Multiplication and Translation Operators on the Fock Spaces for the q-Modified Bessel Function *

We study the multiplication operator M by z2 and the q-Bessel operator Δq,αon a Hilbert spaces Fq,α of entire functions on the disk D( o, ) , 0<q<1 ; and we prove that these operators are adjoint-operators and continuous from Fq,α into itself. Next, we study a generalized translation operators on Fq,α .

on a Hilbert spaces   of entire functions on the disk ; and we prove that these operators are adjoint-operators and continuous from dz and the multiplication operator by , and proved that these operators are densely defined, closed and adjoint-operators on (see [1]).
z Next, the Hilbert space is called Segal-Bargmann space or Fock space and it was the aim of many works [2].
In 1984, Cholewinski [3] introduced a Hilbert space    of even entire functions on , where the inner product is weighted by the modified Macdonald function.On   the Bessel operator and the multiplication by are densely defined, closed and adjoint-operators.
In this paper, we consider the - modified Bessel function:  q , q are given later in Section 2. We define the -Fock space as the Hilbert space of even entire functions z az of center o and radius

:
; Let f and g be in , the inner product is given by g Next, we consider the multiplication operator M by and the q -Bessel operator , and we prove that these operators are continuous from * Author partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503.


into itself, and satisfy: Then, we prove that these operators are adjointoperators on , q   : , , , = , ; , Lastly, we define and study on the Fock space   q , the -translation operators: and the generalized multiplication operators: Using the previous results, we deduce that the operators into itself, and satisfy: ;

The -Fock Spaces
Let and be real numbers such that ; the q-shifted factorial are defined by Jackson [5] defined the q-analogue of the Gamma function as and tends to x   when tends to 1 .In particular, for , we have The q-combinatorial coefficients are defined for The -derivative of a suitable function f (see [6]) is given by Taking account of the paper [4] and the same way, we define the -I  modified Bessel function by : If we put . In [4], the authors study in great detail the -Bessel operator denoted by : .
is the unique analytic solution of the q-problem: The constants , satisfy the following relation: is the Hilbert space of even entire functions where is given by (2) .
The inner product in , agrees with the generalized Fock space associated to the Bessel operator (see [3]).

Multiplication and -Bessel Operators on
On   , we consider the multiplication operators M and given by We denote also by   the -Bessel operator defined for entire functions on We write By straightforward calculation we obtain the following result.

  
Remark 3: The Lemma 2 is the analogous commutation rule of Cholewinski [3].When , is the identity operator.
Lemma 3:  , and and from (4), we obtain = ; ; 2 n n q a n q z On the other hand from (6), we have and which completes the proof of the Lemma.

Generalized Multiplication and Translation Operators on
In this section, we study a generalized multiplication and translation operators on , we define: -The -translation operators on ; -The generalized multiplication operators on ; .; I , the function q satisfies the following product formulas: .; = ; ; z I q w I z q I w q .; = ; ; z M I q w I wz q I w q , we obtain the generalized translation operator given in ( [3], page 181).

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On the other hand from (1) and (2), we get