数理解析研究所講究録
第 1819 巻 2012 年 183-192

183

The -Meixner self-adjoint operators on the-deformed Fock space
$q$

$q$

Hiroaki Yoshida
Department of Information Sciences, Ochanomizu University,
2-1-1, Otsuka, Bunkyo, Tokyo 112-8610 Japan

1. Introduction

The Schr\"odinger algebra plays an important role in mathematical physics and its
applications. It has been introduced and studied as the algebra of symmetries of the$Schr6$ dinger equation (see, for instance, [1], [2]). Although, in general, the Schr\"odinger
algebra can be considered in $(n+1)$ -dimensional (space-time), in this paper, we treat
the centrally extended Schr\"odinger algebra of $n=1.$
It was noticed that the $(1+1)$ Schr\"odinger algebra with central extension
canbe embedded into the two-photon algebra [3] (a low-dimensional Wick algebra), which
gives a sort of boson Fock realization of(the two-photon realization) and also helps us
to understand the structure ofsuch as semidirect product of the Heisenberg algebra
and $sl(2)$ .
In the paper [4], the structure of Schr\"odinger algebra
was investigated related to
the representation theory. Especially, they constructed the canonical Appell system andfound a family of the probability distributions associated to the Lie algebraic structureof the Schr\"odinger algebra . Concretely, they construct the Hilbert space on which
certain two commuting operators act as self-adjoint operators with an adjustment of theinner product. Such a self-adjointization is important because it yields a probabilistic
interpretation of these operators as random variables in non-commutative probability
space.
$\mathcal{S}_{1}$$S_{1}$$S_{1}$$S_{1}$$S_{1}$

2. The Schr\"odinger algebra$S_{1}$and its boson realization

The centrally extended $(1+1)$ Schr\"odinger algebra
is a six-dimensional Lie algebra$M$$D,$
generated by the operators, $K,$ $G,$
with the following non-trivial
commutation relations (see [5]):$S_{1}$$P_{x},$$P_{t},$$[P_{t}, G]=P_{x}, [P_{x}, K]=G, [D, G]=G, [P_{x}, D]=P_{x},$$[P_{t}, D]=2P_{t}, [D, K]=2K, [P_{t}, K]=D, [P_{x}, G]=M,$(2.1)

where the last commutation relation corresponds to the property of the central extension.
For simplicity, if we say Schr\"odinger algebra in this paper, then it means to be centrally

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extended.

has the following vector fields
It is also known that the Schr\"odinger algebra
realization with the multiplication and the partial differentiation by and :$S_{1}$$x$$K=t^{2} \partial_{t}+tx\partial_{x}+\frac{m}{2}x^{2}-dt$

conformal transformation,
Galilei boost,
spatial translation,$G=t\partial_{x}+mx$$P_{x}=\partial_{x}$

$t$

(2.2)

dilation,$D=2t\partial_{t}+x\partial_{x}-d$time translation,mass,$P_{t}=\partial_{t}$$M=m1$

where $m$ and are given parameters, and 1 is the identity operator.
The operators $\{M, G, P_{x}\}$ span a Heisenberg-Weyl subalgebra, and $\{K, D, P_{t}\}$span an $sl(2)$ subalgebra. Indeed the Schr\"odinger algebra can be decomposed into the.
semidirect product as$d$$S_{1}\cong \mathcal{H}\oplus_{s}sl(2)$and be the boson creation and the boson annihilation operators on theLetsymmetric (boson) Fock space, respectively, that is, these two operators satisfy the. The two-photon algebra (see, for instance,
canonical commutation relation[3] is generated by$a^{\uparrow}$$a$$[a, a^{\uparrow}]=1$$h_{6}$$)$$N=aa\dagger, A_{+}=a^{\uparrow}, A_{-}=a,$(2.3)$M=1, B_{+}=(a\dagger)^{2}, B_{-}=a^{2}.$, and $N$ are the double creation, the double annihilation, and theNamely,number operators, respectively. The non-trivial commutation relations among thesegenerators are$B_{+},$$B_{-}$$[A_{-}, A_{+}]=M, [B_{-}, B_{+}]=4N+2M,$$[N, A_{+}]=A_{+}, [N, A_{-}]=-A_{-},$(2.4)$[N, B_{+}]=2B_{+}, [N, B_{-}]=-2B_{-},$$[A_{+}, B_{-}]=-2A_{-}, [A_{-}, B_{+}]=2A_{+},$

from which we can have an embedding of the Schr\"odinger algebra
explicitly as follows:
algebra$S_{1}$

into the two photon$h_{6}$$K= \frac{1}{2}B_{+}, G=A_{+}, P_{x}=A_{-},$(2.5)$D=N+ \frac{1}{2}M, P_{t}=\frac{1}{2}B_{-}, M=M.$

Combining with (2.3), this embedding gives the two-photon realization of the
Schr\"odinger algebra$S_{1}.$

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Here we shall remind the commuting operators of our interest. In [4] they
investigated the following two commuting operators
andobtained by the adjoint
action of the exponentiating : For $\beta>0$$x_{1}$$x_{2}$$P_{t}$$x_{1}=e^{\beta P}tKe^{-\beta P_{t}}=K+\beta D+\beta^{2}P_{t},$(2.6)$x_{2}=e^{\beta P}tGe^{-\beta P_{t}}=G+\beta P_{x}.$They made adjustment an inner product so thatand $G^{*}=\beta P_{x}$ , that is,andare self-adjoint, and gave the probabilistic observations. Especially, the joint
distribution ofandwas determined explicitly based on the Appell system.
$K^{*}=\beta^{2}P_{t}$$x_{1}$$x_{2}$$x_{1}$$x_{2}$The inner product that they assumed was enough for their manipulation but alittle implicit for making an extension to a deformed case. In the next section, we shallintroduce an inner product which makes
be self-adjoint more explicitly, by using a
deformation of the symmetric Fock space. Then we shall find the probability distributionof , in which the boson Fock realization of the Schr\"odinger algebra will be crucial.
$x_{i}$$S_{1}$$x_{i}$

3. The -symmetric Fock space$\beta$We shall slightly change the inner product on the symmetric Fock space, and construct
the -symmetric Fock space, on which
andcan be regarded to be self-adjoint.
$\beta$$x_{1}$$x_{2}$

Let
be a real Hilbert space equipped with the inner product, andbea distinguished unit vector, called vacuum. We denote by
the set of all thefinite linear combinations of the elementary vectors$\mathscr{H}$$\Omega$$\langle\cdot|\cdot\rangle$$\mathcal{F}^{fin}(\mathscr{H})$$\xi_{1}\otimes\cdots\otimes\xi_{n}\in \mathscr{H}^{\otimes n}(n=1,2, \ldots)$

We introduce the inner product $(|)_{\beta}$ on$\mathcal{F}^{fin}(\mathscr{H})$by$( \xi_{1}\otimes\cdots\otimes\xi_{n}|\eta_{1}\otimes\cdots\otimes\eta_{m})_{\beta}=\delta_{m,n}\beta^{n}\sum_{\sigma\in \mathfrak{S}_{n}}\langle\xi_{1}|\eta_{\sigma(1)}\rangle\cdots\langle\xi_{n}|\eta_{\sigma(n)}\rangle,$

where
is the nth symmetric group of permutations.
The strict positivity of the inner productfollows immediately from that of
the inner product on the symmetric (boson) Fock space. Thus we can have the following
definitions:$\mathfrak{S}_{n}$$(\cdot|\cdot)_{\beta}$

Definition 3.1. The -symmetric Fock spaceis given by the completion ofby the inner product, the -creation operator. Given the vector$\beta$$\mathcal{F}^{fin}(\mathscr{H})$$a_{\beta}^{\dagger}(\xi)$$\mathcal{F}_{\beta}(\mathscr{H})$$\xi\in \mathcal{H}$$(\cdot|\cdot)_{\beta}$$\beta$is defined by the canonical left creation that$a_{\beta}^{\dagger}(\xi)\Omega=\xi,$$a_{\beta}^{\dagger}(\xi)\xi_{1}\otimes\cdots\otimes\xi_{n}=\xi\otimes\xi_{1}\otimes\cdots\otimes\xi_{n} n\geq 1,$

and the -annihilation operator
is defined to be the adjoint operator ofrespect to the inner product $(|)_{\beta}$ , that is,$\beta$$a_{\beta}(\xi)$$a_{\beta}(\xi)=(a_{\beta}^{\dagger}(\xi))^{*}$$a_{\beta}^{\dagger}(\xi)$

with

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The followings are direct consequences of the definition:
Proposition 3.2.
(1) The$\beta$-annihilation operator$a_{\beta}(\xi)$acts on the elementary vectors as follows:$a_{\beta}(\xi)\Omega=0, a_{\beta}(\xi)\xi_{1}=\beta\langle\xi|\xi_{1}\rangle\Omega,$$a_{\beta}( \xi)\xi_{1}\otimes\cdots\otimes\xi_{n}=\beta\sum_{k=1}^{n}\langle\xi|\xi_{k}\rangle\xi_{1}\otimes\cdots\otimes\xi_{k}\otimes\cdots\otimes\xi_{n}\vee n\geq 2,$

should be deleted from tensor product.
(2) The -creation and the -annihilation operators satisfy thecommutation relation
where$\xi_{i}\vee$means that$\xi_{i}$$\beta$$\beta$$\beta$-scaled canonical$a_{\beta}(\xi)a_{\beta}^{\dagger}(\eta)-a_{\beta}^{\dagger}(\eta)a_{\beta}(\xi)=\beta\langle\xi|\eta\rangle 1$

We shall work on the -symmetric Fock space of one-mode, and employ the, and $N$ in (2.3)
creation and the -annihilation to construct the operators
instead of the boson creation and the boson annihilation operators. Under the sameembedding in (2.5), we can obtain the self-adjointization of on the -symmetric Fockspace, namely,$\beta-$$\beta$$A\pm,$$\beta$$B_{\pm}$$\beta$$x_{i}$$x_{1}^{(\beta)}=\frac{1}{2}(a_{\beta}^{\dagger})^{2}+\frac{1}{2}(a_{\beta})^{2}+a_{\beta}^{\dagger}a_{\beta}+\frac{1}{2}1,$$x_{2}^{(\beta)}=a_{\beta}^{\dagger}+a_{\beta},$where, for the unit base vector of the one-mode -symmetric Fock space, we shalland , respectively.
simply denotebyand$\beta$$\xi$$a_{\beta}^{\dagger}(\xi)$$a_{\beta}(\xi)$$a_{\beta}^{\dagger}$$a_{\beta}$

Now we shall find the probability distributions of the self-adjoint operatorswith respect to the vacuum expectation on the -symmetric Fock space. We will apply
the theory of orthogonal polynomials to finding the probability distributions.
$x_{i}^{(\beta)}$$\beta$Proposition 3.3. For $\beta>0$ , we define the sequence of polynomials
the recurrence formula$P_{0}^{(\beta)}(X)=1,$$\{P_{n}^{(\beta)}(X)\}_{n\geq 0}$by$P_{1}^{(\beta)}(X)=X- \frac{1}{2},$(3.1)$P_{n+1}^{(\beta)}(X)=(X-( \frac{1}{2}+2\beta n))P_{n}^{(\beta)}(X)-\frac{\beta^{2}2n(2n-1)}{4}P_{n-1}^{(\beta)}(X)$and$n\geq 1,$

stands for the unit base vector for the one-mode -symmetric Fock space.
Then we obtain$\beta$$\xi$$P_{n}^{(\beta)}( x_{1}^{(\beta)})\Omega=\frac{1}{2^{n}}\xi^{\otimes 2n} n\geq 0,$

where$\Omega$is the vacuum vector, and we use the convention that$\xi^{\otimes 0}=\Omega.$

This proposition can be proved by induction without much difficulties.

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In order to determine the measure , we shall reformulate the three-steps recurrence
relation in (3.1) as follows: Divide by, then we have$\mu$$\beta^{n+1}$$\frac{P_{n+1}^{(\beta)}(X)}{\beta^{n+1}}=(\frac{1}{\beta}(X-\frac{1}{2}+\frac{\beta}{2})-(\frac{1}{2}+2n))\frac{P_{n}^{(\beta)}(X)}{\beta^{n}}-n(n-\frac{1}{2})\frac{P_{n-1}^{(\beta)}(X)}{\beta^{n-1}},$and change the variable $X$ by

$Y$

as

$Q_{n}(Y)= \frac{P_{n}(\beta Y+(\frac{1}{2}-\frac{\beta}{2}))}{\beta^{n}}.$Consequently, we obtain the sequence of the monic polynomials $\{Q_{n}(Y)\}$ which satisfies
the recurrence relation$Q_{0}(Y)=1, Q_{1}(Y)=Y- \frac{1}{2},$$Q_{n+1}(Y)=(Y-( \frac{1}{2}+2n))Q_{n}(Y)-n(n-\frac{1}{2})Q_{n-1}(Y) n\geq 1.$

This recurrence relation is known as one for the Laguerre polynomials of the parameter(see, for instance, [7, Sec. 2.11]), and the corresponding orthogonalizing
probability measure can be given by the density function$\alpha=-\frac{1}{2}$$g(t)= \frac{1}{\Gamma(\frac{1}{2})}\frac{e^{-t}}{\sqrt{t}}I_{t\geq 0},$where

$I_{t\geq 0}$

is the indicate function on $\{t|t\geq 0\}.$Remark 3.4. By the form of the density function $g(t)$ , we can find that it is in thetype of gamma distributions. More precisely, it is given as, that is, thedilation of the chi-square distribution with 1 degree of freedom, because the density ofthe distribution
is given by$\frac{1}{2}\chi^{2}(1)$$\frac{1}{2}$$\chi^{2}(1)$$f(t)= \frac{1}{\Gamma(\frac{1}{2})}\frac{e^{-t/2}}{\sqrt{2t}}I_{t\geq 0}.$Since the variable $X$ in the sequence of polynomials
is related to$\{Q_{n}(Y)\}$ by, the orthogonalizing probability measure for$\{P_{n}^{(\beta)}(X)\}$$X= \beta Y+(\frac{1}{2}-\frac{\beta}{2})$is given by the -dilation and the$\beta$$Y$

in$\{P_{n}^{(\beta)}(X)\}$$( \frac{1}{2}-\frac{\beta}{2})$-right translation of one for $\{Q_{n}(Y)\}.$

Theorem 3.5. The probability distribution of the operator
with respect to thevacuum expectation, namely, the orthogonalizing probability measure for the sequenceof polynomials
defined in (3.1), is given by$x_{1}^{(\beta)}$$\mu$$\{P_{n}^{(\beta)}(X)\}$$d\mu=\frac{2}{\beta}f(\frac{2}{\beta}(x-(\frac{1}{2}-\frac{\beta}{2})))I_{t\geq\frac{1}{2}-g}dt$

188, and $dt$ is the Lebesgue measure on . Namely,
where is the density function of
the distribution can be written symbolically by$\mathbb{R}$$\chi^{2}(1)$

$f$

$\mu$

$\frac{\beta}{2}\chi^{2}(1)+(\frac{1}{2}-\frac{\beta}{2})$.

with respect
andRemark 3.6. In [4], they investigated the joint distribution of
to the vacuum expectation. We shall, here, pay our attention upon the algebraic relation, namely,
andbetween the operators$x_{2}^{(\beta)}$$x_{1}^{(\beta)}$$x_{2}^{(\beta)}$$x_{1}^{(\beta)}$$\frac{1}{2}(x_{2}^{(\beta)})^{2}=\frac{1}{2}(a_{\beta}^{\dagger}+a_{\beta})^{2}$(3.2)$= \frac{1}{2}(a_{\beta}^{\dagger})^{2}+\frac{1}{2}(a_{\beta})^{2}+\frac{1}{2}(a_{\beta}^{\dagger}a_{\beta}+a_{\beta}a_{\beta}^{\dagger})$$= \frac{1}{2}(a_{\beta}^{\dagger})^{2}+\frac{1}{2}(a_{\beta})^{2}+a_{\beta}^{\dagger}a_{\beta}+\frac{\beta}{2}1,$where, in the last equality, we have used the commutation relation
we have the algebraic relation$[a_{\beta}, a_{\beta}^{\dagger}]=\beta 1$. Hence,$x_{1}^{(\beta)}=\frac{1}{2}(x_{2}^{(\beta)})^{2}+(\frac{1}{2}-\frac{\beta}{2})1.$

via orthogonal
Although we have derived the distribution of the operator
polynomials, we can obtain it directly by using the above algebraic relation. Becausewith respect to theit is known that the distribution of the field operator, the centeredvacuum expectation on the -symmetric Fock space is given byis distributed according, the operator
Gaussian of variance$x_{1}^{(\beta)}$$x_{2}^{\beta}=a_{\beta}^{\dagger}+a_{\beta}$$\mathcal{N}(0, \beta)$$\beta$$((x_{2}^{(\beta)})^{2}\Omega|\Omega)_{\beta}=\beta$. Therefore we can find thattoalgebraic relation (3.2).
$\chi^{2}(1)$$x_{2}^{(\beta)}$$\frac{1}{\beta}(x_{2}^{(\beta)})^{2}$

has the distribution$\frac{\beta}{2}\chi^{2}(1)+(\frac{1}{2}-\frac{\beta}{2})$by the4. The case of the -deformed Fock space$q$that we have considered in the previous section, ismore of a scaling than a deformation. Here we shall use the -deformation of symmetric. The -deformed symmetric Fock space wasinstead of
Fock space
introduced in [8], which gives an interpolation between the symmetric (boson) and theanti-symmetric (fermion) Fock spaces and, especially, the case $q=0$ of which yields the
canonical model in the free probability theory (see, for instance, [9]).

The -symmetric Fock space$\beta$$\mathcal{F}_{\beta}(\mathscr{H})$$q$$\mathcal{F}_{\beta}(\mathscr{H})$$\mathcal{F}_{q}(\mathscr{H})$$q$

We shall assume to be non-negative, that is, we restrict to $0\leq q<1$ . We
and theconsider the one-mode case and simply denote the -creation operator
andfor the unit base vector of one-mode by-annihilation operator
respectively.
$q$$a_{q}^{\uparrow}(\xi)$

$q$

$q$

$a_{q}(\xi)$$\xi$$a_{q}\dagger$$a_{q},$

189
Then we shall give the -deformation of
as follows:$q$$A_{\pm},$$B\pm$

and $N$ in (2.3) by using$a_{q}\dagger$and$a_{q}$$A_{+}^{(q)}=a_{q}^{\dagger}, A_{-}^{(q)}=a_{q}, B_{+}^{(q)}=(a_{q}^{\uparrow})^{2}, B_{-}^{(q)}=(a_{q})^{2},$$N^{(q)}=a_{q}^{\}}a_{q}, M=1,$

where $M$ remains undeformed as the scalar operator. The above operators satisfy thefollowing non-trivial commutation relations, cf. (2.4):
Lemma 4.1.
$A_{-}^{(q)}A_{+}^{(q)}- qA_{+}^{(q)}A_{-}^{(q)}=M,$$N^{(q)}A_{+}^{(q)}- qA_{+}^{(q)}N^{(q)}=A_{+}^{(q)},$$N^{(q)}A_{-}^{(q)}-q^{-1}A_{-}^{(q)}N^{(q)}=-q^{-1}A_{-}^{(q)},$$B_{-}^{(q)}B_{+}^{(q)}- q^{4}B_{+}^{(q)}B_{-}^{(q)}=q(1+q)^{2}N^{(q)}+(1+q)M,$$N^{(q)}B_{+}^{(q)}- q^{2}B_{+}^{(q)}N^{(q)}=(1+q)B_{+}^{(q)},$$N^{(q)}B_{-}^{(q)}-q^{-2}B_{-}^{(q)}N^{(q)}=-q^{-2}(1+q)B_{-}^{(q)},$$A_{+}^{(q)}B_{-}^{(q)}-q^{-2}B_{-}^{(q)}A_{+}^{(q)}=-q^{-2}(1+q)A_{-}^{(q)},$$A_{-}^{(q)}B_{+}^{(q)}- q^{2}B_{+}^{(q)}A_{-}^{(q)}=(1+q)A_{+}^{(q)}.$

We can see Lemma 4.1 by,direct calculation, similar manipulations concerning withthe above commutation relations can be also found in [10], [11], [12], in which square or
higher powers of -white noise analysis are investigated.
$q$

As we mentioned in Section 1, the classical Schr\"odinger algebra is generated by the$M$ , which have the boson Fock realization as in (2.5). Taking$D,$operators, $K,$ $G,$
in account the commutation relations in Lemma 4.1, we shall give the -deformation of$D,$$K,$ $G,$
by using the -Fock space instead of the symmetric one. Namely, we$P_{x},$$P_{t},$$q$$P_{x},$$P_{t}$$q$put$G^{(q)}=A_{+}^{(q)}, P_{x}^{(q)}=A_{-}^{(q)}, K^{(q)}= \frac{1}{1+q}B_{+}^{(q)}, P_{t}^{(q)}=\frac{1}{1+q}B_{-}^{(q)}$$D^{(q)}=qN^{(q)}+ \frac{1}{1+q}M.$Now we shall consider the -deformation of the operators
andand find their$G,$probability distributions. We give the -deformation by replacing $K,$ $D,$in(2.6) to the -deformed ones, where no scaling parameter is imposed, that is, $\beta=1.$
Hence we have$q$$x_{1}$$q$$q$$x_{1}^{(q)}=K^{(q)}+D^{(q)}+P_{t}^{(q)},$$x_{2}^{(q)}=G^{(q)}+P_{x}^{(q)}.$$x_{2}$$P_{t},$$P_{x}$

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on the -Fock space and its probability
is the field operator
The operator
distribution with respect to the vacuum expectation is rather well-known as the qGaussian and investigated by many authors, for instance, [8], [14], [15].
Therefore we shall pay our attention upon the operator$a_{q}^{\dagger}+a_{q}$$x_{2}^{(q)}$$q$$x_{1}^{(q)}=\frac{1}{1+q}B_{+}^{(q)}+\frac{1}{1+q}B_{-}^{(q)}+qN^{(q)}+\frac{1}{1+q}M,$and determine its probability distribution with respect to the vacuum expectation on
the -Fock space.
Similar to the -symmetric case, we will seek the sequence of polynomials which
are orthogonal with respect to the distribution of$q$$\beta$$x_{1}^{(q)}.$

Proposition 4.2.

recurrence formula$P_{0}^{(q)}(X)=1,$

We define the sequence of polynomials$\{P_{n}^{(q)}(X)\}_{n\geq 0}$$P_{1}^{(q)}(X)=X-\perp$’$1+q$$P_{n+1}^{(q)}(X)=(X-( \frac{1}{1+q}+q[2n]_{q}))P_{n}^{(q)}(X)-\frac{[2n]_{q}[2n-1]_{q}}{(1+q)^{2}}P_{n-1}^{(q)}(X)$and let$\xi$

by the$n\geq 1.$

be the unit base vector for the one-mode -Fock space. Then we obtain$q$$P_{n}^{(q)}( x_{1}^{(q)})\Omega=\frac{1}{(1+q)^{n}}\xi^{\otimes 2n} n\geq 0,$

where$\Omega$is the vacuum vector, and we use the convention that$\xi^{\otimes 0}=\Omega.$, we will
In order to describe the probability distribution of the operator
recall the -deformed Meixner polynomials and fix the notations of the corresponding
probability measures.
The classical infinitely divisible distributions of the Meixner family, that is,
Gaussian, Poisson, gamma, Pascal, and (pure) Meixner types, can be determined as
the orthogonalizing probability measures for the sequence of polynomials given by the:, andfollowing recurrence relations with 4 parameters,$x_{1}^{(q)}$$q$$\kappa_{1},$$P_{0}(X)=1,$$\kappa_{2},$$\gamma$$\delta(\kappa_{2}>0, \delta\geq 0)$$P_{1}(X)=X-\kappa_{1},$$P_{n+1}(X)=(X-(\kappa_{1}+\gamma n))P_{n}(X)-(\kappa_{2}+\delta(n-1))nP_{n-1}(X)$(4.1)$n\geq 1.$

correspond to the mean and the variance of the distribution,
andThe parametersrespectively. The sequence of polynomials $\{P_{n}(X)\}$ given by (4.1) is called the (classical)
Meixner polynomials.
$\kappa_{1}$$\kappa_{2}$One of -deformations of the Meixner polynomials is given as replacing the integers
in the Szeg\"o-Jacobi parameters in the recurrence relation (4.1) to the -integers, andwe will refer the corresponding orthogonalizing probability measures as the -deformed
Meixner distributions.
$q$

$q$

$q$

191
Definition 4.3. Let
be given constants with $0\leq q<1,$
Then the -deformed Meixner distribution onof parameters
is defined
to be the unique probability measure
on for which the sequence ofpolynomials
given by the following recurrence relation are orthogonal:$q,$$\delta$$\kappa_{1},$$\kappa_{2},$$\kappa_{2}>0,$$\gamma,$$\mathbb{R}$$q$$\delta\geq 0.$$\delta$$\kappa_{1},$$\mu(q;\kappa_{1}, \kappa_{2}, \gamma, \delta)$$\kappa_{2},$$\gamma,$$\mathbb{R}$$\{P_{n}\}$$P_{0}(X)=1,$$P_{1}^{(q)}(X)=X-\kappa_{1},$,$P_{n+1}(X)=(X-(\kappa_{1}+\gamma[n]_{q}))P_{n}(X)-(\kappa_{2}+\delta[n-1]_{q})[n]{}_{q}P_{n-1}(X)$(4.2)$n\geq 1.$

Remark 4.4. Although the polynomials defined in (4.2) are affine transformation of
Al-Salam-Chihara polynomials [16], we shall adopt the above parameterization foremphasizing the relation to the five types of infinitely divisible distributions.
By comparing the Szeg\"o-Jacobi parameters for the orthogonal polynomials in
Proposition 4.2 with those in (4.2), we can derive the following theorem, for moredetails see [20]:Theorem 4.5.

The probability distribution of the operator
with respect to thevacuum expectation on the -Fock space is given as the deformed Meixner distribution$x_{1}^{(q)}$$q$$\mu(q^{2};\frac{1}{1+q}, \frac{1}{1+q}, q(1+q), q)$

where we should note that the deformation pammeter
distributions is replaced to$q$,

for the -deformed Meixner$q$$q^{2}.$

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