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 Advances in Pure Mathematics, 2011, 1, 221-227 doi:10.4236/apm.2011.14039 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Multiplication and Translation Operators on the Fock Spaces for the q-Modified Bessel Function* Fethi Soltani Higher College of Technology and Informatics, Street of the Employers, Tunis, Tunisia E-mail: fethisoltani10@yahoo.com Received February 28, 2011; revised April 8, 2011; accepted April 20, 2011 Abstract M by and the -Bessel operator 2zWe study the multiplication operator q,q,q on a Hilbert spaces  of entire functions on the disk 1q,1Do0< <1q,q, ; and we prove that these operators are adjoint-operators and continuous from ,q into itself. Next, we study a generalized translation operators on qq I. Keywords: Generalized -Fock Spaces, -Modified Bessel Function, -Bessel Operator, Multiplication Operator, -Translation Operators qq=0nnn1. Introduction In 1961, Bargmann [1] introduced a Hilbert space of entire functions =fzaz on  such that 20:!nnfan2 On this space the author studied the differential operator =dDdz and the multiplication operator by , and proved that these operators are densely defined, closed and adjoint-operators on (see [1]). zNext, 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 2d212d:=,>1 2ddzzz 2zq Iand the multiplication by are densely defined, closed and adjoint-operators. In this paper, we consider the - modified Bessel function:  222=0 2;: ;nnnxIxq bq where 22;nbqq,q are given later in Section 2. We define the -Fock space 2=0=nnn as the Hilbert space of even entire functions zaz on the disk f1,1Do q of center o and radius 11q, and such that ,22220:;qnnnfabq,q Let f and g be in , such that n2=0=nnfzaz2=0=nnn and zcz, the inner product is given by g,220,= ;qnn nnfgacb q Next, we consider the multiplication operator M by and the q-Bessel operator ,q2z on the Fock space ,q,q, and we prove that these operators are continuous from *Author partially supported by DGRST project 04/UR/15-02 and CMCU program 1 0 G 1 50 3 .  into itself, and satisfy: 222 F. SOLTANI ,11qffq,,qq ,,11qqfqMf Then, we prove that these operators are adjoint- operators on ,q: ,,,=,qqMf gfg,,;,qqf g,q Lastly, we define and study on the Fock space q, the -translation operators: 1/2 2,:= ;; ,zqTf wIzqf wwz1,1D oq and the generalized multiplication operators: 12 2:;; zMfwI zMqfwwz1,,.1Do q Using the previous results, we deduce that the operato rs zT and zM, for 1,1zDo q,q, are continuous from  into itself, and satisfy: ,||1qzzTf I,2;qq fq ,,2;qqq fq||1zzMf Iqq,αaq0< <1q,=1,2,,n  2. The -Fock Spaces Let and be real numbers such that ; the q-shifted factorial are defined by  10=0;:1, ;:1niniaq aqaq  Jackson [5] defined the q-analogue of the Gamma function as  1;:=1 ,;xqxqqxqqq0, 1, 2,x It satisfies the functional equation  11=1xqqxxq , 1=1qq and tends to x when tends to 1. In particular, for , we have q=1,2,n;1=1nnqqq,nkqn The q-combinatorial coefficients are defined for =0,,kn,  , by  ;1:=;;1 1qnqqqknkqq nnkqqqqkn k qqDf (1) The -derivative of a suitable functio n f (see [6]) is given by :,01qfx fqxDf xxqx 0= 0qDf f provided exists. 0ffqDf xand If is differentiable then tends to fx as 1qq. Taking account of the pap er [4] and the same way, we define the -I modified Bessel functio n by  222=0 2;: ;nnnxIxq bq  where 2222221(1) 1;: 1nqqnqqn nbq  (2) If we put 221=;nnUbq, then 211,11nnUqUqq IThus, the - modified Bessel function is defined on 21,1DoqI and tends to the  modified Bessel function as 1qq . In [4], the authors study in great detail the -Bessel operator denot ed by 2,[2 1]:qqq qfxDfx Dfqxx  where 211[21] :1qqqq The -Bessel operator tends to the Bessel operator 1q as 2.;. Lemma 1: 1) The function Iq,1,1Do q, is the unique analytic solution of the q-problem: 2,=,0=1 0=0qqyxyxyand Dy (3) Copyright © 2011 SciRes. APM F. SOLTANI Copyright © 2011 SciRes. APM 223n 2) For , we have 22,22( 1);;nqnbqzzbq22(1)=,1nnn2n 3) The constants , satisfy the following relation: 2;nbq2222 ;nnqqbq22 ;=2222bqn n Let 12 ,. The q-Fock space q is the Hilbert space of even entire functions 2=0=nnnfzaz on 1,1Do q, such that 2;