1. Introduction and preliminaries

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1 t m Mathematical Publications DOI: /tmmp Tatra Mt. Math. Publ. 66 (216), 81 9 THE q-gamma WHITE NOISE Hakeem A. Othman ABSTRACT. For <q<1and<α<1, we construct the infinite dimensional q-gamma white noise measure γ α,q by using the Bochner-Minlos theorem. Then we give the chaos decomposition of an L 2 space with respect to the measure γ α,q via an isomorphism with the 1-mode type interacting Fock space associated to the q-gamma measure. 1. Introduction and preliminaries White noise can be informally regarded as a stochastic process which is independent at different times and is identically distributed with zero mean and infinite variance. T. H i d a introduced the theory of white noise in Nowadays, this theory has been applied to stochastic integration, stochastic partial differential equation, stochastic variational equation, infinite dimensional harmonic analysis, Dirichlet forms, quantum field theory, Feynman integral and quantum probability. For the non-gaussian white noise analysis, Y. I t o constructed the Poissonian counterpart of Hida s theory and K o n d r a t i e v et al. [8] established a purely non-gaussian distribution theory in infinite dimensional analysis by means of a normalized Laplace transform, and B a r h o u m i et al. [1] developed the introduction of infinite dimensional Gegenbauer white noise. On the other hand, a q-deformation of some orthogonal polynomials and their associated measure were introduced. We put for n N, [n] q : 1 qn 1 q 1+q + q2 + + q n 1 ([] q : ). The q-factorial is [n] q!:[1] q [2] q...[n] q, [] q!:1. c 216 Mathematical Institute, Slovak Academy of Sciences. 21 M a t h e m a t i c s Subject Classification: 6H4, 46A32, 46F25, 46G2. Keywords: q-gamma distribution, q-laguerre polynomials, q-gamma noise space, 1-mode type interacting Fock space. 81

2 HAKEEM A. OTHMAN Another quite frequently used symbol is the q-analogue of the Pochhammer symbol: n 1 (a; q) : 1 and (a; q) n : (1 aq j ), n 1, 2,...,. It is easy to see that for all integers n (a; q) n (See [7], [9]). The q-gamma function j (a; q) (aq n ; q). Γ q (x) (q; q) (1 q) 1 x, <q<1, x, 1, 2,... (q x ; q) was introduced by T h o m a c in 1869 and later by J a c k s o n in 194. In 1847, H e i n e gave an equivalent definition, but without the factor (1 q) 1 x. When x n +1withn a nonnegative integer, this definition reduces to Γ q (n +1) 1(1+q)(1 + q + q 2 )+ +(1+q + q q n 1 ) [n] q!, which clearly approaches n! as q 1. The q-gamma distribution is given by Γ q ( α) dμ α,q (x) Γ(α +1)Γ( α) xα e q ( x)i ],+ [ (x) dx and we can easily verify that its Fourier transform is given by: μ α,q (λ) Γ q ( α) Γ(α +1)Γ( α) n e ixλ x α e q ( x) dx Γ(α + n +1)Γ( α n)γ q ( α) Γ(α +1)Γ( α)γ q ( α n) (iλ) n. n! The present paper elaborates an introduction of the q-gamma white noise. The outline of our paper is as follows. In Section 2, we use the q-gamma function to construct a standard nuclear triplet E α,q H α,q : L 2 (I,μ α,q ) by the Bochner-Minlos theorem. In Section 3, we define the q-gamma white noise measure γ α,q on (, B(E α,q )) and we define the q-type wick product by means of an orthogonal system of infinite dimensional q-laguerre polynomials, and we investigate the chaos decomposition of the space L 2(, B(),γ α,q ). 82

3 THE q-gamma WHITE NOISE 2. q-gamma white noise measure Let { Γ q ( α) dμ σ Γ(α+1)Γ( α) σ x α e α,q(x) α+1 q ( x σ )I ],+ [(x) dx, if σ, μ α,q (dx) δ (x) if σ and < α < 1. Note that forσ 1, we have the standard q-gamma distribution: Γ q ( α) μ α,q (dx) Γ(α +1)Γ( α) xα e q ( x)i ],+ [ (x) dx. In [7], the q-laguerre polynomials are defined by L α n (x; q) (qα+1 ; q) n 1 φ 1 (q n ; q α+1 ; q; xq n+α+1 ), (q; q) n where the basic hypergeometric series r φ s have the following forms: [ ] a1,a 2,...,a r rφ s (a 1,a 2,...,a r ; b 1,b 2,...,b s ; q, z) r φ s ; q, z b 1,b 2,...,b s [ (a 1 ; q) n (a 2 ; q) n,...,(a r ; q) n n 1+s r ] (q; q) n (b 1 ; q) n (b 2 ; q) n,...,(b s ; q) n ( 1) n 2 z n q with ( ) n n(n 1). 2 2 They are orthogonal with respect to μ α,q since they verify L α m (x; q)lα n (x; q) dμ α,q(x) (qα+1 ; q) n (q; q) n q n δ m,n. Therefore, if we put ( (q h n,q (x) L α α+1 ) n (x; q) ; q) 1 2 n, (q; q) n q n we obtain an orthonormal basis {h n,q, n N} for the Hilbert space H α,q L 2 (I,μ α,q ), with I ], + [. Now, define the operator A q,onh α,q by ( Aq L α n(., q) ) ( ) ( ) x 1 q (x) xdql 2 α α+1 n q ; q q α+1 x D q L α 1 q n(x; q), where D q f(x) f(qx) f(x) x(q 1). It is easy to show that for all n N, the function h n,q is an eigenvector of A q,namely A q h n,q λ q,n h n,q, 83

4 HAKEEM A. OTHMAN with λ q,n q α+1 n/2 [n] q. Moreover, for any p>, A p q is the Hilbert-Schmidt operator satisfying: A p q 2 HS λ 2p q,n <. n For each p R, define the norm. p on H α,q by f p A p q f λ 2p n q,n f,h n,q 2 1/2, f H α,q, where. and.,. are the norm and the inner product of H α,q, respectively. For p, let E p,α,q be the Hilbert space consisting of all f H α,q with f q <, and E p,α,q, the completion of H α,q with respect to. p.sincea 1 q is of the Hilbert-Schmidt type, identifying H α,q with its dual space we come to real standard nuclear triplet E α,q : E p,α,q H α,q E p,α,q : E α,q. p p Lemma 2.1. The function C(ξ) ˆμ α,q ( ξ ), ξ E α,q is a characteristic function, where ξ ξ(x) dx. I P r o o f. Obviously, C is continuous on E α,q and C() 1. We shall prove that C is positive definite. For z 1,...,z n C and ξ 1,...,ξ n E α,q,wecanseethat z j z k C(ξ j ξ k ) z j z k ˆμ α,q ( ξ j ξ k ) j,k1 where l j (x) z j e ix ξ j. j,k1 j,k1 z j z k j,k1 j1 e ix ξ j e ix ξ k d μα,q (x) l j (x)l k (x) d μα,q (x) 2 l j (x) d μα,q (x), Definition 2.2. The probability measure over E α,q, denoted by γ α,q, is defined by its Fourier transform (via the Bochner-Minlos Theorem) given by: E α,qe i ω,ϕ dγ α,q (ω) Γ(α + n +1)Γ( α n)γ α,q ( α) (i ϕ ) n. Γ(α +1)Γ( α)γ q ( α n) n! n 84

5 THE q-gamma WHITE NOISE We denote γ α,q the q-gamma noise measure and the space (E α,q,γ α,q )bythe q-gamma space. Proposition 2.3. For ξ E α,q s.t ξ let X ξ be the random variable defined on (E α,q,b(),γ α,q ) by X ξ (ω) : ω, ξ, where B() is the σ-algebra on. ThenX ξ has a q-gamma distribution with parameters α and ξ. Proof. Wehave e iλxξ(ω) dγ α,q (ω) C(λξ) μ(λ ξ ) μ ξ α,q(λ), λ R. It proves that the distribution of the random variable X ξ measure μ α,q ξ on R. is the probability Lemma 2.4. For all ξ H α,q we have ω, ξ k dγ α,q (ω) Γ(α + k +1)Γ( α k)γ q( α) Γ(α +1)Γ( α)γ q ( α k)k! ξ k. P r o o f. We know that x k dμ α,q (x) (ˆμ α,q(λ)) (k) i k Γ(α + k +1)Γ( α k)γ q( α). λ Γ(α +1)Γ( α)γ q ( α k)k! Then, we get ω, ξ k dγ q (ω) y k dμ ξ α,q(y) ξ k x k dμ α,q (x) Γ(α + k +1)Γ( α k)γ q( α) Γ(α +1)Γ( α)γ q ( α k)k! ξ k and it completes the proof. 85

6 HAKEEM A. OTHMAN 3. q-gamma isomorphism Let us begin by introducing the q-type Gamma wick product. Definition 3.1. For ω E α,q and n, 1, 2,...,we define the q-type Gamma wick product: : ω n : q (E α,q) n as follows: ( : ω n : q, ϕ n ϕ n ω, ϕ Lα n ϕ ), ; q ϕ E α,q. (3.2) Lemma 3.3. For all ξ E α,q such that ξ,then : ω n : q, ξ n : ω m : q, ξ m dγ α,q (ω) ξ 2n (qα+1 ; q) n (q; q) n q δ n m,n. (3.4) Proof. For ξ H α,q,ξ, the image of the q-gamma white noise measure γ α,q under the map ξ ω ω, R, ω E α,q ξ is the q-gamma distribution μ α,q.thenwehave : ω n : q, ξ n : ω m : q, ξ m dγ α,q (ω) ξ n+m L α n (x; q)lα m (x; q) dμ α,q(x) ξ 2n This gives the desired statement. (q; q) n q n δ m,n. Let us denote F α,q (H α,q ) the following 1 mode type interacting Fock space: F α,q (H α,q ): (q; q) n q n H n α,q. Thus F α,q (H α,q ) consists of sequences f (f,f 1,...) such that for any n N f n H n α,q and 86 f 2 F α,q (H α,q ) (q; q) n q n f n 2 H α,q n <.

7 THE q-gamma WHITE NOISE Theorem 3.5. For each F L 2 (,γ α,q), there exists a unique sequence f (fn ) F α,q(h α,q ) such that F :. n : q, f n (3.6) in the L 2 -sense. On the other hand, for any f (f n ) F α,q (H α,q ), (3.6) defines a function on L 2 (,γ α,q). In that case, F 2 L 2 (,γ α,q) (q; q) n q n f n 2 H α,q n f 2 F α,q (H α,q ). The following unitary operator is called the q-gamma isometry I : F q (H α,q ) L 2 (,γ α,q ), (f n ) F. P r o o f. It is easy to see that the set { } P(E α,q) φ, φ(ω) : ω k : q, φ k,φ k Eα,q, k ω E α,q, n N k of smooth continuous polynomials on E α,q is continuously and densely embedded in L 2 (E α,q,γ α,q). Then, for any F L 2 (E α,q,γ α,q) there exists a unique sequence f (fn ) F α,q(h α,q ) such that that F :. n : q, f n. We need to show for two polynomials φ, ψ given by, respectively, φ(ω) : ω n : q, φ n, ψ(ω) : ω n : q, ψ n φ(ω)ψ(ω) dγ q (ω) (q; q) n q n φ n,ψ n. (3.7) In particular, the L 2 -norm of φ with respect to γ α,q is given by φ 2 (q; q) n q n φ n 2. (3.8) 87

8 HAKEEM A. OTHMAN For this and since φ n and ψ n are linear combinations of elements of the form ξ n,ξ E α,q, it suffices to show that for ξ 1,ξ 2 E α,q, : ω n : q, ξ1 n : ω m : q, ξ2 m dγ α,q (ω) ξ 1,ξ 2 n (qα+1 ; q) n (q; q) n q δ n m,n. (3.9) It is sufficient to prove the identity under the assumption ξ η 1. Taking a unit vector ζ E α,q such that ξ, ζ,wemaywrite η βξ + λζ, β 2 + λ 2 1, and we have : ω n : q, η n k ( ) n β n k λ k : ω (n k) : q, ξ (n k) : ω k : q, ζ k. k Then we get : ω m : q, ξ m : ω n : q, η n dγ α,q (ω) ( ) n β n k λ k : ω m : q, ξ m : ω (n k) : q, ξ (n k) k ko E α,q : ω k : q, ζ k dγ α,q (ω). On the other hand, by using the independence of the two random variables., ξ and., ζ, weobtain : ω m : q, ξ m : ω (n k) : q, ξ (n k) : ω k : q, ζ k dγ α,q (ω) : ω m : q, ξ m : ω (n k) : q, ξ (n k) dγ α,q (ω) : ω k : q, ζ k dγ α,q (ω). Therefore, the last integral is equal to unless k and is 1 if k. Hence, : ω m q, ξ m : ω n : q, η n dγ α,q (ω) β n (qα+1 ; q) n (q; q) n q n : ω m : q, ξ m : ω n : q, ξ n dγ α,q (ω). 88

9 THE q-gamma WHITE NOISE We conclude that : ω m : q, ξ m : ω n : q, η n dγ α,q (ω) β n (qα+1 ; q) n (q; q) n q δ n m,n. Since β ξ, η, we have completed the proof of (3.9). Then, because of this and φ n and ψ n are linear combinations of elements of the form ξ n, ξ E α,q for all φ n,ψ n E n α,q we have: : ω n : q, φ n : ω m : q, ψ m dγ α,q (ω) (qα+1 ; q) n (q; q) n q n φ n,ψ n δ m,n. It follows from (3.7) and (3.8) that ( 2 F 2 L 2 (E α,q,γ α,q) : ω n : q, f n ) dγ α,q (ω) (q; q) n q n f 2 F α,q (H α,q ). The second part of the Theorem 3.5 is straightforward. f n,f n Hα,q n Remark 3.1. In this paper we constructed the infinite dimensional q-gamma white noise, in particular the q-gamma Gel fand triple was obtained. We can use it to develop a new theory of the q-gamma white noise as analogue of Hida s theory in the white noise setting. Also a space of holomorphic functions on the complexification of E α,q with θ-exponential growth which is the analogue of the Gaussian case (where θ is the Young function) can be introduced. We expect to develop a new quantum q-gamma white noise operator theory as an analogue of the quantum white noise operator theory, (see [2] [6], [11] [13]). REFERENCES [1] BARHOUMI, A. OUERDIANE, H. RIAHI, A.: Infinite dimensional Gegenbauer functionals, Banach Center Publ. 78 (27), [2] BARHOUMI, A. LANCONELLI, A. RGUIGUI, H.: QWN-convolution operators with application to differential equations, Random Oper. Stochastic Equations 22 (214), [3] BARHOUMI, A. OUERDIANE, H. RGUIGUI, H.: QWN-Euler operator and associated Cauchy problem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (212), 1254, 2 p. 89

10 HAKEEM A. OTHMAN [4] BARHOUMI, A. OUERDIANE, H. RGUIGUI, H.: Stochastic heat equation on algebra of generalized functions, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (212), 12526, 18 p. [5] BARHOUMI, A. BEN AMMOU, B. K. RGUIGUI, H.: Operator theory: quantum white noise approach, Quantum Stud. Math. Found. 2 (215), [6] GANNOUN, R. HCHAICHI, R. OUERDIANE, H. REZGUI, A.: Un théorème de dualité entre espace de fonction holomorphes à croissance exponentielle, J. Funct. Anal. 171 (2), [7] GASPER, G. RAHMAN, M.: Basic Hypergeometric Series, in: Encyclopedia Math. Appl., Vol. 34, Cambridge Universty Press, Cambridge, 199. [8] KONDRATIV, YU. G. SILVA, J. L. STREIT, L. US, G. F.: Analysis on Poisson and Gamma space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), [9] VAN LEEUWEN, H. MAASSEN, H.: A q-deformation of the Gauss distribution. J. Math. Phys. 36 (1995), [1] OBATA, N.: White Noise Calculus and Fock Space, in: Lecture Notes in Math., Vol. 1577, Springer, Berlin, [11] OUERDIANE, H. RGUIGUI, H.: QWN-conservation operator and associated wick differential equation, Commun. Stoch. Anal. 6 (212), [12] RGUIGUI, H.: Quantum Ornstein-Uhlenbeck semigroups, Quantum Stud. Math. Found. 2 (215), [13] RGUIGUI, H.: Quantum λ-potentials associated to quantum Ornstein-Uhlenbeck semigroups, Chaos Solitons Fractals 73 (215), Received August 26, 216 Department of Mathematics Al-Qunfudah Center for Scientific Research Al-Qunfudhah University College Umm Al-Qura University KSA Department of Mathematics Rada a College of Education and Science Albida University Albida YEMEN hakimalbdoie@yahoo.com 9