta})$ of square
$\pi_{\sigma}^{\beta}$-integrable complex functions. The kernels
$C_{n}^{\sigma,\beta}(\cdot)$, $n\in\mathbb{N}_{0}$, possess certain
remarkable properties involving the Stirling operators of the first and second
kind as well as the falling factorials $(w)_{n}$,
$w\in\mathcal{D}^{\prime}_{\mathbb{C}}$ introduced in [11]. We refer to
Proposition 5.2 and Appendix A.2 for more details and results. Other choice of
generating functions like $\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;\cdot)$
is also possible (see the beginning of Section 5), but at $\beta=1$, the
corresponding system of polynomials do not coincide with the classical
Charlier polynomials. Thus, our natural choice goes to the modified Wick
exponential generating function
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)$.

On the other hand, the construction of the generalized dual Appell system
$\mathbb{Q}^{\sigma,\beta,\alpha}$ turns out to be very appealing since it
involves a differential operator of infinite order on the space of polynomials
$\mathcal{P}(\mathcal{D}^{\prime})$ over $\mathcal{D}^{\prime}$ and the
adjoints of the Stirling operators. This careful choice of the system
$\mathbb{Q}^{\sigma,\beta,\alpha}$ leads us to the so-called biorthogonal
property between the two systems $\mathbb{P}^{\sigma,\beta,\alpha}$ and
$\mathbb{Q}^{\sigma,\beta,\alpha}$, see Theorem 5.7.

The generalized Appell system
$\mathbb{A}^{\sigma,\beta,\alpha}:=(\mathbb{P}^{\sigma,\beta,\alpha},\mathbb{Q}^{\sigma,\beta,\alpha})$
is used to introduce a family of test function spaces
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$, $0\leq\kappa\leq 1$, which
are nuclear spaces and continuously embedded in $L^{2}(\pi_{\sigma}^{\beta})$.
The dual space of $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$ is given by
the general duality theory as
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$. In this way, we obtain the
chain of continuous embeddings

$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}\subset
L^{2}(\pi_{\sigma}^{\beta})\subset(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}.$

A typical example of a test function is the modified Wick exponential
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$
(see Example 6.1) given as a convergent series in terms of the system
$\mathbb{P}^{\sigma,\beta,\alpha}$ while a particular element in
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$ is given by the generalized Radon-
Nikodym derivative $\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)$,
$w\in\mathcal{N}^{\prime}_{\mathbb{C}}$. Moreover, the generalized function
$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)$ plays the role of the
generating function of the system $\mathbb{Q}^{\sigma,\beta,\alpha}$, that is,

$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)=\sum_{k=0}^{\infty}\frac{1}{k!}Q_{k}^{\text{$\pi_{\sigma}^{\beta},\alpha$}}((-w)_{k}).$

The spaces $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\pm\kappa}$ may be
characterized in terms of an integral transform, called the
$S_{\pi_{\sigma}^{\beta}}$-transform. It turns out that all these spaces
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\pm\kappa}$, $0\leq\kappa\leq 1$, are
universal in the sense that the $S_{\pi_{\sigma}^{\beta}}$-transform of their
elements are entire functions (for $0\leq\kappa<1)$ or holomorphic functions
($\kappa=1$) and independent of the measure $\pi_{\sigma}^{\beta}$, see
Theorem 6.6. This feature is well known in non-Gaussian analysis.

The paper is organized as follows. In Section 2, we recall some known concepts
of nuclear spaces and their tensor products. As a motivation to the
generalization of fPm to infinite dimensions, we discuss its finite
dimensional version in Section 3. We show that the monic polynomials
$C_{n}^{\beta}(x)$, $n\in\mathbb{N}_{0}$, obtained using Gram-Schmidt
orthogonalization process to the monomials $x^{n}$,
$n\in\mathbb{\mathbb{N}}_{0}$ are orthogonal in
$L^{2}(\pi_{\vec{\lambda},\beta}^{2})$ ($\pi_{\vec{\lambda},\beta}^{2}$ is the
Poisson measure in 2-dimensions) if, and only if, $\beta=1$. In Section 4, we
define the fPm $\pi_{\sigma}^{\beta}$ in infinite dimensions as a probability
measure on
$(\mathcal{D}^{\prime},\mathcal{C}_{\sigma}(\mathcal{D}^{\prime}))$, where
$\mathcal{C}_{\sigma}(\mathcal{D}^{\prime})$ is the $\sigma$-algebra generated
by the cylinder sets. We also discuss the concept of configuration space
$\Gamma$ and then using the Kolmogorov extension theorem we define a unique
measure $\pi_{\sigma}^{\beta}$ on the configuration space
$(\Gamma,\mathcal{B}(\Gamma))$ whose characteristic function coincides with
that of $\pi_{\sigma}^{\beta}$ on the distribution space
$\mathcal{D}^{\prime}$. In Section 5, we introduce the generalized Appell
system associated with the fPm $\pi_{\sigma}^{\beta}$. This includes the
system of generalized Appell polynomials and the dual Appell system which are
biorthogonal with respect to the fPm $\pi_{\sigma}^{\beta}$. Finally in
Section 6, we construct the test and generalized function spaces associated to
the fPm $\pi_{\sigma}^{\beta}$ and provide some properties as well as its
characterization theorems.

For completeness, in the Appendices A.1–A.3 we provide certain concepts and
results already known in the literature, particularly, the Kolmogorov
extension theorem on the configuration space and Stirling operators in
infinite dimensions.

##  2 Tensor Powers of Nuclear Spaces

We first consider nuclear Fréchet spaces (i.e., a complete metrizable locally
convex space) that may be characterized in terms of projective limits of a
countable number of Hilbert spaces, see e.g., [6], [7], [12], [16] and [30]
for more details and proofs.

Let $\mathcal{H}$ be a real separable Hilbert space with inner product
$(\cdot,\cdot)$ and corresponding norm $|\cdot|$. Consider a family of real
separable Hilbert space $\mathcal{H}_{p}$, $p\in\mathbb{N}$ with Hilbert norm
$|\cdot|_{p}$ such that the space $\bigcap_{p\in\mathbb{N}}\mathcal{H}_{p}$ is
dense in each $\mathcal{H}_{p}$, and

$\dots\subset\mathcal{H}_{p}\subset\dots\subset\mathcal{H}_{1}\subset\mathcal{H}$

with the corresponding system of norms being ordered, i.e.,

$|\cdot|\leq|\cdot|_{1}\leq\dots\leq|\cdot|_{p}\leq\dots\quad p\in\mathbb{N}.$

Now we assume that the space
$\mathcal{N}=\bigcap_{p\in\mathbb{N}}\mathcal{H}_{p}$ is nuclear (i.e., for
each $p\in\mathbb{N}$ there is a $q>p$ such that the canonical embedding
$\mathcal{H}_{q}\hookrightarrow\mathcal{H}_{p}$ is of Hilbert-Schmidt class)
and on $\mathcal{N}$ we fix the _projective limit topology_ , i.e., the
coarsest topology on $\mathcal{N}$ with respect to which each canonical
embedding $\mathcal{N}\hookrightarrow\mathcal{H}_{p}$, $p\in\mathbb{N}$, is
continuous. With respect to this topology, $\mathcal{N}$ is a Fréchet space
and we use the notation

$\mathcal{N}=\underset{p\in\mathbb{N}}{\mathrm{pr\,lim}}\mathcal{H}_{p}$

to denote the space $\mathcal{N}$ endowed with the corresponding projective
limit topology. Such a topological space is called a _projective limit_ or a
_countable limit of the family_ $(\mathcal{H}_{p})_{p\in\mathbb{N}}$.

Let us denote by $\mathcal{H}_{-p}$, $p\in\mathbb{N}$, the dual of
$\mathcal{H}_{p}$ with respect to the space $\mathcal{H}$, with the
corresponding Hilbert norm $|\cdot|_{-p}$. By the general duality theory, the
dual space $\mathcal{N}^{\prime}$ of $\mathcal{N}$ with respect to
$\mathcal{H}$ can then be written as

$\mathcal{N}^{\prime}:=\bigcup_{p\in\mathbb{N}}\mathcal{H}_{-p}$

with the _inductive limit topology_ , i.e., the finest topology on
$\mathcal{N}^{\prime}$ with respect to which all the embeddings
$\mathcal{H}_{-p}\hookrightarrow\mathcal{N}^{\prime}$ are continuous. This
topological space is denoted by

$\mathcal{N}^{\prime}=\underset{p\in\mathbb{N}}{\mathrm{ind\,lim}}\mathcal{H}_{-p}$

and is called an _inductive limit of the family_
$(\mathcal{H}_{-p})_{p\in\mathbb{N}}$. In this way we have obtained the chain
of spaces

$\mathcal{N}\mathcal{\subset H}\subset\mathcal{N}^{\prime}$

called a _nuclear triple_ or _Gelfand triple_. The dual pairing
$\langle\cdot,\cdot\rangle$ between $\mathcal{N}$ and $\mathcal{N}^{\prime}$
is then realized as an extension of the inner product $(\cdot,\cdot)$ on
$\mathcal{H}$, i.e.,

$\langle g,\xi\rangle=(g,\xi),\quad g\in\mathcal{H},\xi\in\mathcal{N}.$

The tensor product of the Hilbert spaces $\mathcal{H}_{p}$, $p\in\mathbb{N}$,
is denoted by $\mathcal{H}_{p}^{\otimes n}$. We keep the notation
$|\cdot|_{p}$ for the Hilbert norm on this space. The subspace of
$\mathcal{H}_{p}^{\otimes n}$ of symmetric elements is denoted by
$\mathcal{H}_{p}^{\hat{\otimes}n}$. The _$n$_ -th tensor power
$\mathcal{N}^{\otimes n}$ of $\mathcal{N}$ and the _$n$_ -th symmetric tensor
power $\mathcal{N}^{\hat{\otimes}n}$ of $\mathcal{N}$ are the nuclear Fréchet
spaces given by

$\mathcal{N}^{\otimes
n}:=\underset{p\in\mathbb{N}}{\mathrm{pr\,lim}}\mathcal{H}_{p}^{\otimes
n}\mathrm{\quad
and}\quad\mathcal{N}^{\hat{\otimes}n}:=\underset{p\in\mathbb{N}}{\mathrm{pr\,lim}}\mathcal{H}_{p}^{\hat{\otimes}n}.$

Furthermore, if $\mathcal{H}_{-p}^{\otimes n}$ (resp.,
$\mathcal{H}_{-p}^{\hat{\otimes}n}$ ) denotes the dual space of
$\mathcal{H}_{p}^{\otimes n}$ (resp., $\mathcal{H}_{p}^{\hat{\otimes}n}$) with
respect to $\mathcal{H}^{\otimes n}$, then the dual space
$(\mathcal{N}{}^{\otimes n})^{\prime}$ of $\mathcal{N}^{\otimes n}$ with
respect to $\mathcal{H}^{\otimes n}$ and the dual space
$(\mathcal{N}{}^{\hat{\otimes}n})^{\prime}$ of $\mathcal{N}^{\hat{\otimes}n}$
with respect to $\mathcal{H}^{\otimes n}$ can be written as

$(\mathcal{N}{}^{\otimes
n})^{\prime}=\underset{p\in\mathbb{N}}{\mathrm{ind\,lim}}\mathcal{H}_{-p}^{\otimes
n}\mathrm{\quad
and}\quad(\mathcal{N}{}^{\hat{\otimes}n})^{\prime}=\underset{p\in\mathbb{N}}{\mathrm{ind\,lim}}\mathcal{H}_{-p}^{\hat{\otimes}n},$

respectively. As before we use the notation $|\cdot|_{-p}$ for the norm on
$\mathcal{H}_{-p}^{\otimes n}$, $p\in\mathbb{N}$, and
$\langle\cdot,\cdot\rangle$ for the dual pairing between
$(\mathcal{N}{}^{\otimes n})^{\prime}$ and $\mathcal{N}^{\otimes n}$. Thus we
have defined the nuclear triples

$\mathcal{N}^{\otimes n}\subset\mathcal{H}^{\otimes
n}\subset(\mathcal{N}{}^{\otimes
n})^{\prime}\quad\mathrm{and}\quad\mathcal{N}^{\hat{\otimes}n}\subset\mathcal{H}^{\hat{\otimes}n}\subset(\mathcal{N}{}^{\hat{\otimes}n})^{\prime}.$

To all the real spaces in this section, we may also consider their
complexifications which will be distinguished by a subscript $\mathbb{C}$,
i.e., the complexification of $\mathcal{H}$ is
$\text{$\mathcal{H}$}_{\mathbb{C}}$ and so on. This means that for
$h\in\text{$\mathcal{H}$}_{\mathbb{C}}$, we have $h=h_{1}+ih_{2}$ where
$h_{1},h_{2}\in\text{$\mathcal{H}$}$.

Let us now introduce spaces of entire functions which will be used later in
the characterization theorems in Section 6. Let
$\mathcal{E}_{2^{-l}}^{k}(\mathcal{H}_{-p,\mathbb{C}})$ denote the set of all
entire functions on $\mathcal{H}_{-p,\mathbb{C}}$ of growth $k\in[1,2]$ and
type $2^{-l}$, $p,l\in\mathbb{Z}$. This is a linear space with norm

$n_{p,l,k}(\varphi)=\sup_{w\in\mathcal{H}_{-p,\mathbb{C}}}|\varphi(w)|\exp(-2^{-l}|z|_{-p}^{k}),\;\varphi\in\mathcal{E}_{2^{-l}}^{k}(\mathcal{H}_{-p,\mathbb{C}}).$

The space of entire functions on $\mathcal{N}^{\prime}_{\mathbb{C}}$ of growth
$k$ and minimal type is naturally introduced by

$\mathcal{E}_{\min}^{k}(\mathcal{N}^{\prime}_{\mathbb{C}}):=\underset{p,l\in\mathbb{N}}{\mathrm{pr\,lim}}\mathcal{E}_{2^{-l}}^{k}(\mathcal{H}_{-p,\mathbb{C}}),$

see e.g., [24, 6]. We will also need the space of entire functions on
$\mathcal{N}_{\mathbb{C}}$ of growth $k$ and finite type given by

$\mathcal{E}_{\max}^{k}(\mathcal{N}{}_{\mathbb{C}}):=\underset{p,l\in\mathbb{N}}{\mathrm{ind\,lim}}\mathcal{E}_{2^{l}}^{k}(\mathcal{H}_{p,\mathbb{C}}).$

##  3 Finite Dimensional Fractional Poisson Measure

In this section we discuss the finite dimensional version of the fractional
Poisson measure as a motivation to its generalization to infinite dimensions.
The one dimensional version of the fractional Poisson analysis was studied in
[5].

At first we introduce the Mittag-Leffler function $E_{\beta}$ with parameter
$\beta\in(0,1]$. The Mittag-Leffler function is an entire function defined on
the complex plane by the power series

$E_{\beta}(z):=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\beta n+1)},\quad
z\in\mathbb{C}.$ (3.1)

The Mittag-Leffler function plays the same role for the fPm as the exponential
function plays for Poisson measure. Note that for $\beta=1$ we have
$E_{1}(z)=\mathrm{e}^{z}$.

For any $0<\beta\leq 1$, the fPm $\pi_{\lambda,\beta}$ on $\mathbb{N}_{0}$ (or
$\mathbb{R})$ with rate $\lambda>0$ is defined for any
$B\in\mathscr{P}(\mathbb{N}_{0})$ by

$\pi_{\lambda,\beta}(B):=\sum_{k\in
B}\frac{\lambda^{k}}{k!}E_{\beta}^{(k)}(-\lambda),$

where $E_{\beta}^{(k)}(z):=\frac{d^{k}}{dz^{k}}E_{\beta}(z)$ is the $k$-th
derivative of the Mittag-Leffler $E_{\beta}$ function. In particular, if
$B=\\{k\\}\in\mathscr{P}(\mathbb{N}_{0})$, $k\in\mathbb{N}_{0}$, we obtain

$\pi_{\lambda,\beta}(\\{k\\}):=\frac{\lambda^{k}}{k!}E_{\beta}^{(k)}(-\lambda).$

The Laplace transform of the measure $\pi_{\lambda,\beta}$ is given for any
$z\in\mathbb{C}$ by

$l_{\pi_{\lambda,\beta}}(z)=\int_{\mathbb{R}}\mathrm{e}^{zx}\,\mathrm{d}\pi_{\lambda,\beta}(x)=\sum_{k=0}^{\infty}\frac{\big{(}\mathrm{e}^{z}\lambda\big{)}^{k}}{k!}E_{\beta}^{(k)}\big{(}-\lambda\big{)}=E_{\beta}\big{(}\lambda(\mathrm{e}^{z}-1)\big{)}.$
(3.2)

######  Remark 3.1.

The measure $\pi_{\lambda t^{\beta},\beta}$ corresponds to the marginal
distribution of the fractional Poisson process
$N_{\lambda,\beta}=(N_{\lambda,\beta}(t))_{t\geq 0}$ with parameter $\lambda
t^{\beta}>0$ defined on a probability space $(\Omega,\mathcal{F},P)$. Thus, we
obtain

$\pi_{\lambda
t^{\beta},\beta}(\\{k\\})=P(N_{\lambda,\beta}(t)=k)=\frac{(\lambda
t^{\beta})^{k}}{k!}E_{\beta}^{(k)}(-\lambda t^{\beta}),\quad
k\in\mathbb{N}_{0}.$

######  Remark 3.2.

The fractional Poisson process $N_{\lambda,\beta}$ was proposed by O. N. Repin
and A. I. Saichev [34]. Since then, it was studied by many authors see for
example [25, 27, 28, 14, 37, 4, 32, 29, 8] and references therein.

A remarkable property of the fPm is that $\pi_{\lambda,\beta}$ is given as a
mixture of Poisson measures with respect to a probability measure
$\nu_{\beta}$ on $\mathbb{R}_{+}:=[0,\infty)$. That probability measure
$\nu_{\beta}$ is absolutely continuous with respect to the Lebesgue measure on
$\mathbb{R}_{+}$ with a probability density $W_{-\beta,1-\beta}$, that is, the
Wright function. The Laplace transform of the measure $\nu_{\beta}$ (or its
density $W_{-\beta,1-\beta}$) is given by

$\int_{0}^{\infty}\mathrm{e}^{-\tau
z}\,\mathrm{d}\nu_{\beta}(\tau)=\int_{0}^{\infty}\mathrm{e}^{-z\tau}W_{-\beta,1-\beta}(\tau)\,\mathrm{d}\tau=E_{\beta}(-z),$
(3.3)

for any $z\in\mathbb{C}$ such that $\mathrm{Re}(z)\geq 0$, see [15, Cor. A.5].
Equation (3.3) is called the complete monotonicity property of the Mittag-
Leffler function, see [33]. More precisely, we have the following lemma.

######  Lemma 3.3.

For $0<\beta\leq 1$, the fPm $\pi_{\lambda,\beta}$ is an integral (or mixture)
of Poisson measure $\pi_{\lambda}$ with respect to the probability measure
$\nu_{\beta}$, i.e.,

$\pi_{\lambda,\beta}=\int_{0}^{\infty}\pi_{\lambda\tau}\,\mathrm{d\nu_{\beta}(\tau),}\quad\forall\lambda>0.$
(3.4)

###### Proof.

For $\beta=1$, we have $\nu_{1}=\delta_{1}$, the Dirac measure at 1, and the
result is clear. For $0<\beta<1$, we denote the right hand side of (3.4) by
$\mathrm{\mu}:=\int_{0}^{\infty}\mathrm{\pi}_{\lambda\tau}W_{-\beta,1-\beta}(\tau)\,\mathrm{d}\tau$.
We compute the Laplace transform of $\mu$ and use Fubini’s theorem to obtain

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{zx}\,\mathrm{d}\mu(x)$
$\displaystyle=\int_{0}^{\infty}\mathrm{e}^{zx}\int_{0}^{\infty}\mathrm{d\pi}_{\lambda\tau}(x)W_{-\beta,1-\beta}(\tau)\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{\infty}\left(\int_{0}^{\infty}\mathrm{e}^{zx}\mathrm{d\pi}_{\lambda\tau}(x)\right)W_{-\beta,1-\beta}(\tau)\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{\infty}\mathrm{e}^{\tau\lambda(\mathrm{e}^{z}-1)}W_{-\beta,1-\beta}(\tau)\,\mathrm{d}\tau$
$\displaystyle=E_{\beta}(\lambda(e^{z}-1)).$

Thus, we conclude that the Laplace transforms of $\mu$ and
$\pi_{\lambda,\beta}$ (cf. (3.2)) coincide. The result follows by the
uniqueness of the Laplace transform. ∎

######  Theorem 3.4 (Moments of $\pi_{\lambda,\beta},$ cf. [26]).

The fPm $\pi_{\lambda,\beta}$ has moments of all order. More precisely, the
$n$-th moment of the measure $\pi_{\lambda,\beta}$ is given by

$m_{\lambda,\beta}(n):=\int_{\mathbb{R}}x^{n}\,\mathrm{d}\pi_{\lambda,\beta}(x)=\sum_{m=0}^{n}\frac{m!}{\Gamma(m\beta+1)}S(n,m)\lambda^{m},$
(3.5)

where $S(n,m)$ is the Stirling number of the second kind.

Here are the first few moments of the measure $\pi_{\lambda,\beta}$:

$\begin{array}[]{rcl}m_{\lambda,\beta}(0)&=&1,\\\
m_{\lambda,\beta}(1)&=&{\displaystyle\frac{\lambda}{\Gamma(\beta+1)},}\\\
m_{\lambda,\beta}(2)&=&{\displaystyle\frac{\lambda}{\Gamma(\beta+1)}+\frac{2\lambda^{2}}{\Gamma(2\beta+1)},}\\\
m_{\lambda,\beta}(3)&=&{\displaystyle\frac{\lambda}{\Gamma(\beta+1)}+\frac{6\lambda^{2}}{\Gamma(2\beta+1)}+\frac{6\lambda^{3}}{\Gamma(3\beta+1)}.}\end{array}$

When $\beta=1$, these moments become the moments of the Poisson measure.

In addition to the Poisson measure $\pi_{\lambda}$ and fPm
$\pi_{\lambda,\beta}$ in $\mathbb{N}_{0}$ we also need the two dimensional
version of both of these measures in $\mathbb{N}_{0}^{2}$ or $\mathbb{R}^{2}$,
the reason for that is clear after Corollary 3.5. The $d$-dimensional Poisson
measure is given by

$\pi_{\vec{\lambda}}^{d}(\\{k_{1},\dots,k_{d}\\})=\prod_{i=1}^{d}\frac{\lambda_{i}^{k_{i}}}{k_{i}!}\mathrm{e}^{-\lambda_{i}}.$

The Laplace transform of $\pi_{\vec{\lambda}}^{2}$ is given by

$l_{\pi_{\vec{\lambda}}^{2}}(z)=\int_{\mathbb{R}^{2}}\mathrm{e}^{(x,s)}\,\mathrm{d}\pi_{\vec{\lambda}}^{2}(x)=\exp\big{(}\lambda_{1}(\mathrm{e}^{s_{1}}-1)+\lambda_{2}(\mathrm{e}^{s_{2}}-1)\big{)}$
(3.6)

where $s=(s_{1},s_{2})\in\mathbb{R}^{2}$. For any $0<\beta\leq 1$,
$\vec{\lambda}\in(\mathbb{R}_{+}^{*})^{2},$ then a possible fractional
generalization of $\pi_{\vec{\lambda}}^{2}$, denoted by
$\pi_{\vec{\lambda},\beta}^{2}$, is given, via its Laplace transform, by
replacing the first exponential function on the right hand side of (3.6) by
the Mittag-Leffler function. More precisely, the Laplace transform of
$\pi_{\vec{\lambda},\beta}^{2}$ is given by

$l_{\pi_{\vec{\lambda},\beta}^{2}}(s)=\int_{\mathbb{R}^{2}}\mathrm{e}^{(x,s)}\,\mathrm{d}\pi_{\vec{\lambda},\beta}^{2}(x)=E_{\beta}\big{(}\lambda_{1}(\mathrm{e}^{s_{1}}-1)+\lambda_{2}(\mathrm{e}^{s_{2}}-1)\big{)},$
(3.7)

where $s=(s_{1},s_{2})\in\mathbb{R}^{2}$.

The moments of the measure $\pi_{\vec{\lambda},\beta}^{2}$, denoted by
$m_{\vec{\lambda},\beta}^{2}(n_{1},n_{2})$, can be obtained by applying
$\frac{d^{n_{1}}}{ds_{1}^{n_{1}}}\frac{d^{n_{2}}}{ds_{2}^{n_{2}}}$,
$n_{1},n_{2}\in\mathbb{N}_{0}$, to Equation (3.7) and then evaluating at
$s_{1}=s_{2}=0$. As an example, here we compute the moments
$m_{\vec{\lambda},\beta}^{2}(1,1)$ and $m_{\vec{\lambda},\beta}^{2}(1,2)$ of
the measure $\pi_{\vec{\lambda},\beta}^{2}$ needed later on:

$\displaystyle
m_{\vec{\lambda},\beta}^{2}(1,1)=\int_{\mathbb{R}^{2}}x_{1}x_{2}\,\mathrm{d}\pi_{\vec{\lambda},\beta}^{2}(x_{1,}x_{2})$
$\displaystyle=\frac{2\lambda_{1}\lambda_{2}}{\Gamma(2\beta+1)},$
$\displaystyle
m_{\vec{\lambda},\beta}^{2}(1,2)=\int_{\mathbb{R}^{2}}x_{1}x_{2}^{2}\,\mathrm{d}\pi_{\vec{\lambda},\beta}^{2}(x_{1,}x_{2})$
$\displaystyle=\frac{2\lambda_{1}\lambda_{2}}{\Gamma(2\beta+1)}+\frac{6\lambda_{1}\lambda_{2}^{2}}{\Gamma(3\beta+1)}.$

We apply the Gram-Schmidt orthogonalization process to the monomials $x^{n}$,
$n\in\mathbb{\mathbb{N}}_{0},$ to obtain monic polynomials $C_{n}^{\beta}(x)$
with $\deg C_{n}^{\beta}(x)=n$ with respect to the inner product

$(p,q)_{\pi_{\lambda,\beta}}:=\int_{\mathbb{R}}p(x)q(x)\,\mathrm{d}\pi_{\lambda,\beta}(x).$

These polynomials are determined by the moments of the measure
$\pi_{\lambda,\beta}$. The first few of these polynomials are given by

$\displaystyle C_{0}^{\beta}(x)$ $\displaystyle=1,$ $\displaystyle
C_{1}^{\beta}(x)$
$\displaystyle=x-(x,C_{0}^{\beta})_{\pi_{\lambda,\beta}}C_{0}^{\beta}(x)=x-m_{\lambda,\beta}(1),$
$\displaystyle C_{2}^{\beta}(x)$
$\displaystyle=x^{2}-(x^{2},C_{0}^{\beta})_{\pi_{\lambda,\beta}}C_{0}^{\beta}(x)-\left(x^{2},\frac{C_{1}^{\beta}}{\|C_{1}^{\beta}\|_{\pi_{\lambda,\beta}}^{2}}\right)_{\pi_{\lambda,\beta}}C_{1}^{\beta}(x),$
$\displaystyle=x^{2}-A(\beta,\lambda)x-m_{\lambda,\beta}(2)+A(\beta,\lambda)m_{\lambda,\beta}(1),$

where

$A(\beta,\lambda)=\frac{m_{\lambda,\beta}(3)-m_{\lambda,\beta}(1)m_{\lambda,\beta}(2)}{m_{\lambda,\beta}(2)-(m_{\lambda,\beta}(1))^{2}}.$

When $\beta=1$, the measure $\pi_{\lambda,1}$ becomes the Poisson measure
$\pi_{\lambda}$ and the polynomials $C_{n}^{1}(x),$ $n\in\mathbb{N}_{0}$, are
the classical Charlier polynomials.

######  Corollary 3.5.

For $\beta\in(0,1]$ it holds

$\int_{\mathbb{R}^{2}}C_{1}^{\beta}(x_{1})C_{2}^{\beta}(x_{2})\,\mathrm{d}\pi_{\vec{\lambda},\beta}^{2}(x_{1},x_{2})=0$

if, and only if, $\beta=1.$

###### Proof.

When $\beta=1$, we have the well known orthogonal property of the Charlier
polynomials, that is,

$\int_{\mathbb{R}^{2}}C_{1}(x_{1})C_{2}(x_{2})\,\mathrm{d}\pi_{\vec{\lambda}}^{2}(x_{1},x_{2})=\int_{\mathbb{R}}C_{1}(x_{1})\,\mathrm{d}\pi_{\lambda}(x_{1})\int_{\mathbb{R}}C_{2}(x_{2})\,\mathrm{d}\pi_{\lambda}(x_{2})=0.$

On the other hand, for $\beta\in(0,1)$ we have

$\displaystyle\int_{\mathbb{R}^{2}}C_{1}^{\beta}(x_{1})C_{2}^{\beta}(x_{2})\,\mathrm{d}\pi_{\vec{\lambda},\beta}^{2}(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}^{2}}\left(x_{1}-m_{\lambda_{1},\beta}(1)\right)\big{(}x_{2}^{2}-A(\beta,\lambda_{2})x_{2}-m_{\lambda_{2},\beta}(2)+A(\beta,\lambda_{2})m_{\lambda_{2},\beta}(1)\big{)}\,\mathrm{d}\pi_{\vec{\lambda},\beta}^{2}(x_{1},x_{2})$
$\displaystyle=m_{\vec{\lambda},\beta}^{2}(1,2)-A(\beta,\lambda_{2})m_{\vec{\lambda},\beta}^{2}(1,1)-m_{\lambda_{1},\beta}(1)m_{\lambda_{2},\beta}(2)+A(\beta,\lambda_{2})m_{\lambda_{1},\beta}(1)m_{\lambda_{2},\beta}(1).$
(3.8)

Equation (3.3.3) defines a function $F(\beta,\lambda_{1},\lambda_{2})$ which
is not equal from to zero for every $\beta\in(0,1)$, see Figure 3.1. ∎

Figure 1: The graph of the function $F(\cdot,\lambda_{1},\lambda_{2})$ with
$\vec{\lambda}=(1,1),(2,3),(1,2)$.

Having in mind the above results, this motivate us to introduce a biorthogonal
system of the fPm in higher dimension.

##  4 Infinite Dimensional Fractional Poisson Measure

After the above preparation, we are ready to define the fPm in infinite
dimensions. We define the fPm in the linear space $\mathcal{D}^{\prime}$ and
then a more careful analysis shows that fPm is indeed a probability measure on
the configuration space $\Gamma$ over $\mathbb{R}^{d}$.

###  4.1 Fractional Poisson Measure on the Linear Space $\mathcal{D}^{\prime}$

Let $\vec{\lambda}=(\lambda_{1},\dots,\lambda_{d})\in(\mathbb{R}_{+}^{*})^{d}$
and $z=(z_{1},\dots,z_{d})\in\mathbb{R}^{d}$ be given. The $d$-dimensional
Poisson measure has characteristic function given by

$C_{\pi_{\lambda}^{d}}(z)=\int_{\mathbb{R}^{d}}\mathrm{e}^{\mathrm{i}(x,z)}\,\mathrm{d}\pi_{\vec{\lambda}}^{d}(x)=\exp\left(\sum_{k=1}^{d}\lambda_{k}(\mathrm{e}^{\mathrm{i}z_{k}}-1)\right).$
(4.1)

Let us consider a _Radon measure_ $\sigma$ on
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$, that is,
$\sigma(\Lambda)<\infty$ for every
$\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$, where
$\mathcal{B}_{c}(\mathbb{R}^{d})$ the family of all
$\mathcal{B}(\mathbb{R}^{d})$-measurable sets with compact closure. Elements
of $\mathcal{B}_{c}(\mathbb{R}^{d})$ are called finite volumes. Here, we
assume $\sigma$ to be non-degenerate (i.e., $\sigma(O)>0$ for all non-empty
open sets $O\subset\mathbb{R}^{d}$) and non-atomic (i.e.,
$\sigma(\left\\{x\right\\})=0$ for every $x\in\mathbb{R}^{d}$). In addition,
we always assume that $\sigma(\mathbb{R}^{d})=\infty$. Let
$\mathcal{D}:=\mathcal{D}(\mathbb{R}^{d})$ be the space of
$C^{\infty}$-functions with compact support in $\mathbb{R}^{d}$ and
$\mathcal{D}^{\prime}:=\mathcal{D}^{\prime}(\mathbb{R}^{d})$ be the dual of
$\mathcal{D}$ with respect to the Hilbert space
$L^{2}(\sigma):=L^{2}(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\sigma)$. In
this way, we obtain the triple

$\mathcal{D}\subset L^{2}(\sigma)\subset\mathcal{D}^{\prime}.$ (4.2)

The infinite-dimensional generalization of the Poisson measure with intensity
measure $\sigma$, denoted by $\pi_{\sigma}$, is obtained by generalizing the
characteristic function (4.1) to

$C_{\pi_{\sigma}}(\varphi):=\int_{\mathcal{D}^{\prime}}\mathrm{e}^{\mathrm{i}\langle
w,\varphi\rangle}\,\mathrm{d}\pi_{\sigma}(w)=\exp\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\mathrm{\,d}\sigma(x)\right),\;\varphi\in\mathcal{D}.$
(4.3)

This is achieve through the Bochner-Minlos theorem (see e.g. [6]) by showing
that $C_{\pi_{\sigma}}$ is the Fourier transform of a measure on the
distribution space $\mathcal{D}^{\prime}$, see [2] and references therein.
Now, using the fact that the Mittag-Leffler function is a natural
generalization of the exponential function, one conjectures that the
characteristic functional

$C_{\pi_{\sigma}^{\beta}}(\varphi):=\int_{\mathcal{D}^{\prime}}\mathrm{e}^{\mathrm{i}\langle
w,\varphi\rangle}\,\mathrm{d}\pi_{\sigma}^{\beta}(w)=E_{\beta}\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\mathrm{\,d}\sigma(x)\right),\quad\varphi\in\mathcal{D},$
(4.4)

defines an infinite-dimensional version of the fPm, denoted by
$\pi_{\sigma}^{\beta}$. However, since the Mittag-Leffler function does not
satisfy the semigroup property of the exponential, it is not obvious that this
is the Fourier transform of a measure on $\mathcal{D}^{\prime}$. Hence, we use
the Bochner-Minlos theorem to show that $C_{\pi_{\sigma}^{\beta}}$ is the
Fourier transform of a probability measure $\pi_{\sigma}^{\beta}$ on the
distribution space $\mathcal{D}^{\prime}$.

######  Theorem 4.1.

For each $0<\beta\leq 1$ fixed, the functional $C_{\pi_{\sigma}^{\beta}}$ in
Equation (4.4) is the characteristic functional on $\mathcal{D}$ of a
probability measure $\pi_{\sigma}^{\beta}$ on the distribution space
$\mathcal{D}^{\prime}$.

###### Proof.

Using the properties of the Mittag-Leffler function, the functional
$C_{\pi_{\sigma}^{\beta}}$ is continuous and $C_{\pi_{\sigma}^{\beta}}(0)=1$
follow directly. To show that the functional $C_{\pi_{\sigma}^{\beta}}$ is
positive definite, we use the complete monotonicity property of $E_{\beta}$,
$0<\beta<1$, see (3.3). For any $\varphi_{i}\in\mathcal{D}$,
$z_{i}\in\mathbb{C}$, $i=1,\dots,n$, using Equation (4.3), we obtain

$\displaystyle\sum_{k,j=1}^{n}C_{\pi_{\sigma}^{\beta}}(\varphi_{k}-\varphi_{j})z_{k}\bar{z}_{j}$
$\displaystyle=\int_{0}^{\infty}\sum_{k,j=1}^{n}\mathrm{e}^{\tau\int_{\mathbb{R}^{d}}(\mathrm{e}^{\mathrm{i}(\varphi_{k}-\varphi_{j})}-1)\,\mathrm{d}\sigma(x)}z_{k}\bar{z}_{j}\,\mathrm{d}\nu_{\beta}(\tau)$
$\displaystyle=\int_{0}^{\infty}\sum_{k,j=1}^{n}C_{\pi_{\tau\sigma}}(\varphi_{k}-\varphi_{j})z_{k}\bar{z}_{j}\,\mathrm{d}\nu_{\beta}(\tau).$

Using the definition of $C_{\pi_{\tau\sigma}}$, the integrand of the last
integral may be written as

$\sum_{k,j=1}^{n}C_{\pi_{\tau\sigma}}(\varphi_{k}-\varphi_{j})z_{k}\bar{z}_{j}=\int_{\mathcal{D}^{\prime}}\left|\sum_{k=1}^{n}\mathrm{e}^{\mathrm{i}\langle
w,\varphi_{k}\rangle}z_{k}\right|^{2}\mathrm{d}\pi_{\tau\sigma}(w)\geq 0.$

This implies that $C_{\pi_{\sigma}^{\beta}}$ is positive-definite. Thus by the
Bochner-Minlos theorem, $C_{\pi_{\sigma}^{\beta}}$ is the characteristic
functional of a probability measure $\pi_{\sigma}^{\beta}$ on the measurable
space $(\mathcal{D}^{\prime},\mathcal{C}_{\sigma}(\mathcal{D}^{\prime}))$. ∎

######  Remark 4.2.

By the analytic property of the Mittag-Leffler function one may write (4.4)
for any $\varphi\in\mathcal{D}$ such that
$\mathrm{supp}\,\varphi\subset\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$, as

$\displaystyle C_{\pi_{\sigma}^{\beta}}(\varphi)$
$\displaystyle=E_{\beta}\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\mathrm{\,d}\sigma(x)\right)=E_{\beta}\left(\int_{\Lambda}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\mathrm{\,d}\sigma(x)\right)$
$\displaystyle=E_{\beta}\left(\int_{\Lambda}\mathrm{e}^{\mathrm{i}\varphi(x)}\mathrm{\,d}\sigma(x)-\sigma(\Lambda)\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}\left(-\sigma(\Lambda)\right)}{n!}\left(\int_{\Lambda}\mathrm{e}^{\mathrm{i}\varphi(x)}\,\mathrm{d}\sigma(x)\right)^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}\left(-\sigma(\Lambda)\right)}{n!}\int_{\Lambda^{n}}\mathrm{e}^{\mathrm{i}(\varphi(x_{1})+\dots+\varphi(x_{n}))}\,\mathrm{d}\sigma^{\otimes
n}(x_{1},\dots,x_{n}),$

where $\sigma^{\otimes n}=\sigma\otimes\dots\otimes\sigma$ is a measure
defined on the Cartesian space
$(\mathbb{R}^{d})^{n}:=\mathbb{R}^{d}\times\dots\times\mathbb{R}^{d}.$ In the
Poisson case, we have $\exp\left(-\sigma(\Lambda)\right)$ instead of
$E_{\beta}^{(n)}\left(-\sigma(\Lambda)\right)$, for all $n\in\mathbb{N}_{0}$,
while the rest of the terms are the same. Hence, the main difference between
these measures ($\pi_{\sigma}^{\beta}$ and $\pi_{\sigma}$) is the different
weight given in each $n$-particle space. In Subsection 4.3 we show that,
indeed, the support of the measure $\pi_{\sigma}^{\beta}$ is a subset of
$\mathcal{D}^{\prime}$, called the configuration space over $\mathbb{R}^{d}$.

We may now generalize the result of Lemma 3.3 to the present infinite
dimensional setting.

######  Lemma 4.3.

For $0<\beta\leq 1$, the fPm $\pi_{\sigma}^{\beta}$ is an integral (or
mixture) of Poisson measure $\pi_{\sigma}$ with respect to the probability
measure $\nu_{\beta}$, i.e.,

$\pi_{\sigma}^{\beta}=\int_{0}^{\infty}\pi_{\tau\sigma}\,\mathrm{d\nu_{\beta}(\tau)}.$
(4.5)

###### Proof.

For $\beta=1$, the result is clear as in the proof of Lemma 3.3. For
$0<\beta<1$, we use the representation (3.3) of the Mittag-Leffler function,
the characteristic functional (4.4) of $\pi_{\sigma}^{\beta}$ can be rewritten
as

$C_{\pi_{\sigma}^{\beta}}(\varphi)=\int_{0}^{\infty}\exp\left(-\tau\int_{\mathbb{R}^{d}}(1-\mathrm{e}^{\mathrm{i}\varphi(x)})\mathrm{\,d}\sigma(x)\right)\mathrm{d}\nu_{\beta}(\tau)$

with the integrand being the characteristic function of the Poisson measure
$\pi_{\tau\sigma}$, $\tau>0$. This implies that the characteristic functional
(4.4) coincides with the characteristic functional of the measure
$\int_{0}^{\infty}\pi_{\tau\sigma}\,\mathrm{d}\nu_{\beta}(\tau)$. The result
follows by the uniqueness of the characteristic functional. ∎

The fPm $\pi_{\sigma}^{\beta}$ is indeed a probability measure on
$(\mathcal{D}^{\prime},\mathcal{C}_{\sigma}(\mathcal{D}^{\prime}))$. In what
follows, we are going to find an appropriate support for
$\pi_{\sigma}^{\beta}$.

###  4.2 Configuration Space

Recall that $\mathcal{B}(\mathbb{R}^{d})$ denotes the Borel $\sigma$-algebra
on $\mathbb{R}^{d}$ and $\mathcal{B}_{c}(\mathbb{R}^{d})$ the system of all
sets in $\mathcal{B}(\mathbb{R}^{d})$ which are bounded and have compact
closure. Below we recall the configuration space over $\mathbb{R}^{d}$ and
related concepts, see [2, 22] for more details.

######  Definition 4.4.

The infinite configuration space __ $\Gamma:=\Gamma_{\mathbb{R}^{d}}$ over
$\mathbb{R}^{d}$ is defined as the set of all locally finite subsets from
$\mathbb{R}^{d}$, that is,

$\Gamma:=\big{\\{}\gamma\subset\mathbb{R}^{d}:|\gamma\cap\Lambda|<\infty\thinspace\text{for
every }\thinspace\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})\big{\\}},$

where $|B|$ denotes the cardinality of the set $B$. The elements of the space
$\Gamma$ are called _configurations_.

Let $C_{0}(\mathbb{R}^{d})$ denote the class of all real-valued continuous
functions on $\mathbb{R}^{d}$ with compact support and
$\mathcal{M}^{+}:=\mathcal{M}^{+}(\mathbb{R}^{d})$ (resp.
$\mathcal{M}_{\mathbb{N}_{0}}^{+}:=\mathcal{M}_{\mathbb{N}_{0}}^{+}(\mathbb{R}^{d})$)
denote the space of all positive (resp. positive integer-valued) Radon
measures on $\mathcal{B}(\mathbb{R}^{d})$.

######  Definition 4.5.

Each configuration $\gamma\in\Gamma$ can be identified with a non-negative
integer-valued Radon measure as follows

$\Gamma\ni\gamma\mapsto\sum_{x\in\gamma}\delta_{x}\in\mathcal{M}_{\mathbb{N}_{0}}^{+}\subset\mathcal{M}^{+},$

where $\delta_{x}$ is the Dirac measure at $x\in\mathbb{R}^{d}$ and
$\sum_{x\in\emptyset}\delta_{x}:=0$ (zero measure). The space $\Gamma$ can be
endowed with the topology induced by the vague topology on $\mathcal{M}^{+}$,
i.e., the weakest topology on $\Gamma$ with respect to which all mappings

$\Gamma\ni\gamma\mapsto\langle\gamma,f\rangle:=\langle
f\rangle_{\gamma}:=\int_{\mathbb{R}^{d}}f(x)\,\mathrm{d}\gamma(x)=\sum_{x\in\gamma}f(x)\in\mathbb{R}$

are continuous for any $f\in C_{0}(\mathbb{R}^{d})$.

######  Definition 4.6.

Let $\mathcal{B}(\Gamma)$ be the Borel $\sigma$-algebra corresponding to the
vague topology on $\Gamma$.

  1. 1.

The $\sigma$-algebra $\mathcal{B}(\Gamma)$ is generated by the sets of the
form

$C_{\Lambda,n}=\left\\{\gamma\in\Gamma\,|\,|\gamma\cap\Lambda|=n\right\\}\\!,$
(4.6)

where $\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$, $n\in\mathbb{N}_{0}$, and
the set $C_{\Lambda,n}$ is a Borel set of $\Gamma$, that is,
$C_{\Lambda,n}\in\mathcal{B}(\Gamma)$. Sets of the form (4.6) are called
_cylinder sets_.

  2. 2.

For any $B\subset\mathbb{R}^{d}$, we introduce a function
$N_{B}:\Gamma\rightarrow\mathbb{N}_{0}$ such that

$\displaystyle N_{B}(\gamma):$ $\displaystyle=|\gamma\cap
B|,\quad\gamma\in\Gamma.$

Then $\mathcal{B}(\Gamma)$ is the minimal $\sigma$-algebra with which all
functions $N_{\Lambda}$, $\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$, are
measurable.

######  Definition 4.7.

Let $Y\in\mathcal{B}(\mathbb{R}^{d})$ be given. The space __ of configurations
contained in _ $Y$_ is denoted by _$\Gamma_{Y}$ , _i.e.,

$\Gamma_{Y}:=\left\\{\gamma\in\Gamma\mid|\gamma\cap(\mathbb{R}^{d}\setminus
Y)|=0\right\\}.$

The $\sigma$-algebra $\mathcal{B}(\Gamma_{Y})$ may be introduced in a similar
way

$\mathcal{B}(\Gamma_{Y}):=\sigma\left(\left\\{N_{\Lambda}{\upharpoonright}_{\Gamma_{Y}}\mid\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})\right\\}\right),$

where $N_{\Lambda}{\upharpoonright}_{\Gamma_{Y}}$ denotes the restriction of
the mapping $N_{\Lambda}$ to $\Gamma_{Y}$.

######  Definition 4.8.

Let $Y\in\mathcal{B}(\mathbb{R}^{d})$ be given. The space of n-point
configurations $\Gamma_{Y}^{(n)}$ over a set $Y$ is the subset of $\Gamma_{Y}$
defined by

$\Gamma_{Y}^{(n)}:=\\{\gamma\in\Gamma_{Y}\mid|\gamma|=n\\},n\in\mathbb{N},\quad\Gamma_{Y}^{(0)}:=\\{\emptyset\\}.$

A topological structure may be introduced on $\Gamma_{Y}^{(n)}$,
$n\in\mathbb{N}$, through a natural surjective mapping from

$\widetilde{Y^{n}}=\\{(x_{1},\dotsc,x_{n})\mid x_{k}\in Y,x_{k}\neq
x_{j}\thinspace\text{if}\thinspace k\neq j\\}$

onto $\Gamma_{Y}^{(n)}$ defined by

$\mathrm{sym}_{Y}^{n}:\widetilde{Y^{n}}\longrightarrow\Gamma_{Y}^{(n)}$
$(x_{1},\dotsc,x_{n})\mapsto\\{x_{1},\dotsc,x_{n}\\}.$

Indeed, using the mapping $\mathrm{sym}_{\Lambda}^{n},$ one constructs a
bijective mapping between $\Gamma_{Y}^{(n)}$ and the symmetrization
$\widetilde{Y^{n}}/S_{n}$ of $\widetilde{Y^{n}},$ where $S_{n}$ is the
permutation group over $\\{1,\dotsc,n\\}.$ In this way, $\mathrm{sym}_{Y}^{n}$
induces a metric on $\Gamma_{Y}^{(n)}$. A set $U\subset\Gamma_{Y}^{(n)}$ is
open in this topology if, and only if, the inverse image
$(\mathrm{sym}_{Y}^{n})^{-1}(U)$ is open in $\widetilde{Y^{n}}.$ We denote by
$\mathcal{B}(\Gamma_{Y}^{(n)})$ the corresponding Borel $\sigma$-algebra and
$\mathcal{T}_{Y}^{(n)}$ the associated topology on $\Gamma_{Y}^{(n)}$.

For $\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d}),$ each space $\Gamma_{\Lambda}$
can be described by the disjoint union

$\Gamma_{\Lambda}=\bigsqcup_{n=0}^{\infty}\Gamma_{\Lambda}^{(n)}.$

In particular, this representation provides an equivalent description of the
$\sigma$-algebra $\mathcal{B}(\Gamma_{\Lambda})$ as the $\sigma$-algebra of
the disjoint union of the $\sigma$-algebras
$\mathcal{B}(\Gamma_{\Lambda}^{(n)})\thinspace,n\in\mathbb{N}_{0}.$ The
corresponding topology is denoted by $\mathcal{T}_{\Lambda}$ such that
$(\Gamma_{\Lambda},\mathcal{T}_{\Lambda})$ is a topological space for each
$\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$.

For each $\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$ and any pair
$\Lambda_{1},\Lambda_{2}\in\mathcal{B}_{c}(\mathbb{R}^{d})$ such that
$\Lambda_{1}\subset\Lambda_{2},$ let us consider the natural measurable
projections

$\displaystyle p_{\Lambda}:\Gamma\longrightarrow\Gamma_{\Lambda}$
$\displaystyle
p_{\Lambda_{1},\Lambda_{2}}:\Gamma_{\Lambda_{2}}\longrightarrow\Gamma_{\Lambda_{1}}$
(4.7) $\displaystyle\gamma\mapsto\gamma\cap\Lambda$
$\displaystyle\gamma\mapsto\gamma\cap\Lambda_{1}.$

We use now the concept of the projective limit in order to show that the
measurable space $(\Gamma,\mathcal{B}(\Gamma))$ coincides with the projective
limit. More precisely, we have the following theorem.

######  Theorem 4.9.

The family
$\left\\{(\Gamma_{\Lambda},\mathcal{B}(\Gamma_{\Lambda})),p_{\Lambda_{1},\Lambda_{2}},\mathcal{B}_{c}(\mathbb{R}^{d})\right\\}$
is a projective system of measurable spaces with ordered index set
$(\mathcal{B}_{c}(\mathbb{R}^{d}),\subset)$ and the measurable space
$(\Gamma,\mathcal{B}(\Gamma))$ is (up to an isomorphism) the projective limit
together with the family of maps
$p_{\Lambda}:\Gamma\longrightarrow\Gamma_{\Lambda}$ for any
$\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$. In addition, we have the
following commutative diagram.

$\textstyle{\Gamma^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I}$$\scriptstyle{P_{\Lambda_{2}}}$$\scriptstyle{P_{\Lambda_{1}}}$$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\Lambda_{2}}}$$\scriptstyle{p_{\Lambda_{1}}}$$\textstyle{\Gamma_{\Lambda_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\Lambda_{2},\Lambda_{1}}}$$\textstyle{\Gamma_{\Lambda_{1}}}$

###### Proof.

It is clear from the construction above that the maps
$p_{\Lambda_{1},\Lambda_{2}}$,
$\Lambda_{1},\Lambda_{2}\in\mathcal{B}_{c}(\mathbb{R}^{d})$ are measurable and
satisfies

$p_{\Lambda_{1},\Lambda_{2}}\circ
p_{\Lambda_{2},\Lambda_{3}}=p_{\Lambda_{1},\Lambda_{3}},\quad\Lambda_{1}\subset\Lambda_{2}\subset\Lambda_{3}\>\>\mathrm{in\>\>}\mathcal{B}_{c}(\mathbb{R}^{d}).$

As a result,
$\left\\{(\Gamma_{\Lambda},\mathcal{B}(\Gamma_{\Lambda})),p_{\Lambda_{1},\Lambda_{2}},\mathcal{B}_{c}(\mathbb{R}^{d})\right\\}$
is a projective system. On the other hand, it is easy to see from (4.7) that
the following relation

$p_{\Lambda_{1}}=p_{\Lambda_{1},\Lambda_{2}}\circ
p_{\Lambda_{2}},\quad\Lambda_{1}\subset\Lambda_{2}\>\>\mathrm{in\>\>}\mathcal{B}_{c}(\mathbb{R}^{d})$

holds. By definition of $\mathcal{B}(\Gamma)$, the family of maps
$p_{\Lambda}$, $\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$ satisfy the
conditions in a projective limit of measurable spaces. This concludes the
proof. ∎

###  4.3 Fractional Poisson Measure on $\Gamma$

Recall from Section 4.1 the measure $\sigma$ on the underlying measurable
space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ and the product measure
$\sigma^{\otimes n}$ on
$((\mathbb{R}^{d})^{n},\mathcal{B}((\mathbb{R}^{d})^{n}))$, for each
$n\in\mathbb{N}$. Then $\sigma^{\otimes
n}((\mathbb{R}^{d})^{n}\setminus\widetilde{(\mathbb{R}^{d})^{n}})=0$, since
$\sigma$ is non-atomic. It follows that $\sigma^{\otimes
n}(B{}^{n}\setminus\widetilde{B^{n}})=0$, for every
$B\in\mathcal{B}(\mathbb{R}^{d})$. For each
$\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$, let us consider the restriction
of $\sigma^{\otimes n}$ to
$(\widetilde{\Lambda^{n}},\mathcal{B}(\widetilde{\Lambda^{n}}))$, which is a
finite measure, and then define the image measure $\sigma_{\Lambda}^{(n)}$ on
$(\Gamma_{\Lambda}^{(n)},\mathcal{B}(\Gamma_{\Lambda}^{(n)}))$ under the
mapping $\mathrm{sym}_{\Lambda}^{n}$ by

$\sigma_{\Lambda}^{(n)}:=\sigma^{\otimes
n}\circ(\mathrm{sym}_{\Lambda}^{n})^{-1}.$

For $n=0$, we set $\sigma_{\Lambda}^{(0)}(\left\\{\emptyset\right\\}):=1$.
Now, for each $0<\beta<1$, one may define a probability measure
$\pi_{\sigma,\Lambda}^{\beta}$ on
$(\Gamma_{\Lambda},\mathcal{B}(\Gamma_{\Lambda}))$ by

$\pi_{\sigma,\Lambda}^{\beta}:=\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}(-\sigma(\Lambda))}{n!}\sigma_{\Lambda}^{(n)}.$
(4.8)

Note that $E_{\beta}^{(n)}(-\sigma(\Lambda))\geq 0$, $n\in\mathbb{N}_{0}$ due
to (3.3). In addition, we have

$\displaystyle\pi_{\sigma,\Lambda}^{\beta}(\Gamma_{\Lambda})$
$\displaystyle=\pi_{\sigma,\Lambda}^{\beta}\left(\bigsqcup_{n=0}^{\infty}\Gamma_{\Lambda}^{(n)}\right)=\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}(-\sigma(\Lambda))}{n!}\sigma_{\Lambda}^{(n)}(\Gamma_{\Lambda}^{(n)})$
$\displaystyle=\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}(-\sigma(\Lambda))}{n!}\sigma(\Lambda)^{n}=E_{\beta}(0)=1.$

The family
$\\{\pi_{\sigma,\Lambda}^{\beta}\mid\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})\\}$
of probability measures yields a probability measure on
$(\Gamma,\mathcal{B}(\Gamma))$. In fact, this family is consistent in the
sense that the measure $\pi_{\sigma,\Lambda_{1}}^{\beta}$ is the image measure
of $\pi_{\sigma,\Lambda_{2}}^{\beta}$ under $p_{\Lambda_{1},\Lambda_{2}}$,
that is,

$\pi_{\sigma,\Lambda_{1}}^{\beta}=\pi_{\sigma,\Lambda_{2}}^{\beta}\circ
p_{\Lambda_{1},\Lambda_{2}}^{-1},\quad\forall\Lambda_{1},\Lambda_{2}\in\mathcal{B}_{c}(\mathbb{R}^{d}),\Lambda_{1}\subset\Lambda_{2}.$

By the Kolmogorov extension theorem on configuration space (see Appendix A.1)
the family
$\big{\\{}\pi_{\sigma,\Lambda}^{\beta}\mid\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})\big{\\}}$
determines uniquely a measure $\pi_{\sigma}^{\beta}$ on
$(\Gamma,\mathcal{B}(\Gamma))$ such that

$\pi_{\sigma,\Lambda}^{\beta}=\pi_{\sigma}^{\beta}\circ
p_{\Lambda}^{-1},\quad\forall\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d}).$

Actually, we don’t need the whole family of local sets
$\mathcal{B}_{c}(\mathbb{R}^{d})$ instead, a sub-family
$\mathcal{J}_{\mathbb{R}^{d}}$ with an abstract concept of local sets, see
(I1)–(I3) in Appendix A.1.

Let us now compute the characteristic functional of the measure
$\pi_{\sigma}^{\beta}$. Given $\varphi\in\mathcal{D}$, we have
$\mathrm{supp\,}\varphi\subset\Lambda$ for some
$\Lambda\in\mathcal{B}_{c}(\mathbb{R}^{d})$, such that

$\langle\gamma,\varphi\rangle=\langle
p_{\Lambda}(\gamma),\varphi\rangle,\hskip 7.22743pt\forall\leavevmode\nobreak\
\gamma\in\Gamma.$

Thus,

$\int_{\Gamma}\mathrm{e}^{\mathrm{i}\langle\gamma,\varphi\rangle}\mathrm{d}\pi_{\sigma}^{\beta}(\gamma)=\int_{\Gamma_{\Lambda}}\mathrm{e}^{\mathrm{i}\langle\gamma,\varphi\rangle}\mathrm{d}\pi_{\sigma,\Lambda}^{\beta}(\gamma)$

and the infinite divisibility (4.8) of the measure
$\pi_{\sigma,\Lambda}^{\beta}$ yields for the right-hand side of the above
equality

$\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}(-\sigma(\Lambda))}{n!}\int_{\Lambda^{n}}\mathrm{e}^{\mathrm{i}(\varphi(x_{1})+\dots+\varphi(x_{n}))}\,\mathrm{d}\sigma^{\otimes
n}(x_{1},\dots,x_{n})=\sum_{n=0}^{\infty}\frac{E_{\beta}^{(n)}(-\sigma(\Lambda))}{n!}\left(\int_{\Lambda}\mathrm{e}^{\mathrm{i}\varphi(x)}\,\mathrm{d}\sigma(x)\right)^{n}$

which corresponds to the Taylor expansion of the function

$E_{\beta}\left(\int_{\Lambda}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\,\mathrm{d}\sigma(x)\right)=E_{\beta}\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\,\mathrm{d}\sigma(x)\right).$

Hence, for any $\varphi\in\mathcal{D}$, we obtain

$\int_{\Gamma}\mathrm{e}^{\mathrm{i}\langle\gamma,\varphi\rangle}\,\mathrm{d}\pi_{\sigma}^{\beta}(\gamma)=E_{\beta}\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\mathrm{i}\varphi(x)}-1)\,\mathrm{d}\sigma(x)\right).$
(4.9)

######  Remark 4.10.

  1. 1.

The characteristic functional of the measure $\pi_{\sigma}^{\beta}$ given in
(4.9) coincides with the characteristic functional (4.4) of the measure
$\pi_{\sigma}^{\beta}$ on the distribution space $\mathcal{D}^{\prime}$. The
functional (4.9) shows that the measure $\pi_{\sigma}^{\beta}$ is supported on
generalized functions of the form
$\sum_{x\in\gamma}\delta_{x}\in\mathcal{D}^{\prime},$ $\gamma\in\Gamma$.

  2. 2.

Note that $\Gamma\subset\mathcal{D}^{\prime}$ but in contrast to $\Gamma,$
$\mathcal{D}^{\prime}$ is a linear space. Since
$\pi_{\sigma}^{\beta}(\Gamma)=1$, the measure space
$(\mathcal{D}^{\prime},\mathcal{C}_{\sigma}(\mathcal{D}^{\prime}),\pi_{\sigma}^{\beta})$
can, in this way, be regarded as a linear extension of the fractional Poisson
space $(\Gamma,\mathcal{B}(\Gamma),\pi_{\sigma}^{\beta}).$

##  5 Generalized Appell System

In this section we introduce the generalized Appell system associated with the
fPm $\pi_{\sigma}^{\beta}$. First we consider the analytic continuation of the
characteristic functional $C_{\pi_{\sigma}^{\beta}}$ to
$\mathcal{D}_{\mathbb{C}}:=\mathcal{D}\oplus\mathrm{i}\mathcal{D}$. By
definition, an element $\varphi\in\mathcal{D}_{\mathbb{C}}$ decomposes into
$\varphi=\varphi_{1}+\mathrm{i}\varphi_{2}$,
$\varphi_{1},\varphi_{2}\in\mathcal{D}$. Hence, computing
$C_{\pi_{\sigma}^{\beta}}(-\mathrm{i}\varphi)$, $\varphi\in\mathcal{D}$,
yields the Laplace transform of the measure $\pi_{\sigma}^{\beta}$, that is,

$l_{\pi_{\sigma}^{\beta}}(\varphi):=C_{\pi_{\sigma}^{\beta}}(-\mathrm{i}\varphi)=E_{\beta}\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\varphi(x)}-1)\,\mathrm{d}\sigma(x)\right).$

In particular, choosing $\beta=1$ we obtain the Laplace transform of the
classical Poisson measure $\pi_{\sigma}:=\pi_{\sigma}^{1}$ with intensity
$\sigma$ on the configuration space $\Gamma$. For more details, we refer to
[12, 20, 17, 18, 3] and reference therein.

The following two properties are satisfied by the fPm $\pi_{\sigma}^{\beta}$,
$0<\beta\leq 1$.

(A1)

The measure $\pi_{\sigma}^{\beta}$ has an analytic Laplace transform in a
neighborhood of zero, that is, the mapping

$\mathcal{D_{\mathbb{C}}}\ni\varphi\mapsto
l_{\pi_{\sigma}^{\beta}}(\varphi)=\int_{\mathcal{D}^{\prime}}\mathrm{e}^{\langle
w,\varphi\rangle}\,\mathrm{d}\pi_{\sigma}^{\beta}(w)=E_{\beta}\left(\int_{\mathbb{R}^{d}}(\mathrm{e}^{\varphi(x)}-1)\,\mathrm{d}\sigma(x)\right)\in\mathbb{C}$

is holomorphic in a neighborhood $\mathcal{U}\subset\mathcal{D}_{\mathbb{C}}$
of zero.

(A2)

For any nonempty open subset $\mathcal{U}\subset\mathcal{D}^{\prime}$ it
should hold that $\pi_{\sigma}^{\beta}(\mathcal{U})>0$.

The assumption (A1) guarantees the existence of the moments of all order of
the measure $\pi_{\sigma}^{\beta}$ while (A2) guarantees the embedding of the
test function space on $L^{2}(\pi_{\sigma}^{\beta})$, see e.g., Section 3 in
[19]. In addition, the Laplace transform $l_{\pi_{\sigma}^{\beta}}(\varphi)$
of the measure $\pi_{\sigma}^{\beta}$ has the decomposition in terms of the
moment kernels $M_{n}^{\sigma,\beta}$ (by the kernel theorem) given by

$l_{\pi_{\sigma}^{\beta}}(\varphi)=\sum_{n=0}^{\infty}\frac{1}{n!}\langle
M_{n}^{\sigma,\beta},\varphi^{\otimes
n}\rangle,\qquad\varphi\in\mathcal{D}_{\mathbb{C}},\,M_{n}^{\sigma,\beta}\in\big{(}\mathcal{D}_{\mathbb{C}}^{{}^{\hat{\otimes}n}}\big{)}^{\prime}.$
(5.1)

###  5.1 Generalized Appell Polynomials

In this subsection we follow [23] to introduce the system of Appell
polynomials associated with the fPm $\pi_{\sigma}^{\beta}$. Let us consider
the triple (4.2) such that

$\mathcal{D}\subset\mathcal{N}\subset
L^{2}(\sigma)\subset\mathcal{N}^{\prime}\subset\mathcal{D}^{\prime}$ (5.2)

as described in Section 2. Also, the chain (5.2) holds for the tensor product
of these spaces.

Then we introduce the normalized exponential
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;z)$ by

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;w)=\frac{\mathrm{e}^{\langle
w,\varphi\rangle}}{l_{\pi_{\sigma}^{\beta}}(\varphi)},\quad
w\in\mathcal{D}^{\prime}_{\mathbb{C}},\,\varphi\in\mathcal{D}_{\mathbb{C}}.$
(5.3)

Since $l_{\pi_{\sigma}^{\beta}}(0)=1$ and $l_{\pi_{\sigma}^{\beta}}$ is
holomorphic, there exists a neighborhood
$\mathcal{U}_{0}\subset\mathcal{D}_{\mathbb{C}}$ of zero, such that
$l_{\pi_{\sigma}^{\beta}}(\varphi)\neq 0$ for all $\varphi\in\mathcal{U}_{0}$.
For $\varphi\in\mathcal{U}_{0}$, the normalized exponential
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;z)$ can be expanded in a power
series and then we use the polarization identity in order to apply the kernel
theorem to obtain

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;w)=\sum_{n=0}^{\infty}\frac{1}{n!}\langle
P_{n}^{\sigma,\beta}(w),\varphi^{\otimes n}\rangle,\quad
w\in\mathcal{D}^{\prime}_{\mathbb{C}},\;\varphi\in\mathcal{U}_{0},$ (5.4)

for suitable
$P_{n}^{\sigma,\beta}(w)\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$.
The family

$\mathbb{P}^{\sigma,\beta}=\left\\{\langle
P_{n}^{\sigma,\beta}(\cdot),\varphi^{(n)}\rangle\mid\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},n\in\mathbb{N}_{0}\right\\}$
(5.5)

is called the _Appell system_ associated to the fPm _ $\pi_{\sigma}^{\beta}$_.
Let us now consider the transformation
$\alpha:\mathcal{D}_{\mathbb{C}}\longrightarrow\mathcal{D}_{\mathbb{C}}$
defined on a neighborhood
$\mathcal{U}_{\alpha}\subset\mathcal{D}_{\mathbb{C}}$ of zero, by

$\alpha(\varphi)(x)=\log(1+\varphi(x)),\quad\varphi\in\mathcal{U}_{\alpha},\>x\in\mathbb{R}^{d}.$

Note that for $\varphi=0\in\mathcal{D_{\mathbb{C}}}$, we have
$\alpha(\varphi)=0$. Also, $\mathcal{U}_{\alpha}$ is chosen in such a way that
$\alpha$ is invertible and holomorphic on $\mathcal{U}_{\alpha}$. Then
$\alpha$ can be expanded as

$\alpha(\varphi)=\sum_{n=1}^{\infty}\frac{1}{n!}\widehat{\mathrm{d}^{n}\alpha(0)}(\varphi)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\varphi^{n}}{n},$
(5.6)

where

$\widehat{\mathrm{d}^{n}\alpha(0)}(\varphi)=\frac{\partial^{n}}{\mathrm{\partial}t_{1}\dots\mathrm{\partial}t_{n}}\alpha(t_{1}\varphi+\dots+t_{n}\varphi)|_{t_{1}=\dots=t_{n}=0}$

for all $n\in\mathbb{N}$. For the inverse function $g_{\alpha}$ of $\alpha$,
we have

$(g_{\alpha}\varphi)(x)=\mathrm{e}^{\varphi(x)}-1,\qquad\varphi\in\mathcal{V}_{\alpha}\subset\mathcal{D}_{\mathbb{C}},\>x\in\mathbb{R}^{d}$

for some neighborhood $\mathcal{V}_{\alpha}$ of zero in
$\mathcal{D}_{\mathbb{C}}$. A similar procedure as before yields the
decomposition

$g_{\alpha}(\varphi)=\sum_{n=1}^{\infty}\frac{1}{n!}\widehat{\mathrm{d}^{n}g_{\alpha}(0)}(\varphi)=\sum_{n=1}^{\infty}\frac{\varphi^{n}}{n!}.$
(5.7)

Now using the function $\alpha$, we introduce the modified normalized
exponential $\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);x)$ as

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w):=\frac{\exp(\langle
w,\alpha(\varphi)\rangle)}{l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))}=\frac{\exp(\langle
w,\log(1+\varphi)\rangle)}{E_{\beta}\left(\int_{\mathbb{R}^{d}}\varphi(x)\mathrm{\,d}\sigma(x)\right)}=\frac{\exp(\langle
w,\log(1+\varphi)\rangle)}{E_{\beta}\left(\langle\varphi\rangle_{\sigma}\right)}$
(5.8)

for $\varphi\in\mathcal{U}^{\prime}_{\alpha}\subset\mathcal{U_{\alpha}}$,
$w\in\mathcal{D}_{\mathbb{C}}^{{}^{\prime}}$. Since $l_{\pi_{\sigma}^{\beta}}$
is holomorphic on a neighborhood of zero, for each fixed
$w\in\mathcal{D}^{\prime}_{\mathbb{C}}$,
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\cdot);w)$ is a holomorphic function
on some neighborhood
$\mathcal{U}^{\prime}_{\alpha}\subset\mathcal{U}_{\alpha}$ of zero. Then we
have the map
$\mathcal{D}_{\mathbb{C}}\ni\varphi\mapsto\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w)$
which admits a power series

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w)=\sum_{n=0}^{\infty}\frac{1}{n!}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{\otimes
n}\rangle,\quad\varphi\in\mathcal{U}^{\prime}_{\alpha},\;w\in\mathcal{D}_{\mathbb{C}}^{{}^{\prime}},$
(5.9)

where the kernels
$C_{n}^{\sigma,\beta}:\mathcal{D}^{\prime}_{\mathbb{C}}\rightarrow(\mathcal{D}_{\mathbb{C}}^{{}^{\hat{\otimes}n}})^{\prime}$,
$n\in\mathbb{N}$, $C_{0}^{\sigma,\beta}=1$. By Equation (5.9), it follows that
for any $\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$,
$n\in\mathbb{N}_{0}$, the function

$\mathcal{D}^{\prime}_{\mathbb{C}}\ni w\mapsto\langle
C_{n}^{\sigma,\beta}(w),\varphi^{(n)}\rangle$

is a polynomial of order $n$ on $\mathcal{D}^{\prime}_{\mathbb{C}}$.

######  Definition 5.1.

The family

$\mathbb{P}^{\sigma,\beta,\alpha}=\left\\{\langle
C_{n}^{\sigma,\beta}(\cdot),\varphi^{(n)}\rangle\mid\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},n\in\mathbb{N}_{0}\right\\}$

is called the _generalized Appell system_ associated to the fPm _
$\pi_{\sigma}^{\beta}$ _or the _ $\mathbb{P}^{\sigma,\beta,\alpha}$-system._

In the following proposition we collect some properties of the kernels
$C_{n}^{\sigma,\beta}(\cdot)$ which appeared in [21] but specific to the
measure _$\pi_{\sigma}^{\beta}$_.

######  Proposition 5.2.

For $z,w\in\mathcal{D}_{\mathbb{C}}^{\prime}$, $n\in\mathbb{N}_{0}$, the
following properties hold

(P1)

$C_{n}^{\sigma,\beta}(w)=\sum_{m=0}^{n}\mathbf{s}(n,m)^{*}P_{m}^{\sigma,\beta}(w)$,
where $\mathbf{s}(n,m)$ is the Stirling operator of the first kind defined in
(A.6) in Appendix A.2.

(P2)

$w^{\otimes
n}=\sum_{k=0}^{n}\sum_{m=0}^{k}\binom{n}{k}\mathbf{S}(k,m)^{*}C_{m}^{\sigma,\beta}(w)\hat{\otimes}M_{n-k}^{\sigma,\beta}$,
where $\mathbf{S}(n,m)$ is the Stirling operator of the second kind defined in
(A.7) in Appendix A.2 and
$M_{n}^{\sigma,\beta}\in(\mathcal{D}_{\mathbb{C}}^{{}^{\hat{\otimes}n}})^{\prime}$
are the moment kernels of $\pi_{\sigma}^{\beta}$ given in (5.1).

(P3)

_$C_{n}^{\sigma,\beta}(z+w)=\sum_{k+l+m=n}\frac{n!}{k!l!m!}C_{k}^{\sigma,\beta}(z)\hat{\otimes}C_{l}^{\sigma,\beta}(w)\hat{\otimes}M_{m}^{\sigma,\beta,\alpha}$
, _where
$M_{m}^{\sigma,\beta,\alpha}\in(\mathcal{D}_{\mathbb{C}}^{{}^{\hat{\otimes}n}})^{\prime}$
is determined by

$l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))=\sum_{m=0}^{\infty}\frac{1}{m!}\langle
M_{m}^{\sigma,\beta,\alpha},\varphi^{\otimes
m}\rangle,\quad\varphi\in\mathcal{D}_{\mathbb{C}}.$ (5.10)

(P4)

_$C_{n}^{\sigma,\beta}(z+w)=\sum_{k=0}^{n}\binom{n}{k}C_{k}^{\sigma,\beta}(z)\hat{\otimes}(w)_{n-k}$
, _where $(w)_{n}$ is the falling factorial on
$\mathcal{D}^{\prime}_{\mathbb{C}}$ determined by (A.1).

(P5)

$C_{n}^{\sigma,\beta}(w)=\sum_{k=0}^{n}\binom{n}{k}\sum_{m=0}^{n-k}C_{k}^{\sigma,\beta}(0)\hat{\otimes}\big{(}\mathbf{s}(n-k,m)^{*}w^{\otimes
m}\big{)}$.

(P6)

_$\mathbb{E}_{\pi_{\sigma}^{\beta}}(\langle
C_{n}^{\sigma,\beta}(\cdot),\varphi^{(n)}\rangle)=\delta_{n,0}$ , _where
$\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$, $\delta_{n,k}$ is
the Kronecker delta function and $\mathbb{E}_{\pi_{\sigma}^{\beta}}(\cdot)$ is
the expectation with respect to the measure _$\pi_{\sigma}^{\beta}$._

(P7)

For all $p^{\prime}>p$ such that the embedding
$\mathcal{H}_{p^{\prime}}\hookrightarrow\mathcal{H}_{p}$ is a Hilbert-Schmidt
operator and for all $\varepsilon>0$ there exist $C_{\varepsilon}>0$ such that

$|C_{n}^{\sigma,\beta}(w)|_{-p^{\prime}}\leq
C_{\varepsilon}n!\varepsilon^{-n}\exp(\varepsilon|w|_{-p}),\quad
w\in\mathcal{H}_{-p^{\prime},\mathbb{C}},\,n\in\mathbb{N}_{0}.$

###### Proof.

(P1) In view of Equation (5.4), we have

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w)=\frac{\exp(\langle
w,\alpha(\varphi)\rangle)}{l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))}=\sum_{m=0}^{\infty}\frac{1}{m!}\langle
P_{m}^{\sigma,\beta}(w),\alpha(\varphi)^{\otimes m}\rangle.$ (5.11)

Using Equation (A.11), we obtain

$\displaystyle\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w)$
$\displaystyle=\sum_{m=0}^{\infty}\Bigg{\langle}P_{m}^{\sigma,\beta}(w),\sum_{n=m}^{\infty}\frac{1}{n!}\mathbf{s}(n,m)\varphi^{\otimes
n}\Bigg{\rangle}$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{n=m}^{\infty}\frac{1}{n!}\Bigg{\langle}\mathbf{s}(n,m)^{*}P_{m}^{\sigma,\beta}(w),\varphi^{\otimes
n}\Bigg{\rangle}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\Bigg{\langle}\sum_{m=0}^{n}\mathbf{s}(n,m)^{*}P_{m}^{\sigma,\beta}(w),\varphi^{\otimes
n}\Bigg{\rangle}.$

On the other hand, using the equality (5.9) and comparing both series for
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi),w)$ gives

$C_{n}^{\sigma,\beta}(w)=\sum_{m=1}^{n}\mathbf{s}(n,m)^{*}P_{m}^{\sigma,\beta}(w).$

(P2) Similar as in the proof of (P1), we use Equation (5.9) and the fact that
$g_{\alpha}$ is the inverse of $\alpha$ to obtain

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;w)=\sum_{m=0}^{\infty}\frac{1}{m!}\langle
C_{m}^{\sigma,\beta}(w),g_{\alpha}(\varphi)^{\otimes m}\rangle.$ (5.12)

Using Equation (A.10) we replace $g_{\alpha}(\varphi)^{\otimes m}$ in the
above Equation (5.12) and making some standard manipulations yields

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;z)=\sum_{m=0}^{\infty}\Bigg{\langle}C_{m}^{\sigma,\beta}(w),\sum_{n=m}^{\infty}\frac{1}{n!}\mathbf{S}(n,m)\varphi^{\otimes
n}\Bigg{\rangle}=\sum_{n=0}^{\infty}\frac{1}{n!}\Bigg{\langle}\sum_{m=0}^{n}\mathbf{S}(n,m)^{*}C_{m}^{\sigma,\beta}(w),\varphi^{\otimes
n}\Bigg{\rangle}.$

On the other hand, comparing the above series for
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi,w)$ and the Equation (5.4), we
obtain

$P_{n}^{\sigma,\beta}(w)=\sum_{m=0}^{n}\mathbf{S}(n,m)^{*}C_{m}^{\sigma,\beta}(w).$
(5.13)

By Equation (5.3), we have the equality

$\mathrm{e}^{\langle
w,\varphi\rangle}=\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;w)l_{\pi_{\sigma}^{\beta}}(\varphi).$
(5.14)

Now using the equations (5.4) and (5.1), we obtain the equation

$\sum_{n=0}^{\infty}\frac{1}{n!}\langle w^{\otimes n},\varphi^{\otimes
n}\rangle=\sum_{n=0}^{\infty}\frac{1}{n!}\Bigg{\langle}\sum_{k=0}^{n}\binom{n}{k}P_{n}^{\sigma,\beta}(w)\hat{\otimes}M_{n-k}^{\sigma,\beta},\varphi^{\otimes
n}\Bigg{\rangle}$

which implies that

$w^{\otimes
n}=\sum_{k=0}^{n}\binom{n}{k}P_{n}^{\sigma,\beta}(w)\hat{\otimes}M_{n-k}^{\sigma,\beta}.$
(5.15)

The claim follows by applying Equation (5.13) to Equation (5.15).

(P3) By definition of the modified normalized exponential, we have

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);z+w)=\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);z)\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w)l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi)).$

For $l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))$, we use the decomposition
(5.10) such that the above equation yields

$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\langle
C_{n}^{\sigma,\beta}(z+w),\varphi^{\otimes n}\rangle$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\langle
C_{k}^{\sigma,\beta}(z),\varphi^{\otimes
k}\rangle\sum_{l=0}^{\infty}\frac{1}{l!}\langle
C_{l}^{\sigma,\beta}(w),\varphi^{\otimes
l}\rangle\sum_{m=0}^{\infty}\frac{1}{m!}\langle
M_{m}^{\sigma,\beta,\alpha},\varphi^{\otimes m}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\Bigg{\langle}\sum_{k+l+m=n}\frac{n!}{k!l!m!}C_{k}^{\sigma,\beta}(z)\hat{\otimes}C_{l}^{\sigma,\beta}(w)\hat{\otimes}M_{m}^{\sigma,\beta,\alpha},\varphi^{\otimes
n}\Bigg{\rangle}.$

Thus, the result follows by comparing the coefficients in both sides of the
equation.

(P4) Again, by definition of the modified normalized exponential, we have

$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);z+w)=\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);z)\exp(\langle
w,\alpha(\varphi)\rangle).$

By Equations (A.1) and (5.9), we have

$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\langle
C_{n}^{\sigma,\beta}(z+w),\varphi^{\otimes n}\rangle$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\langle
C_{k}^{\sigma,\beta}(z),\varphi^{\otimes
k}\rangle\sum_{m=0}^{\infty}\frac{1}{m!}\langle(w)_{m},\varphi^{\otimes
m}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\Bigg{\langle}\sum_{k=0}^{n}\binom{n}{k}C_{k}^{\sigma,\beta}(z)\hat{\otimes}(w)_{n-k},\varphi^{\otimes
n}\Bigg{\rangle}.$

Thus the assertion follows immediately by comparing the coefficients in both
sides of the equation.

(P5) The result follows from (P4) at $z=0$ and (A.9).

(P6) Note that for $\varphi\in\mathcal{D}_{\mathbb{C}}$, we have

$\sum_{n=0}^{\infty}\frac{1}{n!}\mathbb{E_{\pi_{\sigma}^{\beta}}}(\langle
C_{n}^{\sigma,\beta}(w),\varphi^{\otimes
n}\rangle)=\mathbb{E}_{\pi_{\sigma}^{\beta}}(\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);w))=\frac{\mathbb{E}_{\pi_{\sigma}^{\beta}}(\exp(\langle\cdot,\alpha(\varphi)\rangle))}{l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))}=1.$

By polarization identity and comparison of coefficients, we obtain the result.

(P7) Let $\varepsilon>0$ be given. Then let
$C_{\varepsilon},\sigma_{\varepsilon}>0$ be chosen in such a way that
$|\alpha(\varphi)|_{p}\leq\varepsilon$ and $C_{\varepsilon}\geq
1/|l_{\pi_{\beta}^{\sigma}}(\alpha(\varphi))|$ for
$|\varphi|_{p}=\sigma_{\varepsilon}.$ By definition of
$C_{n}^{\sigma,\beta}(w)$ and the Cauchy formula, we have

$\displaystyle|\langle C_{n}^{\sigma,\beta}(w),\varphi^{\otimes n}\rangle|$
$\displaystyle=|\widehat{\mathrm{d}^{n}\mathrm{e}_{\pi_{\sigma}^{\beta}}(0;w)}(\varphi)|$
$\displaystyle\leq
n!\frac{1}{\sigma_{\varepsilon}^{n}}\left(\sup_{|\varphi|_{p}=\sigma_{\varepsilon}}\frac{\exp\left(|\alpha(\varphi)|_{p}|w|_{-p}\right)}{|l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))|}\right)|\varphi|_{p}^{n}$
$\displaystyle\leq
n!\frac{1}{\sigma_{\varepsilon}^{n}}\left(\sup_{|\varphi|_{p}=\sigma_{\varepsilon}}\frac{1}{|l_{\pi_{\sigma}^{\beta}}(\alpha(\varphi))|}\right)\exp\left(\varepsilon|w|_{-p}\right)|\varphi|_{p}^{n}$
$\displaystyle\leq
C_{\varepsilon}n!\sigma_{\varepsilon}^{-n}\exp\left(\varepsilon|w|_{-p}\right)|\varphi|_{p}^{n}.$

Let $p^{\prime}>p$ be such that $i_{p^{\prime},p}$ is a Hilbert-Schmidt
operator. Then by the kernel theorem, we have

$|C_{n}^{\sigma,\beta}(w)|_{-p^{\prime}}\leq
n!C_{\varepsilon}\exp\left(\varepsilon|w|_{-p}\right)\left(\frac{1}{\sigma_{\varepsilon}}\|i_{p^{\prime},p}\|_{HS}\right)^{n},\quad
w\in\mathcal{H}_{-p,\mathbb{C}}.$

For sufficiently small $\varepsilon$, we fix
$\sigma_{\varepsilon}=\varepsilon\|i_{p^{\prime},p}\|_{HS}$ so that

$|C_{n}^{\sigma,\beta}(w)|_{-p^{\prime}}\leq
n!C_{\varepsilon}\varepsilon^{-n}\exp\left(\varepsilon|w|_{-p}\right).$

This concludes the proof. ∎

###  5.2 Generalized Dual Appell System

In what follows, we use again the approach in [23] of non-Gausian analysis to
introduce the generalized dual Appell system associated with the fPm
$\pi_{\sigma}^{\beta}$.

######  Definition 5.3.

The _space of smooth polynomials_ $\mathcal{P}(\mathcal{D}^{\prime})$ on
$\mathcal{D}^{\prime}$ is the space consisting of finite linear combinations
of monomial functions, that is,

$\mathcal{P}(\mathcal{D}^{\prime}):=\left\\{\varphi(w)=\sum_{n=0}^{N(\varphi)}\langle
w^{\otimes
n},\varphi^{(n)}\rangle\,\bigg{|}\,\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},\;w\in\mathcal{D}^{\prime},\;N(\varphi)\in\mathbb{N}_{0}\right\\}.$

The space $\mathcal{P}(\mathcal{D}^{\prime})$ shall be equipped with the
natural topology, such that the mapping

$I:\mathcal{P}(\mathcal{D}^{\prime})\longrightarrow\bigoplus_{n=0}^{\infty}\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$

defined for any $\varphi(\cdot)=\sum_{n=0}^{\infty}\langle\cdot^{\otimes
n},\varphi^{(n)}\rangle\in\mathcal{P}(\mathcal{D}^{\prime})$ by

$I\varphi=\vec{\varphi}=(\varphi^{(0)},\varphi^{(1)},\dots,\varphi^{(n)},\dots)$

becomes a topological isomorphism from $\mathcal{P}(\mathcal{D}^{\prime})$ to
the topological direct sum of symmetric tensor powers
$\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$ (see [6, 35]). Note that only a
finite number of $\varphi^{(n)}$ is non-zero. With respect to this topology, a
sequence $(\varphi_{m})_{m\in\mathbb{N}}$ of smooth continuous polynomials,
that is, $\varphi_{m}(w)=\sum_{n=0}^{N(\varphi_{m})}\langle w^{\otimes
n},\varphi_{m}^{(n)}\rangle$ converges to
$\varphi(w)=\sum_{n=0}^{N(\varphi)}\langle w^{\otimes
n},\varphi^{(n)}\rangle\in\mathcal{P}(\mathcal{D}^{\prime})$ if, and only if,
the sequence $(N(\varphi_{m}))_{m\in\mathbb{N}}$ is bounded and
$(\varphi_{m}^{(n)})_{m\in\mathbb{N}}$ converges to $\varphi^{(n)}$ in
$\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$ for all $n\in\mathbb{N}_{0}$.

Using Proposition 5.2-(P2), the space of smooth polynomials
$\mathcal{P}(\mathcal{D}^{\prime})$ can also be expressed in terms of the
generalized Appell polynomials associated with the measure
$\pi_{\sigma}^{\beta}$ given by

$\mathcal{P}(\mathcal{D}^{\prime}):=\left\\{\varphi(w)=\sum_{n=0}^{N(\varphi)}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{(n)}\rangle\,\bigg{|}\,\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},w\in\mathcal{D}^{\prime},N(\varphi)\in\mathbb{N}_{0}\right\\}.$

We denote by
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$ the dual
space of $\mathcal{P}(\mathcal{D}^{\prime})$ with respect to
$L^{2}(\pi_{\sigma}^{\beta}):=L^{2}(\mathcal{D}^{\prime},\mathcal{C}_{\sigma}(\mathcal{D}^{\prime}),\pi_{\sigma}^{\beta};\mathbb{C})$
and obtain the triple

$\mathcal{P}(\mathcal{D}^{\prime})\subset
L^{2}(\pi_{\sigma}^{\beta})\subset\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime}).$
(5.16)

The (bilinear) dual pairing
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$ between
$\mathcal{P}(\mathcal{D}^{\prime})$ and
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$ is then
related to the (sesquilinear) inner product on $L^{2}(\pi_{\sigma}^{\beta})$
by

$\langle\\!\langle
F,\varphi\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}=(\\!(F,\bar{\varphi})\\!)_{L^{2}(\pi_{\sigma}^{\beta})},\quad
F\in
L^{2}(\pi_{\sigma}^{\beta}),\;\varphi\in\mathcal{P}(\mathcal{D}^{\prime}),$

where $\bar{\varphi}$ denotes the complex conjugate function of $\varphi$.
Further we introduce the constant function $\bm{1}\in
L^{2}(\pi_{\sigma}^{\beta})\subset\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$
such that $\bm{1}(w)=1$ for all $w\in\mathcal{D}^{\prime}$, so for any
polynomial $\varphi\in\mathcal{P}(\mathcal{D}^{\prime})$,

$\mathbb{E_{\pi_{\sigma}^{\beta}}}(\varphi):=\int_{\mathcal{D}^{\prime}}\varphi(w)\,\mathrm{d}\pi_{\sigma}^{\beta}(w)=\langle\\!\langle\bm{1},\varphi\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}.$

Now, we will describe the distributions in
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$ in a
similar way as the smooth polynomials $\mathcal{P}(\mathcal{D}^{\prime})$,
that is, for any
$\Phi\in\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$, we
find elements
$\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$ and
operators $Q_{n}^{\sigma,\beta,\alpha}$ on
$(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$, such that

$\Phi=\sum_{n=0}^{\infty}Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})\in\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime}).$

To this end, we define first a differential operator $D(\Phi^{(n)})$ depending
on $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$ such
that when applied to the monomials $\langle w^{\otimes
m},\varphi^{(m)}\rangle,$
$\varphi^{(m)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}m}$,
$m\in\mathbb{N}_{0}$, gives

$D(\Phi^{(n)})\langle w^{\otimes
m},\varphi^{(m)}\rangle:=\begin{cases}\frac{m!}{(m-n)!}\langle
w^{\otimes(m-n)}\hat{\otimes}\Phi^{(n)},\varphi^{(m)}\rangle,&\mathrm{for}\>m\geq
n\\\ 0,&\mathrm{otherwise}\end{cases}$

and extend by linearity from the monomials to elements in
$\mathcal{P}(\mathcal{D}^{\prime})$. If we consider the space of Schwartz test
function $\mathcal{S}(\mathbb{R})$ instead of using the space $\mathcal{D}$
with the triple

$\mathcal{S}(\mathbb{R})\subset
L^{2}(\mathbb{R},\mathrm{d}x)\subset\mathcal{S}^{\prime}(\mathbb{R}),$

then for $n=1$ and
$\Phi^{(1)}=\delta_{t}\in\mathcal{S}_{\mathbb{C}}^{\prime}(\mathbb{R})$, the
differential operator $D(\delta_{t})$ coincides with the Hida derivative, see
[16]. Note that $D(\Phi^{(n)})$ is a continuous linear operator from
$\mathcal{P}(\mathcal{D}^{\prime})$ to $\mathcal{P}(\mathcal{D}^{\prime})$
(see [23, Lemma 4.13]) and this enables us to define the dual operator

$D(\Phi^{(n)})^{*}:\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})\longrightarrow\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime}).$

Below we need the evaluation of the operator $D(\Phi^{(n)})$ on the monomials
$\langle P_{m}^{\sigma,\beta}(w),\varphi^{(m)}\rangle$, $m\in\mathbb{N}_{0}$
in (5.5). We state this result in the next proposition and the proof can be
found in [23, Lemma 4.14].

######  Proposition 5.4.

For $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$ and
$\varphi^{(m)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}m}$ we have

$D(\Phi^{(n)})\langle
P_{m}^{\sigma,\beta}(w),\varphi^{(m)}\rangle=\begin{cases}{\displaystyle\frac{m!}{(m-n)!}\langle
P_{m-n}^{\sigma,\beta}(w)\hat{\otimes}\Phi^{(n)},\varphi^{(m)}\rangle,}&\mathit{for}\;m\geq
n\\\ 0,&\mathit{for}\;m<n.\end{cases}$

Now, we set $Q_{n}^{\sigma,\beta}(\Phi^{(n)}):=D(\Phi^{(n)})^{*}\mathbf{1}$
for $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$ and
denote the so-called $\mathbb{Q}^{\sigma,\beta}$-system in
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$ by

$\mathbb{Q}^{\sigma,\beta}:=\left\\{Q_{n}^{\sigma,\beta}(\Phi^{(n)})\mid\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime},\,n\in\mathbb{N}_{0}\right\\}.$

The pair
$\mathbb{A}^{\sigma,\beta}=(\mathbb{P}^{\sigma,\beta},\mathbb{Q}^{\sigma,\beta})$
is called the _Appell system generated by the measure_ $\pi_{\sigma}^{\beta}$.
This system satisfies the biorthogonal property, see [23], given in the
following theorem.

######  Theorem 5.5.

For $\Phi^{(m)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}m})^{\prime}$ and
$\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$ we have

$\langle\\!\langle Q_{n}^{\sigma,\beta}(\Phi^{(m)}),\langle
P_{n}^{\sigma,\beta},\varphi^{(n)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}=\delta_{m,n}n!\langle\Phi^{(n)},\varphi^{(n)}\rangle,\quad
n,m\in\mathbb{N}_{0}.$ (5.17)

However, our aim is to construct the generalized dual Appell system
$\mathbb{Q}^{\sigma,\beta,\alpha}$ such that
$\mathbb{P}^{\sigma,\beta,\alpha}$ and $\mathbb{Q}^{\sigma,\beta,\alpha}$ are
biorthogonal. The reason to do this is because when $\beta=1$ we obtain only
one system of orthogonal polynomials, so-called the Charlier polynomials, see
[18].

First, recall the function $g_{\alpha}(\varphi)$,
$\varphi\in\mathcal{D}_{\mathbb{C}}$ from Section 5.1. By Equation (A.10), we
have

$g_{\alpha}(\varphi)^{\otimes
n}=\sum_{k=n}^{\infty}\frac{n!}{k!}\mathbf{S}(n,k)\varphi^{\otimes k}.$

Then for any
$\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$, we have

$\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes
n}\rangle=\sum_{k=n}^{\infty}\frac{n!}{k!}\langle\Phi^{(n)},\mathbf{S}(k,n)\varphi^{\otimes
k}\rangle=\sum_{k=n}^{\infty}\frac{n!}{k!}\langle\mathbf{S}(k,n)^{*}\Phi^{(n)},\varphi^{\otimes
k}\rangle,$ (5.18)

where
$\mathbf{S}(k,n)^{*}\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})^{\prime}$.
Now, we define the operator $G(\Phi^{(n)})$ by

$G(\Phi^{(n)}):\mathcal{P}(\mathcal{D}^{\prime})\longrightarrow\mathcal{P}(\mathcal{D}^{\prime}),\;\varphi\mapsto
G(\Phi^{(n)})\varphi:=\sum_{k=n}^{\infty}\frac{n!}{k!}D(\mathbf{S}(k,n)^{*}\Phi^{(n)})\varphi.$

Since $D(\mathbf{S}(k,n)^{*}\Phi^{(n)})$ is continuous for any
$\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$, it is
easy to see that $G(\Phi^{(n)})$ is also continuous and so its adjoint
$G(\Phi^{(n)})^{*}:\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})\longrightarrow\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$
exists.

######  Definition 5.6.

For any $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$,
we define the _generalized function_
$Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})\in\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$,
$n\in\mathbb{N}_{0}$, by

$Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}):=G(\Phi^{(n)})^{*}\bm{1}.$ (5.19)

The family

$\mathbb{Q}^{\sigma,\beta,\alpha}:=\left\\{Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})\mid\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime},\,n\in\mathbb{N}_{0}\right\\}$

is said to be the _generalized dual Appell system_
$\mathbb{Q}^{\sigma,\beta,\alpha}$ associated with $\pi_{\sigma}^{\beta}$ or
the $\mathbb{Q}^{\sigma,\beta,\alpha}$-system and the pair
$\mathbb{A}^{\sigma,\beta,\alpha}:=(\mathbb{P}^{\sigma,\beta,\alpha},\mathbb{Q}^{\sigma,\beta,\alpha})$
is called the _generalized Appell system_ generated by the measure
$\pi_{\sigma}^{\beta}$.

The following theorem states the biorthogonal property of the generalized
Appell system $\mathbb{A}^{\sigma,\beta,\alpha}$.

######  Theorem 5.7.

For $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$ and
$\varphi^{(m)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}m}$ we have

$\langle\\!\langle Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}),\langle
C_{m}^{\sigma,\beta},\varphi^{(m)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}=\delta_{n,m}n!\langle\Phi^{(n)},\varphi^{(n)}\rangle,\quad
n,m\in\mathbb{N}_{0}.$

###### Proof.

By Proposition 5.2-(P1), we have

$\langle
C_{m}^{\sigma,\beta},\varphi^{(m)}\rangle=\bigg{\langle}\sum_{i=0}^{m}\mathbf{s}(m,i)^{*}P_{i}^{\sigma,\beta},\varphi^{(m)}\bigg{\rangle}=\sum_{i=0}^{m}\langle
P_{i}^{\sigma,\beta},\mathbf{s}(m,i)\varphi^{(m)}\rangle.$

Then it follows from Proposition 5.4 (noted below with $\star$) and
Proposition 11-(P4) in [23] ($\star$$\star$) that

$\displaystyle\langle\\!\langle
Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}),\langle
C_{m}^{\sigma,\beta},\varphi^{(m)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\langle\\!\langle\mathbf{1},G(\Phi^{(n)})\langle
C_{m}^{\sigma,\beta},\varphi^{(m)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\sum_{i=0}^{m}\langle\\!\langle\mathbf{1},G(\Phi^{(n)})\langle
P_{i}^{\sigma,\beta},\mathbf{s}(m,i)\varphi^{(m)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle\overset{\star}{=}\sum_{i=k}^{m}\sum_{k=n}^{\infty}\frac{n!}{k!}\frac{i!}{(i-k)!}\langle\\!\langle\mathbf{1},\langle
P_{i-k}^{\sigma,\beta}\hat{\otimes}\mathbf{S}(k,n)^{*}\Phi^{(n)},\mathbf{s}(m,i)\varphi^{(m)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\sum_{i=k}^{m}\sum_{k=n}^{\infty}\frac{n!i!}{k!(i-k)!}\mathbb{E}_{\pi_{\sigma}^{\beta}}(\langle
P_{i-k}^{\sigma,\beta}\hat{\otimes}\mathbf{S}(k,n)^{*}\Phi^{(n)},\mathbf{s}(m,i)\varphi^{(m)}\rangle)$
$\displaystyle\overset{\star\star}{=}\sum_{i=k}^{m}\sum_{k=n}^{m}\frac{n!i!}{k!(i-k)!}\delta_{i,k}\langle\mathbf{S}(k,n)^{*}\Phi^{(n)},\mathbf{s}(m,k)\varphi^{(m)}\rangle$
$\displaystyle=\sum_{k=n}^{m}n!\langle\Phi^{(n)},\mathbf{S}(k,n)\mathbf{s}(m,k)\varphi^{(m)}\rangle$
$\displaystyle=\delta_{n,m}n!\langle\Phi^{(n)},\varphi^{(n)}\rangle,$

where the last equality is obtained using Proposition A.9 in Appendix A.2. ∎

######  Remark 5.8.

In Appendix A.3, we provide an alternative proof for the biorthogonal property
of the generalized Appell system $\mathbb{A}^{\sigma,\beta,\alpha}$ using the
$S_{\text{$\pi_{\sigma}^{\beta}$}}$-transform (to be introduced in Section 6)
of the generalized function
$Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})\in\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$.
It is based on the fact that $\exp(\langle z,\varphi\rangle)$ is an
eigenfunction of the generalized function $G(\Phi^{(n)})$.

Using Theorem 5.7, the space
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$ can now be
characterized in a similar way as the space
$\mathcal{P}(\mathcal{D}^{\prime})$. See [21] for the proof of the following
theorem.

######  Theorem 5.9.

For every
$\Phi\in\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$,
there exists a unique sequence $(\Phi^{(n)})_{n\in\mathbb{N}_{0}}$,
$\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$ such that

$\Phi=\sum_{n=0}^{\infty}Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})$

and vice versa, every such series generates a generalized function in
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$.

##  6 Test and Generalized Function Spaces

In this section, we construct the test function space and the generalized
function space associated to the fPm $\pi_{\sigma}^{\beta}$ and study some
properties. Here, we consider a nuclear triple

$\underset{p\in\mathbb{N}}{\mathrm{pr\,lim}}\mathcal{H}_{p}=\mathcal{N}\subset
L^{2}(\sigma)\subset\mathcal{N}^{\prime}=\underset{p\in\mathbb{N}}{\mathrm{ind\,lim}}\mathcal{H}_{-p},$

as described in Section 2 such that

$\mathcal{D}\subset\mathcal{N}\subset
L^{2}(\sigma)\subset\mathcal{N}^{\prime}\subset\mathcal{D}^{\prime}.$

Let $\varphi=\sum_{n=0}^{N}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{(n)}\rangle\in\mathcal{P}(\mathcal{D}^{\prime})$
be given. Then we use the fact that $\mathcal{D}\subset\mathcal{N}$ so that
$\varphi^{(n)}\in\mathcal{N}_{\mathbb{C}}^{\hat{\otimes}n}$. Note that

$\mathcal{N}_{\mathbb{C}}^{\hat{\otimes}n}=\underset{p\in\mathbb{N}}{\mathrm{pr\,lim}}\,\mathcal{H}_{p,\mathbb{C}}^{\hat{\otimes}n}$

and so $\varphi^{(n)}\in\mathcal{H}_{p,\mathbb{C}}^{\hat{\otimes}n}$ for all
$p\in\mathbb{N}$. For each $p,q\in\mathbb{N}$ and $\kappa\in[0,1]$, we
introduce a norm $\|\cdot\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}$ on
$\mathcal{P}(\mathcal{D}^{\prime})$ by

$\|\varphi\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}^{2}:=\sum_{n=0}^{\infty}(n!)^{1+\kappa}2^{nq}|\varphi^{(n)}|_{p}^{2}.$

Let $(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{\kappa}$ be the Hilbert space
obtained by completing the space $\mathcal{P}(\mathcal{D}^{\prime})$ with
respect to the norm $\|\cdot\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}$. The Hilbert
space $(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{\kappa}$ has inner product
given by

$(\\!(\varphi,\psi)\\!)_{q,\pi_{\sigma}^{\beta}}^{\kappa}:=\sum_{n=0}^{\infty}(n!)^{1+\kappa}2^{nq}(\varphi^{(n)},\overline{\psi^{(n)}})_{p},$

and admits the representation

$(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{\kappa}:=\left\\{\varphi=\sum_{n=0}^{\infty}\langle
C_{m}^{\sigma,\beta},\varphi^{(n)}\rangle\in
L^{2}(\pi_{\sigma}^{\beta})\;\bigg{|}\;\|\varphi\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}^{2}=\sum_{n=0}^{\infty}(n!)^{1+\kappa}2^{nq}|\varphi^{(n)}|_{p}^{2}<\infty\right\\}.$

Then the test function space $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$
is defined by

$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}:=\underset{p,q\in\mathbb{N}}{\mathrm{pr\,lim}}(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{\kappa}.$

The test function space $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$ is a
nuclear space which is continuously embedded in $L^{2}(\pi_{\sigma}^{\beta})$.

######  Example 6.1.

The modified normalized exponential given in (5.8) has the norm

$\|\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}^{2}=\sum_{n=0}^{\infty}(n!)^{1+\kappa}2^{nq}\frac{|\varphi|_{p}^{2n}}{(n!)^{2}},\quad\varphi\in\mathcal{N}_{\mathbb{C}}.$

  1. 1.

If $\kappa=0$, we have

$\|\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\|_{p,q,0,\pi_{\sigma}^{\beta}}^{2}=\exp(2^{q}|\varphi|_{p}^{2})<\infty,\qquad\forall\varphi\in\mathcal{N}_{\mathbb{C}}.$

  2. 2.

For $\kappa\in(0,1)$, we use the Hölder inequality with the pair
$(\frac{1}{\kappa},\frac{1}{1-\kappa})$ and obtain

$\displaystyle\|\mathrm{e}_{\pi_{\lambda,\beta}}(\alpha(\varphi);\cdot)\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}^{2}$
$\displaystyle\leq\left(\sum_{n=0}^{\infty}\left(\frac{1}{2^{n\kappa}}\right)^{\frac{1}{\kappa}}\right)^{\kappa}\left(\sum_{n=0}^{\infty}\left(\frac{\left(2^{\kappa}2^{q}|\varphi|_{p}^{2}\right)^{n}}{(n!)^{1-\kappa}}\right)^{\frac{1}{1-\kappa}}\right)^{1-\kappa}$
$\displaystyle=2^{\kappa}\exp\left((1-\kappa)2^{\frac{\kappa+q}{1-\kappa}}|\varphi|_{p}^{\frac{2}{1-\kappa}}\right)<\infty,$

for all $\varphi\in\mathcal{N}_{\mathbb{C}}.$ Thus,
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$,
$\kappa\in[0,1)$.

  3. 3.

For $\kappa=1$, we have

$\|\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\|_{p,q,\kappa,\pi_{\sigma}^{\beta}}^{2}=\sum_{n=0}^{\infty}2^{nq}|\varphi|_{p}^{2n},\quad\varphi\in\mathcal{N}_{\mathbb{C}}.$

Hence, we have
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\notin(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$
if $\varphi\neq 0$, but
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\in(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{1}$
if $2^{q}|\varphi|_{p}^{2}<1$. Moreover, the set

$\\{\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)\mid
2^{q}|\varphi|_{p}^{2}<1,\varphi\in\mathrm{\mathcal{N}_{\mathbb{C}}}\\}$

is total in $(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{1}$.

For $\kappa=1$, we collect below the most important properties of the space
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$, see [21] for the proofs.

######  Theorem 6.2.

$(i)$

$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$ is a nuclear space.

$(ii)$

The topology in $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$ is uniquely defined
by the topology on $\mathcal{N}$, i.e., it does not depend on the choice of
the family of norms $\\{|\cdot|_{p}\\}$, $p\in\mathbb{N}$.

$(iii)$

There exist $p^{\prime},q^{\prime}>0$ such that for all $p\geq p^{\prime}$,
$q\geq q^{\prime}$ the topological embedding
$(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{1}\subset
L^{2}(\pi_{\sigma}^{\beta})$ holds. $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$
is continuously and densely embedded in $L^{2}(\pi_{\sigma}^{\beta})$.

######  Proposition 6.3.

Any test function $\varphi$ in $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$ has
a uniquely defined extension to $\mathcal{N}^{\prime}_{\mathbb{C}}$ as an
element of $\mathcal{E}_{\min}^{1}(\mathcal{N}^{\prime}_{\mathbb{C}})$. For
all $p>p^{\prime}$ such that the embedding
$\mathcal{H}_{p}\hookrightarrow\mathcal{H}_{p^{\prime}}$ is of the Hilbert-
Schmidt class and for all $\varepsilon>0$, $p\in\mathbb{N}$, we obtain the
following bound

$|\varphi(w)|\leq
C\|\varphi\|_{p,q,1,\pi_{\sigma}^{\beta}}\mathrm{e}^{\varepsilon|w|_{-p^{\prime}}},\qquad\varphi\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1},\>w\in\mathcal{H}_{-p,\mathbb{C}},$

where $2^{q}>(\varepsilon\|i_{p^{\prime},p}\|_{HS})^{-2}$ and

$C=C_{\varepsilon}(1-2^{-q}\varepsilon^{-2})^{-1/2}.$

For each $p,q\in\mathbb{N}$ and $\kappa\in[0,1]$, we denote by
$(\mathcal{H}_{-p})_{-q,\pi_{\sigma}^{\beta}}^{-\kappa}$ the Hilbert space
dual of the space $(\mathcal{H}_{p})_{q,\pi_{\sigma}^{\beta}}^{\kappa}$ with
respect to $L^{2}(\pi_{\sigma}^{\beta})$ with the corresponding (Hilbert) norm
$\|\cdot\|_{-p,-q,\kappa,\pi_{\sigma}^{\beta}}$. This space admits the
following representation

$(\mathcal{H}_{-p})_{-q,\pi_{\sigma}^{\beta}}^{-\kappa}:=\left\\{\Phi=\sum_{n=0}^{\infty}Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})\in\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})\,\middle|\,\|\Phi\|_{-p,-q,\kappa,\pi_{\sigma}^{\beta}}^{2}:=\sum_{n=0}^{\infty}(n!)^{1-\kappa}2^{-nq}|\Phi^{(n)}|_{-p}^{2}<\infty\right\\}.$

By the general duality theory, the dual space
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$ of
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$ with respect to
$L^{2}(\pi_{\sigma}^{\beta})$ is then given by

$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}:=\bigcup_{p,q\in\mathbb{N}}(\mathcal{H}_{-p})_{-q,\pi_{\sigma}^{\beta}}^{-\kappa}.$

Since
$\mathcal{P}(\mathcal{D}^{\prime})\subset(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$,
the space $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$ can be viewed as a
subspace of
$\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$ and so we
extend the triple in (5.16) to the chain of spaces

$\mathcal{P}(\mathcal{D}^{\prime})\subset(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}\subset
L^{2}(\pi_{\sigma}^{\beta})\subset(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}\subset\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime}).$

The action of a distribution

$\Phi=\sum_{n=0}^{\infty}Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$

on a test function

$\varphi=\sum_{n=0}^{\infty}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{(n)}\rangle\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$

using the biorthogonal property in Theorem 5.7 is given by

$\langle\\!\langle\Phi,\varphi\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}=\sum_{n=0}^{\infty}n!\langle\Phi^{(n)},\varphi^{(n)}\rangle.$

Now we give two examples of the generalized functions in
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$. For a more generalized case, see
[21].

######  Example 6.4 (Generalized Radon-Nikodym derivative).

We define a generalized function
$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$,
$w\in\mathcal{N}^{\prime}_{\mathbb{C}}$ with the following property

$\langle\\!\langle\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot),\varphi\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}=\int_{\mathcal{N}^{\prime}}\varphi(x-w)\,\mathrm{d}\mu(x),\quad\varphi\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}.$

First, we have to establish the continuity of
$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)$. Let
$w\in\mathcal{H}_{-p,\mathbb{C}}$ be given. Then, if $p\geq p^{\prime}$ is
sufficiently large and $\varepsilon>0$ is small enough, we use Proposition
6.3, that is, there exists $q\in\mathbb{N}$ and $C>0$ such that

$\displaystyle{\displaystyle\left|\int_{\mathcal{N}^{\prime}}\varphi(x-w)\,\mathrm{d}\pi_{\sigma}^{\beta}(x)\right|}$
$\displaystyle\leq{\displaystyle
C\|\varphi\|_{p,q,1,\pi_{\sigma}^{\beta}}\int_{\mathcal{N}^{\prime}}\exp(\varepsilon|x-w|_{-p^{\prime}})\,\mathrm{d}\pi_{\sigma}^{\beta}(x)}$
$\displaystyle{\displaystyle\leq
C\|\varphi\|_{p,q,1,\pi_{\sigma}^{\beta}}\exp(\varepsilon|w|_{-p^{\prime}})\int_{\mathcal{N}^{\prime}}\exp(\varepsilon|x|_{-p^{\prime}})\,\mathrm{d}\pi_{\sigma}^{\beta}(x).}$

Since $\varepsilon$ is sufficiently small, the last integral exists by Lemma 9
from [23]. This implies that
$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$.
Let us show that in $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$ the
generalized function $\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)$ admits
the canonical expansion

$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)=\sum_{k=0}^{\infty}\frac{1}{k!}Q_{n}^{\sigma,\beta,\alpha}((-w)_{k}).$
(6.1)

Note that the right hand side of (6.1) defines an element in
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$. Then it is sufficient to compare
the action of both sides of (6.1) on a total set from
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$. For
$\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$, we use the
biorthogonal property of $\mathbb{P}^{\sigma,\beta,\alpha}$ and
$\mathbb{Q}^{\sigma,\beta,\alpha}$-systems and obtain

$\displaystyle\langle\\!\langle\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot),\langle
C_{n}^{\sigma,\beta},\varphi^{(n)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\bigg{\langle}\\!\\!\\!\bigg{\langle}\sum_{k=0}^{\infty}\frac{1}{k!}Q_{n}^{\sigma,\beta,\alpha}((-w)_{k}),\langle
C_{n}^{\sigma,\beta},\varphi^{(n)}\rangle\bigg{\rangle}\\!\\!\\!\bigg{\rangle}_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\langle(-w)_{n},\varphi^{(n)}\rangle.$

On the other hand, by Proposition 5.2-(P4) and (P6),

$\displaystyle\langle\\!\langle\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot),\langle
C_{n}^{\sigma,\beta},\varphi^{(n)}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle={\displaystyle\int}_{\mathcal{D}^{\prime}}\langle
C_{n}^{\sigma,\beta}(x-w),\varphi^{(n)}\rangle\,\mathrm{d}\pi_{\sigma}^{\beta}(x)$
$\displaystyle{\displaystyle\negthickspace\overset{(P4)}{=}\sum_{k=0}^{\infty}}\binom{n}{k}{\displaystyle\int}_{\mathcal{D}^{\prime}}\langle
C_{n}^{\sigma,\beta}(x)\hat{\otimes}(-w)_{n-k},\varphi^{(n)}\rangle\,\mathrm{d}\pi_{\sigma}^{\beta}(x)$
$\displaystyle{\displaystyle=\sum_{k=0}^{\infty}}\binom{n}{k}\mathbb{E}_{\pi_{\sigma}^{\beta}}(\langle
C_{n}^{\sigma,\beta}(x)\hat{\otimes}(-w)_{n-k},\varphi^{(n)}\rangle)$
$\displaystyle\negthickspace\negthickspace\overset{(P6)}{=}\langle(-w)_{n},\varphi^{(n)}\rangle.$

Thus, we have shown that $\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(w,\cdot)$ is
the generating function of the $\mathbb{Q}^{\sigma,\beta,\alpha}$-system,
i.e.,

$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(-w,\cdot)=\sum_{k=0}^{\infty}\frac{1}{k!}Q_{k}^{\sigma,\beta,\alpha}((w)_{k}).$

######  Example 6.5 (Delta function).

For $w\in\mathcal{N}^{\prime}_{\mathbb{C}}$, we define a distribution by the
following $\mathbb{Q}^{\sigma,\beta,\alpha}$-decomposition:

$\delta_{w}=\sum_{n=0}^{\infty}Q_{n}^{\sigma,\beta,\alpha}(C_{n}^{\sigma,\beta}(w)).$

If $p\in\mathbb{N}$ is large enough and $\varepsilon>0$ is sufficiently small,
by Proposition 5.2-(P7), for any $w\in\mathcal{H}_{-p,\mathbb{C}}$ we have

$\|\delta_{w}\|_{-p,-q,\pi_{\sigma}^{\beta},\alpha}^{2}=\sum_{n=0}^{\infty}2^{-nq}|C_{n}^{\sigma,\beta}(w)|_{-p}^{2}\overset{(P7)}{\leq}C_{\varepsilon}^{2}\exp(2\varepsilon|w|_{-p})\sum_{n=0}^{\infty}\varepsilon^{-2n}2^{-nq}$

which is finite for sufficiently large $q\in\mathbb{N}$. This implies that
$\delta_{w}\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$. Now, for

$\varphi=\sum_{n=0}^{\infty}\langle
C_{n}^{\sigma,\beta},\varphi^{(n)}\rangle\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$

the action of $\delta_{w}$ is given by

$\langle\\!\langle\delta_{w},\varphi\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}=\sum_{n=0}^{\infty}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{(n)}\rangle=\varphi(w)$

using the biorthogonal property of $\mathbb{P}^{\sigma,\beta,\alpha}$ and
$\mathbb{Q}^{\sigma,\beta,\alpha}$-systems. This means that $\delta_{w}$ (in
particular for $w$ real) plays the role of a $\delta$-function (evaluation
map) in the calculus we discuss.

Recall Example 6.1 where the modified normalized exponential
$\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi);\cdot)$ is a test function
in $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$ only if
$2^{q}|\varphi|_{p}^{2}<1$ for $\varphi\in\mathcal{N}_{\mathbb{C}}$. We define
the $S_{\pi_{\sigma}^{\beta}}$-transform of a distribution
$\Phi\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}\subset\mathcal{P}^{\prime}_{\pi_{\sigma}^{\beta}}(\mathcal{D}^{\prime})$
by

$S_{\pi_{\sigma}^{\beta}}\Phi(\varphi):=\langle\\!\langle\Phi,\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi;\cdot)\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$

if $\varphi$ is chosen in the above way. By the biorthogonal property of
$\mathbb{P}^{\sigma,\beta,\alpha}$ and
$\mathbb{Q}^{\sigma,\beta,\alpha}$-systems, we have

$S_{\pi_{\sigma}^{\beta}}\Phi(\varphi)=\sum_{n=0}^{\infty}\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes
n}\rangle.$

Now, we introduce the convolution of a function
$\varphi\in(N)_{\pi_{\sigma}^{\beta}}^{1}$, with respect to the measure
$\pi_{\sigma}^{\beta}$ given by

$C_{\pi_{\sigma}^{\beta}}\varphi(w)=\langle\\!\langle\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(-w,\cdot),\varphi\rangle\\!\rangle_{\pi_{\sigma}^{\beta}},$

where
$\rho_{\pi_{\sigma}^{\beta}}^{\alpha}(-w,\cdot)\in(N)_{\pi_{\sigma}^{\beta}}^{-1}$
is the generalized Radon-Nikodym derivative (see Example 6.4) for any
$w\in\mathcal{N}^{\prime}_{\mathbb{C}}$. If $\varphi$ has the representation

$\varphi=\sum_{n=0}^{\infty}\langle
C_{n}^{\sigma,\beta}(w),\varphi^{(n)}\rangle\in(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1},$

then the action of $C_{\pi_{\sigma}^{\beta}}$ on $\varphi$ is given, for every
$w\in\mathcal{N}^{\prime}_{\mathbb{C}}$, by

$C_{\pi_{\sigma}^{\beta}}\varphi(w)=\sum_{n=0}^{\infty}\langle(-w)_{n},\varphi^{(n)}\rangle\overset{\eqref{eq:Stirling-1st-
inner-product}}{=}\sum_{n=0}^{\infty}\sum_{k=0}^{n}(-1)^{k}\langle w^{\otimes
k},\mathbf{s}(n,k)\varphi^{(n)}\rangle.$

The following is the characterization theorem of the test and generalized
function spaces associated to the fPm which is a standard result for this
approach. For the proof, we refer to [23, 21].

######  Theorem 6.6.

$(i)$

The convolution $C_{\pi_{\sigma}^{\beta}}$ is a topological isomorphism from
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{1}$ on
$\mathcal{E}_{\min}^{1}(\mathcal{N}^{\prime}_{\mathbb{C}})$.

$(ii)$

The $S_{\pi_{\sigma}^{\beta}}$-transform is a topological isomorphism from
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-1}$ on
$\mathrm{Hol}_{0}(\mathcal{N}_{\mathbb{C}})$.

$(iii)$

The $S_{\pi_{\sigma}^{\beta}}$-transform is a topological isomorphism from
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$, $\kappa\in[0,1)$, on
$\mathcal{E}_{\max}^{2/(1-\kappa)}(\mathcal{N}_{\mathbb{C}})$.

##  7 Conclusion and Outlook

In this paper, we constructed the generalized Appell system
$\mathbb{A}^{\sigma,\beta,\alpha}=(\mathbb{P}^{\sigma,\beta,\alpha},\mathbb{Q}^{\sigma,\beta,\alpha})$
associated to the fPm $\pi_{\sigma}^{\beta}$ in infinite dimension. The Appell
polynomials $\mathbb{P}^{\sigma,\beta,\alpha}$ (generated by the modified Wick
exponential) and the dual Appell system $\mathbb{Q}^{\sigma,\beta,\alpha}$ are
biorthogonal to each other, see Theorem 5.7. It turns out that the kernels
$C_{n}^{\sigma,\beta}(\cdot)$ of the system $\mathbb{P}^{\sigma,\beta,\alpha}$
are given in terms of the Stirling operators or in terms of the falling
factorials on $\mathcal{D}^{\prime}_{\mathbb{C}}$, see Proposition 5.2. The
system $\mathbb{P}^{\sigma,\beta,\alpha}$ is used to define the spaces of test
functions $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$, $0\leq\kappa\leq
1$, while $\mathbb{Q}^{\sigma,\beta,\alpha}$ is suitable to describe the
generalized functions spaces $(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$
arising from $\pi_{\sigma}^{\beta}$, see Section 6. The spaces
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{\kappa}$ and
$(\mathcal{N})_{\pi_{\sigma}^{\beta}}^{-\kappa}$ are universal in the sense
that their characterization via $S_{\pi_{\sigma}^{\beta}}$-transform is
independent of the measure $\pi_{\sigma}^{\beta}$ (see Theorem 6.6) as is well
known from non-Gaussian analysis, [23, 21].

In a future work we plan to investigate the stochastic counterpart associated
to the fPm, namely the fractional Poisson process $N_{\lambda}^{\beta}$ in one
and infinite dimensions. In particular, their representations in terms of
known processes as well as possible applications.

##  Appendix A Appendix

###  A.1 Kolmogorov extension theorem on configuration space

In this section, we discuss a version of Kolmogorov extension theorem to the
configuration space $(\Gamma,\mathcal{B}(\Gamma))$. The following definitions
and properties of measurable spaces can be found in [9], [13] and [31].

######  Definition A.1.

Let $(X,\mathcal{A})$ and $(X^{\prime},\mathcal{A}^{\prime})$ be two
measurable spaces.

  1. 1.

The spaces $(X,\mathcal{A})$ and $(X^{\prime},\mathcal{A}^{\prime})$ are
called isomorphic if, and only if, there exists a measurable bijective mapping
$f:X\longrightarrow X^{\prime}$ such that its inverse $f^{-1}$ is also
measurable.

  2. 2.

$(X,\mathcal{A})$ and $(X^{\prime},\mathcal{A}^{\prime})$ are called
$\sigma$-isomorphic if, and only if, there exists a bijective mapping
$F:\mathcal{A}\longrightarrow\mathcal{A}^{\prime}$ between the
$\sigma$-algebras which preserves the operations in a $\sigma$-algebra.

  3. 3.

$(X,\mathcal{A})$ is said to be countable generated if, and only if, there
exists a denumerable class $\mathscr{D}\subset\mathcal{A}$ such that
$\mathscr{D}$ generates $\mathcal{A}$.

  4. 4.

$(X,\mathcal{A})$ is said to be separable if, and only if, it is countably
generated and for each $x\in X$ the set $\left\\{x\right\\}\in\mathcal{A}$.

######  Definition A.2.

Let $(X,\mathcal{A})$ be a countable generated measurable space. Then
$(X,\mathcal{A})$ is called the standard Borel space if, and only if, there
exists a Polish space $(X^{\prime},\mathcal{A}^{\prime})$ (i.e., a metrizable,
complete metric space which fulfills the second axiom of countability and the
$\sigma$-algebra $\mathcal{A}^{\prime}$ coincides with the Borel
$\sigma$-sigma) such that $(X,\mathcal{A})$ and
$(X^{\prime},\mathcal{B}(X^{\prime}))$ are $\sigma$-isomorphic.

######  Example A.3.

  1. 1.

Every locally compact, $\sigma$-compact space is a standard Borel space.

  2. 2.

Polish spaces are standard Borel spaces.

######  Proposition A.4.

  1. 1.

If $(X,\mathcal{A})$ is a countable generated measurable space, then there
exists $E\subset\left\\{0,1\right\\}^{\mathbb{N}}$ such that $(X,\mathcal{A})$
is $\sigma$-isomorphic to $(E,\mathcal{B}(E))$. Thus $(X,\mathcal{A})$ is
$\sigma$-isomorphic to a separable measurable space.

  2. 2.

Let $(X,\mathcal{A})$ and $(X^{\prime},\mathcal{A}^{\prime})$ be separable
measurable spaces. Then $(X,\mathcal{A})$ is $\sigma$-isomorphic to
$(X^{\prime},\mathcal{A}^{\prime})$ if, and only if, they are isomorphic.

The following theorem states some operations under which separable standard
Borel spaces are closed, see [31] and [9].

######  Theorem A.5.

  1. 1.

Countable product, sum, and union are separable standard Borel spaces.

  2. 2.

The projective limit is a separable standard Borel space.

  3. 3.

Any measurable subset of a separable standard Borel space is also a separable
standard Borel space.

We need also a version of Kolmogorov’s extension theorem for separable
standard Borel spaces.

######  Theorem A.6 (cf. [31, Chap. V, Theorem 3.2]).

Let $(X_{n},\mathcal{A}_{n})$, $n\in\mathbb{N}$, be separable standard Borel
spaces. Let $(X,\mathcal{A})$ be the projective limit of the space
$(X_{n},\mathcal{A}_{n})$ relative to the maps $p_{m,n}:X_{n}\longrightarrow
X_{m}$, $m\leq n$. If $\left\\{\mu_{n}\right\\}_{n\in\mathbb{N}}$ is a
sequence of probability measures such that n is a measure on
$(X_{n},\mathcal{A}_{n})$ and $\mu_{m}=\mu_{n}\circ p_{m,n}^{-1}$ for $m\leq
n$. Then there exists a unique measure on $(X,\mathcal{A})$ such that
$\mu_{n}=\mu\circ p_{n}^{-1}$ for all $n\in\mathbb{N}$ where $p_{n}$ is the
projection map from $X$ on $X_{n}$.

This theorem can be extended to an index set $I$ which is a directed set with
an order generating sequence, i.e., there exists a sequence
$(\alpha_{n})_{n\in\mathbb{N}}$ in $I$ such that for every $\alpha\in I$ there
exists $n\in\mathbb{N}$ with $\alpha<\alpha_{n}$. We apply this general
framework to our configuration space $\Gamma$. Assume that $(X,\mathfrak{X})$
is a separable standard Borel space. To use $\mathcal{B}_{c}(X)$ makes this
generality have no sense, hence we have to introduce an abstract concept of
local sets. Let $\mathcal{J}_{X}$ be a subset of $\mathfrak{X}$ with the
properties:

(I1)

$\Lambda_{1}\cup\Lambda_{2}\in\mathcal{J}_{X}$ for all
$\Lambda_{1},\Lambda_{2}\in\mathcal{J}_{X}.$

(I2)

If $\Lambda\in\mathcal{J}_{X}$ and $A\in\mathfrak{X}$ with $A\subset\Lambda$
then $A\in\mathcal{J}_{X}$.

(I3)

There exists a sequence $\left\\{\Lambda_{n}\mid n\in\mathbb{N}\right\\}$ from
$\mathcal{J}_{X}$ with $X=\bigcup_{n\in\mathbb{N}}\Lambda_{n}$ such that if
$\Lambda\in\mathcal{J}_{X}$ then $\Lambda\subset\Lambda_{n}$ for some
$n\in\mathbb{N}$.

We can then construct the configuration space as in Subsection 4.2 taking
$X=\mathbb{R}^{d}$ and replacing $\mathcal{B}(\mathbb{R}^{d})$ by
$\mathcal{J}_{\mathbb{R}^{d}}$. Our aim is to show that
$(\Gamma,\mathcal{B}(\Gamma))$ is a separable standard Borel space and thus by
Theorem A.6 the measure $\pi_{\sigma}^{\beta}$ in Subsection 4.3 exists.

It follows from Theorem A.5 that for any
$\Lambda\in\mathcal{J}_{\mathbb{R}^{d}}$ and for any $n\in\mathbb{N}$, the set
$\Lambda^{n}$ is a separable standard Borel space. Thus, by the same argument
$\widetilde{\Lambda^{n}}/S_{n}$ is also a separable standard Borel space, see
e.g. [36]. Now taking into account the isomorphism between
$\widetilde{\Lambda^{n}}/S_{n}$ and $\Gamma_{\Lambda}^{(n)}$,
$\Gamma_{\Lambda}^{(n)}$ is also a separable standard Borel space as well as
$\Gamma_{\Lambda}$ by Theorem A.5-(1). Therefore, given
$(\Gamma,\mathcal{B}(\Gamma))$ as the projective limit of the projective
system
$\left\\{(\Gamma_{\Lambda},\mathcal{B}(\Gamma_{\Lambda})),p_{\Lambda_{1},\Lambda_{2}},\mathcal{J}_{\mathbb{R}^{d}}\right\\}$
of separable standard Borel spaces, by Theorem A.5-(2),
$(\Gamma,\mathcal{B}(\Gamma))$ is a separable standard Borel space.

###  A.2 Stirling Operators

In this appendix we discuss the Stirling operators which we use in Section 5
related to the Taylor expansion of a holomorphic function typical in Poisson
analysis. For more details and other applications, see [10, 11].

For $n\in\mathbb{N}$ and $k\in\mathbb{N}_{0}$, we define the _falling
factorial_ by

$(k)_{n}:=n!\binom{k}{n}=k(k-1)\cdots(k-n+1).$

The latter expression allows us to define falling factorials as polynomials of
a variable $z\in\mathbb{C}$ replacing $k$ as

$(z)_{n}:=z(z-1)\cdots(z-n+1).$

The generating function of the falling factorials is

$\sum_{n=0}^{\infty}\frac{u^{n}}{n!}(z)_{n}=\exp[z\log(1+u)].$

The _Stirling numbers of the first kind,_ denoted by $s(n,k)$, are defined as
the coefficients of the expansion $(z)_{n}$ in $z$, in explicit,

$(z)_{n}:=\sum_{k=1}^{n}s(n,k)z^{k},$

while the _Stirling numbers of the second kind,_ denoted by $S(n,k)$, are
defined as the coefficients of the expansion $z^{n}$ in $(z)_{k}$, that is,

$z^{n}=\sum_{k=1}^{n}S(n,k)(z)_{k}.$

Let us consider lifting the polynomials $(z)_{n}$ to
$\mathcal{D}_{\mathbb{C}}^{\prime}$. We call these polynomials the falling
factorials on $\mathcal{D}_{\mathbb{C}}^{\prime}$, denoted by $(w)_{n}$, for
$w\in\mathcal{D}_{\mathbb{C}}^{\prime}$ (see [10]). The generating function of
the falling factorials on $\mathcal{D}_{\mathbb{C}}^{\prime}$ is given by

$\exp(\langle
w,\log(1+\varphi)\rangle)=\sum_{n=0}^{\infty}\frac{1}{n!}\langle(w)_{n},\varphi^{\otimes
n}\rangle,\quad\varphi\in\mathcal{D}_{\mathbb{C}},\;w\in\mathcal{D}_{\mathbb{C}}^{\prime}.$
(A.1)

The falling factorial may be written recursively (see Proposition 5.4 in [10])
as follows

$\displaystyle(w)_{0}=1,$ $\displaystyle(w)_{1}=w,$
$\displaystyle(w)_{n}(x_{1},\dots,x_{n})=w(x_{1})(w(x_{2})-\delta_{x_{1}}(x_{2}))$
$\displaystyle\qquad\qquad\times\dots\times(w(x_{n})-\delta_{x_{1}}(x_{n})-\delta_{x_{2}}(x_{n})-\dots-\delta_{x_{n-1}}(x_{n})),$

for $n\geq 2$ and $(x_{1},\dots,x_{n})\in(\mathbb{R}^{d})^{n}$.

Now we define the _Stirling operators of the first kind_ as the linear
operators
$\mathbf{s}(n,k):\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}\longrightarrow\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k}$,
$n\geq k$, satisfying

$\displaystyle\langle(w)_{n},\varphi^{(n)}\rangle$
$\displaystyle=\sum_{k=1}^{n}\langle w^{\otimes
k},\mathbf{s}(n,k)\varphi^{(n)}\rangle,\qquad\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},\;w\in\mathcal{D}_{\mathbb{C}}^{\prime},$
(A.2)

and the _Stirling operators of the second kind_ as the linear operators
$\mathbf{S}(n,k):\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}\longrightarrow\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k}$,
$n\geq k$, satisfying

$\displaystyle\langle w^{\otimes n},\varphi^{(n)}\rangle$
$\displaystyle=\sum_{k=1}^{n}\langle(w)_{k},\mathbf{S}(n,k)\varphi^{(n)}\rangle,\qquad\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},\;w\in\mathcal{D}_{\mathbb{C}}^{\prime}.$
(A.3)

######  Remark A.7.

The Stirling operators $\mathbf{s}(n,k)$ and $\mathbf{S}(n,k)$ introduced in
[11] are defined on the space of measurable, bounded, compactly supported,
symmetric functions, however; in this paper, we define these operators on the
space $\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$ as a consequence of
extending the falling factorials to the space of generalized functions
$\mathcal{D}_{\mathbb{C}}^{\prime}$ rather than using the space of Radon
measures.

Let $n,k\in\mathbb{N}$, $k\leq n$ and $i_{1},\dots,i_{k}\in\mathbb{N}$ such
that $i_{1}+\dots+i_{k}=n$. We define the operator
$\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}\in\mathcal{L}(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n},\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})$
(the space of linear operators from
$\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$ into
$\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k}$) by

$(\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}\varphi^{(n)})(x_{1},\dots,x_{k}):=\frac{1}{k!}\sum_{\iota\in
S_{k}}\varphi^{(n)}(\underbrace{x_{\iota(1)},\dots,x_{\iota(1)}}_{i_{1}},\dots,\underbrace{x_{\iota(k)},\dots,x_{\iota(k)}}_{i_{k}}),$
(A.4)

for $\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$ and
$(x_{1},\dots,x_{k})\in(\mathbb{R}^{d})^{k}$. In particular, we have

$\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}\varphi^{\otimes
n}=\varphi^{i_{1}}\hat{\otimes}\dots\hat{\otimes}\varphi^{i_{k}},\quad\varphi\in\mathcal{D}_{\mathbb{C}}.$
(A.5)

When $k=1$, we denote $\mathbb{D}_{n}^{(n)}:=\mathbb{D}^{(n)}$ such that for
$\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$,

$(\mathbb{D}^{(n)}\varphi^{(n)})(x)=\varphi^{(n)}(x,\dots,x),\quad
x\in\mathbb{R}^{d}.$

The operator $\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}$ is continuous (see [10]
and [11]), that is, its adjoint
$(\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)})^{*}:(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})^{\prime}\longrightarrow(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$
exists and is well-defined. In fact, the operators $\mathbf{s}(n,k)$ and
$\mathbf{S}(n,k)$ can be written explicitly in terms of the operator
$\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}$, (see Proposition 3.7 in [11]) that is,
for any $n,k\in\mathbb{N}$, $k\leq n$,

$\mathbf{s}(n,k)=\frac{n!}{k!}\sum_{i_{1}+\dots+i_{k}=n}\frac{(-1)^{n-k}}{i_{1}\dots
i_{k}}\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}.$ (A.6)

and

$\mathbf{S}(n,k)=\frac{n!}{k!}\sum_{i_{1}+\dots+i_{k}=n}\frac{1}{i_{1}!\dots
i_{k}!}\mathbb{D}_{i_{1},\dots,i_{k}}^{(n)}.$ (A.7)

Hence, the Stirling operators are continuous (see Proposition 3.7 in [11]) and
so their adjoints $\mathbf{s}(n,k)^{*}$ and $\mathbf{S}(n,k)^{*}$ are well
defined, that is,

$\mathbf{s}(n,k)^{*}:(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})^{\prime}\longrightarrow(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}\quad\mathrm{and\quad}\mathbf{S}(n,k)^{*}:(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})^{\prime}\longrightarrow(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$
(A.8)

and satisfy

$\langle
w^{(k)},\mathbf{s}(n,k)\varphi^{(n)}\rangle=\langle\mathbf{s}(n,k)^{*}w^{(k)},\varphi^{(n)}\rangle\quad\mathrm{and\quad}\langle
w^{(k)},\mathbf{S}(n,k)\varphi^{(n)}\rangle=\langle\mathbf{S}(n,k)^{*}w^{(k)},\varphi^{(n)}\rangle,$

for all $w^{(k)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})^{\prime}$ and
$\varphi^{(n)}\in\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n}$. Hence, the
Equations (A.2) and (A.3) imply that

$\displaystyle(w)_{n}$
$\displaystyle=\sum_{k=1}^{n}\mathbf{s}(n,k)^{*}w^{\otimes k},$ (A.9)
$\displaystyle w^{\otimes n}$
$\displaystyle=\sum_{k=1}^{n}\mathbf{S}(n,k)^{*}(w)_{k}.$

######  Proposition A.8 (see [11, Prop. 3.15]).

For each $k\in\mathbb{N}$ and $\xi\in\mathcal{D}_{\mathbb{C}},$

$\sum_{n=k}^{\infty}\frac{1}{n!}\mathbf{S}(n,k)\xi^{\otimes
n}=\frac{1}{k!}(\mathrm{e}^{\xi}-1)^{\otimes k}$ (A.10)

and

$\sum_{n=k}^{\infty}\frac{1}{n!}\mathbf{s}(n,k)\xi^{\otimes
n}=\frac{1}{k!}(\log(1+\xi))^{\otimes k}.$ (A.11)

######  Proposition A.9 (see [11, Prop. 3.19]).

For any $i,n\in\mathbb{N}$,

$\sum_{k=1}^{n}\mathbf{s}(k,i)\mathbf{S}(n,k)=\sum_{k=1}^{n}\mathbf{S}(k,i)\mathbf{s}(n,k)=\delta_{n,i}\mathbf{1}^{(i)},$

where $\mathbf{1}^{(i)}$ denotes the identity operator on
$\mathcal{D}_{\mathbb{C}}^{\otimes i}$.

###  A.3 An Alternative Proof of the Theorem 5.7 (Biorthogonal Property)

In Section 5.2, we proved the biorthogonal property of
$\mathbb{A}^{\sigma,\beta,\alpha}$ using the definitions of
$\mathbb{P}^{\sigma,\beta,\alpha}$ and
$\mathbb{Q}^{\sigma,\beta,\alpha}$-systems. Here, we use a property of the
generalized function $Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})$ using the
$S_{\pi_{\sigma}^{\beta}}$-transform (see Theorem A.11 below) to provide an
alternative proof of the Theorem 5.7.

######  Lemma A.10.

For every $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$,
$z\in\mathcal{D}^{\prime}_{\mathbb{C}}$ and
$\varphi\in\mathcal{D}_{\mathbb{C}}$, we have

$G(\Phi^{(n)})(\exp\langle
z,\varphi\rangle)=\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes
n}\rangle\exp\langle z,\varphi\rangle.$

In other words, the function $\exp\langle z,\varphi\rangle$ is an
eigenfunction of the generalized function $G(\Phi^{(n)})$.

###### Proof.

It follows from (A.8) that
$\mathbf{S}(k,n)^{*}\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}k})^{\prime}$.
We use the definition of the differential operator
$D(\mathbf{S}(k,n)^{*}\Phi^{(n)})$ to the monomial $\langle
z,\varphi\rangle^{m}$, $m\geq k$, to obtain

$\displaystyle D(\mathbf{S}(k,n)^{*}\Phi^{(n)})\langle z,\varphi\rangle^{m}$
$\displaystyle=D(\mathbf{S}(k,n)^{*}\Phi^{(n)})\langle z^{\otimes
m},\varphi^{\otimes m}\rangle$ $\displaystyle=\frac{m!}{(m-k)!}\langle
z^{\otimes(m-k)}\hat{\otimes}\mathbf{S}(k,n)^{*}\Phi^{(n)},\varphi^{\otimes
m}\rangle$ $\displaystyle=\frac{m!}{(m-k)!}\langle
z,\varphi\rangle^{m-k}\langle\mathbf{S}(k,n)^{*}\Phi^{(n)},\varphi^{\otimes
k}\rangle.$

Now, we apply the above result ($\star$) to the Taylor series of the function
$\exp\langle z,\varphi\rangle$ and obtain

$\displaystyle D(\mathbf{S}(k,n)^{*}\Phi^{(n)})(\exp\langle z,\varphi\rangle)$
$\displaystyle=D(\mathbf{S}(k,n)^{*}\Phi^{(n)})\sum_{m=0}^{\infty}\frac{\langle
z,\varphi\rangle^{m}}{m!}$
$\displaystyle\overset{\star}{=}\langle\mathbf{S}(k,n)^{*}\Phi^{(n)},\varphi^{\otimes
k}\rangle\sum_{m=k}^{\infty}\frac{1}{(m-k)!}\langle z,\varphi\rangle^{m-k}$
$\displaystyle=\langle\mathbf{S}(k,n)^{*}\Phi^{(n)},\varphi^{\otimes
n}\rangle\exp\langle z,\varphi\rangle.$

Thus, applying the operator $G(\Phi^{(n)})$ to $\exp\langle z,\varphi\rangle$,
we obtain

$\displaystyle G(\Phi^{(n)})(\exp\langle z,\varphi\rangle)$
$\displaystyle=\sum_{k=n}^{\infty}\frac{n!}{k!}D(\mathbf{S}(k,n)^{*}\Phi^{(n)})(\exp\langle
z,\varphi\rangle)$
$\displaystyle=\sum_{k=n}^{\infty}\frac{n!}{k!}\langle\mathbf{S}(k,n)^{*}\Phi^{(n)},\varphi^{\otimes
k}\rangle\exp\langle z,\varphi\rangle$
$\displaystyle=\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes
n}\rangle\exp\langle z,\varphi\rangle,$

where the last equality is a consequence of Equation (5.18). ∎

######  Theorem A.11.

For $\Phi^{(n)}\in(\mathcal{D}_{\mathbb{C}}^{\hat{\otimes}n})^{\prime}$, the
generalized function $Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})$ satisfies

$S_{\text{$\pi_{\sigma}^{\beta}$}}(Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}))(\varphi)=\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes
n}\rangle,\quad\varphi\in\mathcal{V}_{\alpha}\subset\mathcal{D}_{\mathbb{C}}.$

###### Proof.

Using Lemma A.10, the $S_{\pi_{\sigma}^{\beta}}$-transform of
$Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})$ is given by

$\displaystyle
S_{\pi_{\sigma}^{\beta}}(Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}))(\varphi)$
$\displaystyle=\langle\\!\langle
G(\Phi^{(n)})^{*}\bm{1},\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi,\cdot)\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\langle\\!\langle\bm{1},G(\Phi^{(n)})\mathrm{e}_{\pi_{\sigma}^{\beta}}(\varphi,\cdot)\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle={\displaystyle\frac{1}{l_{\pi_{\sigma}^{\beta}}(\varphi)}\int_{\mathcal{D}^{\prime}}G(\Phi^{(n)})}(\exp\langle
z,\varphi\rangle)\,\mathrm{d}\pi_{\sigma}^{\beta}(z)$
$\displaystyle={\displaystyle\frac{\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes
n}\rangle}{l_{\pi_{\sigma}^{\beta}}(\varphi)}\int_{\mathcal{D}^{\prime}}}\exp\langle
z,\varphi\rangle\,\mathrm{d}\pi_{\sigma}^{\beta}(z)$
$\displaystyle=\langle\Phi^{(n)},g_{\alpha}(\varphi)^{\otimes n}\rangle.\qed$

Now using the above result, we provide an alternative proof of Theorem 5.7.

###### Proof of Theorem 5.7 _(_ Alternative _)_.

The $S_{\pi_{\sigma}^{\beta}}$-transform of
$Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)})$ at $\alpha(\varphi)$ is given by

$\displaystyle
S_{\pi_{\sigma}^{\beta}}(Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}))(\alpha(\varphi))$
$\displaystyle=\langle\\!\langle
Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}),\mathrm{e}_{\pi_{\sigma}^{\beta}}(\alpha(\varphi),\cdot)\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}\langle\\!\langle
Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}),\langle
C_{m}^{\sigma,\beta},\varphi^{\otimes
m}\rangle\rangle\\!\rangle_{\pi_{\sigma}^{\beta}}.$

By Theorem A.11 with $\varphi$ replaced by $\alpha(\varphi)$ we obtain

$S_{\pi_{\sigma}^{\beta}}(Q_{n}^{\sigma,\beta,\alpha}(\Phi^{(n)}))(\alpha(\varphi))=\langle\Phi^{(m)},\varphi^{\otimes
m}\rangle.$

The result follows by a comparison of coefficients and the polarization
identity. ∎

#### Acknowledgments

This work was partially supported by the Center for Research in Mathematics
and Applications (CIMA-UMa) related with the Statistics, Stochastic Processes
and Applications (SSPA) group, through the grant UIDB/MAT/04674/2020 of FCT-
Fundação para a Ciência e a Tecnologia, Portugal and by the Complex systems
group of the Premiere Research Institute of Science and Mathematics (PRISM),
MSU-Iligan Institute of Technology. The financial support of the Department of
Science and Technology – Accelerated Science and Technology Human Resource
Development Program (DOST-ASTHRDP) of the Philippines under the Research
Enrichment (Sandwich) Program is gratefully acknowledged. Also, special thanks
to Prof. Ludwig Streit for recommending relevant reading resources.

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