数理解析研究所講究録 1227 巻 2001 年 218-229

218

Graphical representations of the q-creation andthe q-annihilation operators and set partition
statistics
Hiroaki Yoshida
Department of Information Sciences,
Ochanomizu University, Otsuka, Bunkyo,
Tokyo 112-8610 Japan
Abstract
In this note, we introduce acertain graphical representations of the-creation the -annihilation, the -number, and the scalar operatorson the -Fock space. These are given as the cards on which some flowlines are drawn and increasing and decreasing of the number of lines
correspond to creation and annihilation, respectively. By the cards arrangement technique with these cards, of which machinery is the sameas of the partition function for the lattice spins model in statistical mechanics, we write the moments of the -Poisson random variable using
the number of arc crossings, which is the same set partition statistic
as restricted crossings investigated by P. Biane. This result suggests
another -deformed moments-cumulants relations, which interpolates
between the usual and the free cases exactly. This is the joint workwith Naoko Saitoh at Ochanomizu University.
$q$

$q$

$q$

$q$

$q$

$q$

1The -Fock space$q$

For aHibert space 7{ and $q\in(-1,1)$ , the -Fock spacecan be defined
as follows: (see, for instance, [BS1], [BKS]) Let
be the linear span of, where varies invectors of the form
and we put
for some distinguished vector called vacuum. We consider thesesquilinear formongiven by the sesquilinear extension of$\mathcal{F}_{q}(\mathcal{H})$$q$$\mathcal{F}^{fin}(\mathcal{H})$$\xi_{1}\otimes\cdots\otimes\xi_{n}\in \mathcal{H}^{\otimes n}$$\mathcal{H}^{\otimes 0}\cong \mathrm{C}\Omega$$n$$\mathrm{z}_{\geq 0}$$\Omega$$\langle\cdot|\cdot\rangle_{q}$$\mathcal{F}^{J^{jn}}(\mathcal{H})$$\langle\xi_{1}\otimes\cdots\otimes\xi_{n}|\eta_{1}\otimes\cdots\otimes\eta_{m}\rangle_{q}=\delta_{n,m}\sum_{\pi\in S_{n}}q^{i(\pi)}\langle\xi_{1}|\eta_{\pi(1)}\rangle\cdots\langle\xi_{n}|\eta_{\pi(n)}\rangle$

,

219
where
denotes the symmetric group of permutations of n elements andis the number of the inversions of permutation vr cE S. defined by$S_{n}$$i(\pi)=\{\#(i,j)|1\leq i<j\leq n, \pi(i)>\pi(j)\}$The strict positivity (see [BS2]) of$\langle\cdot|\cdot\rangle_{q}$$i(\cdot)$.

allows the following definitions:Definition 1.1. The -Fock space $F_{q}(?t)$ is the completion ofwithrespect to, and given the vector, we define the -creation operator
and the -annihilation operator $a(\xi)$ on Fq(H) by$q$$\mathcal{F}^{fn}:(\mathcal{H})$$\langle\cdot|\cdot\rangle_{q}$$a^{*}(\xi)$$\xi\in \mathcal{H}$

$q$

$q$

$a^{*}(\xi)\Omega=\xi$,$a^{*}(\xi)\xi_{1}\otimes\cdots\otimes\xi_{n}=\xi\otimes\xi_{1}\otimes\cdots\otimes\xi_{n}$and$a(\xi)\Omega=0$,,$a( \xi)\xi_{1}\otimes\cdots\otimes\xi_{n}=\sum_{i=1}^{n}q^{i-1}\langle\xi|\xi_{i}\rangle\xi_{1}\otimes\cdots\otimes\xi_{i}\otimes\cdots\otimes\xi_{n}\vee$respectively, where the symbol
product.

$\xi_{i}\vee$means that$\xi_{i}$has to be deleted in the tensor

They are adjoints of each other with respect to the scalar product.
Furthermore, it is very important to note that they fulfill the q-commutation$\langle\cdot|\cdot\rangle_{q}$relations,$a(\xi)a^{*}(\eta)-qa^{*}(\eta)a(\xi)=\langle\xi|\eta\rangle\cdot 1$$\xi$

We can easily see from the definition that, forobtain that$a^{*}(\xi)\xi^{\otimes n}$$a(\xi)\xi^{\otimes n}$$=\xi^{\otimes(n+1)}$

$\xi$

,

$\eta\in \mathcal{H}$$\in \mathcal{H}$.

with $||\xi||=1$ , we,$=[n]_{q}\xi^{\otimes(n-1)}$,$(n(n\geq\geq 0)1)$

where we use the convention that. Combining these relations, wehave thatfor
with $||\xi||=1$ and $n\geq 1$ ,$\xi^{\otimes 0}=\Omega$$\xi\in \mathcal{H}$$a^{*}(\xi)a(\xi)\xi^{\otimes n}=[n]_{q}\xi^{\otimes n}$Hence we may regard$a^{*}(\xi)a(\xi)$.

as the -number operator.
$q$

2The -Poisson random variable$q$

In [SY1], we introduced, for $q\in[0,1)$ , the -deformed Poisson distribution as
the orthogonalizing probability measure for acertain -deformation of Charlier
polynomials, which interpolates the usual $(q=1)$ and the free $(q=0)$ Poisson
distributions
$q$

$q$

220
Definition 2.1. For qe [0,1) and A $>0$ , we define the -deformed Poisson
distr ibution of the parameter Aas the orthogonalizing probability measure forthe sequence of polynomials$q$$P_{0}(X)=1$ ,$P_{1}(X)=X-\lambda$,$(n\geq 1)$$P_{n+1}(X)=(X-(\lambda+[n]_{q}))P_{n}(X)-\lambda[n]_{q}P_{n-1}(X)$.

In the subsequent paper [SY2], we gave the -Poisson random variable
on the -Fock space, of which the distribution with respect to the vacuum
expectation is the -deformed Poisson distribution.
$q$

$q$

$q$

Definition 2.2. For$\xi\in \mathcal{H}$with $||\xi||=1$ and$\lambda>0$, we call the operator$(a^{*}(\xi)+\sqrt{\lambda}\cdot 1)(a(\xi)+\sqrt{\lambda}\cdot 1)$$=$$a^{*}(\xi)a(\xi)+\sqrt{\lambda}(a^{*}(\xi)+a(\xi))+\lambda\cdot 1$

the -Poisson random variable of the parameter A.
$q$, the -Gaussian random
It is the sum of the -number operator,, and the scalar operator . 1, which is natural qvariable
deformation compatible with the result of Hudson-Pathasarathy in [HP].
$a^{*}(\xi)a(\xi)$$q$$\sqrt{\lambda}(a^{*}(\xi)+a(\xi))$$q$$\lambda$

3The moments of the -Poisson random variable$q$

moment of the -Poisson random variable of the parameter Acan beThegiven as the monic polynomial in Aof degree without constant. Thus we canmoment as in the form,
put the$n\mathrm{t}\mathrm{h}$

$q$

$n$

$n\mathrm{t}\mathrm{h}$$\phi(x^{n})=\sum_{k_{-}^{-}1}^{n}S_{q}(n, k)\lambda^{k}$,where $S_{q}(n, k)$ is the some constant.
In order to evaluate the constants $S_{q}(n, k)$ , we shall recall the set partition
statistic, the number of restricted crossings which was introduced in [F1] andalso studied in [Bi]. Here, we will call it the number of arc crossings because
we would like to use the terminology arcs.
if
We call the set $\pi=\{B_{1}, B_{2}, \ldots, B_{k}\}$ is the partition of 1, 2, ,ablock ofare disjoint sets, of which union is 1, 2, , . We shall callthe set of partitions
the partition . For $n\geq 1$ , we denote by.
ofDefinition 3.1. Let $\pi=\{B_{1}, B_{2}, \ldots, B_{k}\}$ be apartition in $P(\{1, \ldots,n\})$ . If theblock has more than one elements (i.e. $|Bj|=mj\geq 2$ ), put $B_{j}=$ {&j,i, {1,2,$\{$$\ldots$$\{$$\ldots$$\pi$$\{1, 2, \ldots, n\}$$B_{j}$$n\}$$\mathcal{P}(\{1, 2, \ldots, n\})$$B_{i}\in\pi$$n\}$$B_{i}’ \mathrm{s}$

221
...,
} where $b_{j,1}<b_{j,2}<\ldots<b_{j,m_{j}}$ , then we make $(m_{j}-1)$ connections, successively. We have, of, ...,
like bridges (course, totally $\sum_{j=1}^{k}(]B_{j}|-1)$ connections and we shall call them arcs of thepartition .
is the number:
The number of arc crossings for apartition$b_{j,m_{J}}$$b_{j,1},$$(b_{j,m_{j}-1}, b_{j,m_{j}})$$(b_{j,2}, b_{j,3})$$\mathrm{b}\mathrm{j}|2,$

$\pi$

$\pi$

$\in \mathcal{P}(\{1, \ldots, n\})$,$c_{a}(\pi)=\#$$\{(m_{1}, m_{2},m_{3},m_{4})|\mathrm{o}\mathrm{f}\pi(m_{1},m_{3})\mathrm{a}\mathrm{n}\mathrm{d}1\leq m_{1}<m_{2}(m_{2},m_{4})\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}<m_{3}<m_{4}\leq n$

$\}$

.

Theorem 3.2.
$S_{q}(n, k)=$$q^{c_{a}(\pi)}$$\sum$,$\pi\in P(\{.1,\ldots,n\})$

yr

where

$c_{a}(\pi)$

has precisely$k$blocksdenotes the number of the arc crossings for the partition

$\pi$

.

In order to see the above statement, we shall adopt the cards arrangement
technique which is similar as in [ER] for juggling patterns but we will use
considerably different kinds of cards.
$(i=0,1,2, \ldots)$ for the -creation operNotation 3.3. We prepare the cards
has inflow lines from the left and $(i+1)$ outflow lines toator. The card
the right, where one new line starts from the middle point on the ground level.
level
level goes out to the
For each $j\geq 1$ , the inflow line of thethe creation card of level (See Fig.
without any crossing. We call the card
$C_{i}$$q$$i$$C_{i}$$(j+1)\mathrm{s}\mathrm{t}$$j\mathrm{t}\mathrm{h}$$i$$C_{i}$

3.1).

[Level 3][Level 0]

Fig. 3.1. The creation cards and the weights for the first few levels
Next we shall make the cards for the -annihilation operator. For $i\geq 1$ ,$(j=1,2, \ldots, i)$ which has
inflow lines from the left
we consider the cards, only the inflow line
and $(i-1)$ outflow lines to the right. On the card
level and willthegroundonlevel goes down to the middle pointof thelevel go in horizontallybe annihilated. The lines flowed into lower thanparallel and keep the levels. Hence $(j-1)$ crossings will occur. Moreover if thelevel withoutlevel, it will flow out to theline flow into the$q$$A_{i}^{(j)}$$i$$A_{i}^{(j)}$$j\mathrm{t}\mathrm{h}$$j\mathrm{t}\mathrm{h}$$k(>j)\mathrm{t}\mathrm{h}$$(k-1)\mathrm{s}\mathrm{t}$222
any crossing. We call the cards
the annihilation cards of level (See Fig.
3.2).
We shall give the weight to the card by to the number of the crossingsthat occur on the card, thus the card
has the weight
and the cardhas 1.
$A_{i}^{(j)}$

$i$

$q$

$A_{i}^{(j)}$$q^{j-1}$$C_{i}$

Fig. 3.2. The annihilation cards and the weights of the first few levels
Remark 3.4. The creation cards represent the relations
where the number of lines correspond to the power of tensor products.
The annihilation cards reflect the relations$a^{*}\xi^{\otimes i}=\xi^{\otimes(:+1)}(i\geq 0)$$a\xi^{\otimes i}$,$=$where the annihilating line indicates the position of the factor in the tensor
product which should be deleted

223$(j=1,2, \ldots, i)$ for the
Notation 3.5. For $i\geq 1$ , we consider the cards
has inflow lines and, the same number of,-number operator. The card
level goes down to the middle
lineof theOnly
theinflow
outflow lines.
point on the ground and continue its flow as the first (lowest) line. The inflow
level will be increased only one in upper level and oneslines of lower thanlevel will keep their levels. Hence it will occur $(j-1)$
of higher than
. We call the card
, hence, it has the weightcrossings on the card
the number card of level (See Fig. 3.3).
$N_{i}^{(j)}$$N_{i}^{(j)}$

$q$

$i$

$i$

$j\mathrm{t}\mathrm{h}$$j\mathrm{t}\mathrm{h}$$j\mathrm{t}\mathrm{h}$$N^{(j)}.\cdot$$q^{j-1}$$N_{i}^{(j)}$$i$[Level 2][Level 1]

Fig. 3.3. The number cards and the weights of the first few levels
can be obtained by
Remark 3.6. It is obvious that the number cardwithand the creation cardcomposition of the annihilation cardtheinstance,bygluingFor
theonground.
pointsidentifying the two middle$N_{i}^{(j)}$$A_{i}^{(j)}$$C_{i-1}$

224
cards

A4 and(3)$C_{3}$, we have the card

$N\ovalbox{\tt\small REJECT}^{3)}$4as follows:$\Rightarrow$

Notation 3.7. Furthermore, we shall make the scalar cards $K_{j}(i=0,1,2, \ldots)$for the scalar operator. The card K. has horizontally parallel lines and theshort pole like segment of line at the middle point on the ground without any
the scalar card of levelcrossing, thus it has the weight 1. We call the card
(See Fig. 3.4).
$i$$\cdot$$K_{i}$$i$Fig. 3.4. The scalar cards and the weights of the first few levels
In order to evaluate the yrth moments, we expand$x^{n}=(N+\sqrt{\lambda}a^{*}+\sqrt{\lambda}a+\lambda\cdot 1)^{n}$and consider in amonomial wise. Now we consider the monomial$y=f_{n}f_{n-1}\cdots f_{2}f_{1}$,. 1 . Here we should be, and,is one of ,where eachaware of that the position numbers are labeled from the right. Put the sets as$N$$f_{j}$$N(y)$$=\{j :

$A(y)$$=\{j :$(\sqrt{\lambda}a^{*})$

f_{j}=N\}$$(\sqrt{\lambda}a)$,f_{j}=(\sqrt{\lambda}a)\}$,and$(\lambda$$)$$C(y)$$=\{j :

f_{j}=(\sqrt{\lambda}a^{*})\}$$K(y)$$=\{j :

f_{j}=(\lambda\cdot 1)\}$,,, $kc=\#_{C(y)}$ , $k_{A}=\#_{A(y)}$ , and $k_{K}=\#_{K(y)}$ . It isandtrivial that we have the disjoint union of$\mathrm{d}\mathrm{e}_{1}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}k_{N}=\#_{N(y)}$$\{1, 2, \ldots, n\}=N(y)\cup C(y)\cup A(y)\cup K(y)$225
and $k_{N}+k_{C}+k_{A}+k_{K}=n$ .
For the monomial , we define the level of theby$k\mathrm{t}\mathrm{h}$$y$$\mathrm{L}(\mathrm{k})=\sum_{j=1}^{k-1}\mathrm{x}(\mathrm{j})$where$\mathrm{x}(\mathrm{j})$factor$f_{k}$, $L(k)(k\geq 2)$ ,with $L(1)=0$ ,,is the step function defined by,
1if$\chi(j)=\{$ -1,
$j\in
A(y)$
,
if
0if $j\in N(y)\cup K(y)$ .
$j\in \mathrm{C}(\mathrm{y})$Then, we can easily obtain that the monomial
tation if and only if the following conditions:(a)$L(k)\geq 0$

for every $1\leq k\leq n$ ,(b)$\mathrm{L}(\mathrm{k})\geq 1$

if

(c)

$k\in \mathrm{N}(\mathrm{y})$$\sum_{j=1}^{n}\chi(j)=0$$y$

has non-zero vacuum expec-

, and

.

Especially, we have that $\phi(y)\neq 0$ implies $k_{A}=kc$ .
Let
have

$y$

be the monomial of non-zero vacuum expectation as above, then we$\phi(y)=(\prod_{k\in A(y)\cup N(y)}[L(k)]_{q})\lambda^{(k_{K}+k_{C})}$

where$[x]_{q}$denotes the -number. Because the level$q$$\mathrm{L}\{\mathrm{k}$$(f_{k-1}f_{k-2}\cdots f_{1})\Omega\in \mathbb{C}\xi^{\otimes L(k)}$,) reflects the fact.

For the monomial of non-zero vacuum expectation, we will make the set, of the ordered set 1, 2, , as follows: The partitionof partitions,has $(k_{K}+k_{C})$ blocks where each number in the set $K(y)$ makes ablock of size
blocks starts from anumber in $C(y)$ and ends by a1and each of the rest
$A(y)$, and the intermediate numbers are in the set $N(y)$ . Of course,
number in
we consider the numbers in each block are arranged in increasing order. Thatis,$y$$\Psi_{n}(y)$$\{$$\ldots$$n\}$$k_{C}$$\Psi_{n}(y)=|\pi$$=\{B_{1}, B_{2}, \ldots, B_{k}\}$$\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{b}1\mathrm{o}\mathrm{c}\mathrm{k}B_{j}=\{b_{j\mathrm{J}},b_{j,2},\ldots,b_{j,m}\}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}m_{j}=|B_{j}|\mathrm{a}\mathrm{n}\mathrm{d}b_{j},<b_{j,\ell+1}k=k_{C}+k_{K},\pi \mathrm{i}\mathrm{s}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{f}m_{j}--\mathrm{l}\mathrm{t}^{t}\mathrm{h}\mathrm{e}\mathrm{n}b_{j,1}\in b_{j,m}\in A(y),\mathrm{a}\mathrm{n}\mathrm{d}b_{j,\ell}\in N(\acute{y})\mathrm{f}\mathrm{o}\mathrm{r}\ell\neq \mathrm{A}’(y),\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{f}m_{j}\geq 2\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}b_{j1}\in C(y)1,m_{j}^{J}.j’,’\}$.

226
Proposition 3.8. Let
expectation, where each2,
be the subset of

be a monomial of non-zero vacuum, and (A .1), and,is one of N,
as described above. Then we have that$f_{n}f_{n-\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} j/\ovalbox{\tt\small REJECT}$$f_{2}fr$$(\mathrm{J}_{\ovalbox{\tt\small REJECT}}’)$$I\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} \mathrm{P}(\{1,$$\mathrm{r}\mathrm{z}\})$\ldots ,$\phi(y)=(\sum_{\pi\in\Psi_{n}(y)}q^{c_{a}(\pi)})\lambda^{(k_{K}+k_{C})}$

where$c_{a}(\pi)$$\ovalbox{\tt\small REJECT} \mathrm{I}_{n}^{\ovalbox{\tt\small REJECT}}(\mathrm{y})$$(\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT})$,is the number of arc crossings for the partition

$\pi$

.

For the monomial of non-zero vacuum expectation, we will arrange
the cards along with the following rule: If $k\in C(y)$ , that is, if
position with thethen we will put the creation card of level $L(k)$ at the
$k\in
A(y)$
,
that is, if-multiplicated weight, which is unique. If
position. At
then we will put the annihilation card of level $L(k)$ at thethis time, $L(k)$ cards are available for our arrangement and the weights should. If $k\in N(y)$ , that is, if $f_{k}=N$ then we will use
be also multiplicated by
the number card of level $L(k)$ with the original weight, where $L(k)$ cards arethen the scalar card of levelalso available. If $k\in K(y)$ , that is, if
position.
$L(k)$ with the weight Awill be used at theWe will consider all the admissible cards arrangements then it is trivial
that there are $\prod_{k\in A(y)\cup N(y)}L(k)$ ways of cards arrangements. We also obtaineasily that the sum of the products of the weights for all the admissible cards
arrangements can be given as

Proof.

$y$$f_{k}=(\sqrt{\lambda}a^{*})$$k\mathrm{t}\mathrm{h}$$f_{k}=(\sqrt{\lambda}a)$$\sqrt{\lambda}$$k\mathrm{t}\mathrm{h}$$\sqrt{\lambda}$$f_{k}=(\lambda\cdot 1)$$k\mathrm{t}\mathrm{h}$$\lambda^{k_{K}}(\sqrt{\lambda})^{(k_{C}+k_{A})}(\prod_{k\in A(y)\cup N(y)}[L(k)]_{q})=\lambda^{(k_{K}+k_{C})}(\prod_{k\in A(y)\cup N(y)}[L(k)]_{q})$,because we know $k_{A}=k_{C}$ . By the rule for our cards arrangements, it follows
that the partitions determined by the connected lines in the pattern of theare in aone-t0-0ne corresponabove cards arrangements and the set
dence.
It is not so difficult to see that the crossings appeared in the patterns of
the cards arrangements are nothing else but the arc crossings for the partitions
determined by the connected lines.
We remind, now, how to give the weights to the cards, then it follows that$\Psi_{n}(y)$$\prod_{k\in A(y)\cup N(y)}[L(k)]_{q}=\sum_{\pi\in\Psi_{\hslash}(y)}q^{c_{a}(\pi)}$,

which ends the proof.
Example 3.8. We consider the monomial$(\sqrt{\lambda}a)(\sqrt{\lambda}a)N(\sqrt{\lambda}a^{*})(\lambda\cdot 1)(\sqrt{\lambda}a^{*})=f_{6}f_{5}f_{4}f_{3}f_{2}f_{1}$$\square$,227 227
which has non-zero expectation with the sequence of the levels $\{L(k)\}_{k=1}^{6}=$$\{0,1,1,2,2,1\}$ . This monomial yields the four ways of the cards arrangement(Co,$K_{1},$ $C_{1},$$N_{2}^{(\alpha)},$$A_{2}^{(\beta)},$$A_{1}^{(1)}$),

where $\alpha=1,2$ and $\beta=1,2$ .

We shall list all the admissible ways of the cards arrangements with the corresponding partitions and the products of the weights.
(i)$\pi=\{\{1,6\}, \{2\}, \{3,4,5\}\}$,

Product of weights$=\lambda^{3}$$=\{\{1,5\}, \{2\}, \{3,4,6\}\}$,

Product of weights$=q\lambda^{3}$

Product of weights$=q\lambda^{3}$

(i)

$\pi$

(i)

$\pi$

$=\{\{1,4,5\}, \{2\}, \{3,6\}\}$

,

228
(iv)

$.\pi=\{\{1,4,6\}, \{2\}, \{3,5\}\}$,

Product of weights$=q^{2}\lambda^{3}$

The vacuum expectation of this monomial is $[2]_{q}[2]_{q}[1]_{q}\lambda^{3}=(1+2q+q^{2})\lambda^{3}$ ,which is given by the sum of the products of the weights listed above.
Remark 3.9. The numbers $S_{q}(n,k)$ has been already investigated in [Bi] as acertain -deformation of the Stirling number of the second kind. He introduced
them as the coefficients of the generating function for the continued fraction,
which corresponds to the orthogonal polynomials for our -deformed Poisson
distribution. Hence our card arrangement method is nothing else but thecertain graphical interpretation for his -deformation, which is intermediatedby the -cration and -annihilation operators.
$q$

$q$

$q$

$q$

$q$

4A -deformed moments-cumulants relation$q$

Having the result of the previous section in mind, we should like to propose
the following -deformed moment-cumulant relation:$q$$\mu_{n}=$$\sum_{\pi\in \mathcal{P}(\{1,\ldots,n\})}q^{c_{\Phi}(\pi)}\prod_{i_{-}^{-}1}^{k}\alpha_{|B_{i}|}$,$n\geq 1$,$\pi=\{B_{1\prime}B_{2\prime\cdots\prime}B_{k}\}$is the number of arc crossings of the partition , which interpolates
where
the relations for usual and for free exactly.
Theorem 4.1. The $nth$ moments of our -deformed Poisson random variable.
of parameter Acan be given as the above relation by putting$c_{a}(\pi)$

$\pi$

$q$

$\alpha_{i}=\lambda(i\geq 1)$

This result is compatible with the characterization for the usual Poisson
random variable that the Poisson distribution has the constant cumulant inall orders. Furthermore, the above moments-cumulants relation characterizes
for $i\geq 3$ , which interpolates the usualthe -Gaussian law by putting
Gaussian law and the semi-circle one$q$$\alpha_{i}=0$229
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$q$

$q$

$q$

$q$