Multi-variable orthogonal polynomials Abdallah Dhahri Department of Mathematics Faculty of Sciences of Tunis University of Tunis El-Manar 1060 Tunis,Tunisia [email protected] Abstract We characterize the atomic probability measure on Rd which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are di- agonal matrices. Finally, as a consequence of the multiple Jacobi orthogonal polynomials case, we give the Jacobi sequences of the Gegenbauer, Chebyshev and Legendre orthogonal polynomials. 1 Introduction Let µ be a probability measure on R with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the sequence 1, x2,...,xn,... to get a { } sequence P (x); n =0, 1,... of orthogonal polynomials in L2(µ), where P (x)=1 { n } 0 and Pn(x) is a polynomial of degree n with leading coefficient 1. It is well-known that these polynomials Pn’s satisfy the recursion formula: (x αn)Pn(x)= Pn+1(x)+ ωnPn−1(x), n 0. arXiv:1401.5434v1 [math.FA] 21 Jan 2014 − ≥ where α R,ω 0 for n 0 and P = 0 by convention. The sequences (α ) and n ∈ n ≥ ≥ −1 n n (wn)n are called the Jacobi sequences associated to the probability measure µ (cf [8], [10], [13]). In the multi-dimensional case (cf [9],[11], [12],[14]) the formulations of these results are recently given by identifying the theory of multi-dimensional orthogonal polynomials with the theory of symmetric interacting Fock spaces (cf [1]). The multi-dimensional analogue of positive numbers wn (resp. real numbers αn) are the positive definite ma- trices (resp. Hermitean matrices). 1 In this paper, we characterize the atomic probability measures on Rd which having a finite number of atoms. Moreover, we give the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials and we prove that they are diagonal matrices. As a corollary of the Jacobi case, we give the explicit forms of the ones associated to the Gegenbaur, Chebyshev and Legendre orthogonal polynomials. This paper is organized as follows. In section 2 we recall the basic properties of the complex polynomial algebra in d commuting indeterminates and we give the multi- dimensional Favard Lemma. The characterization of the atomic probability measures on Rd which having a finite number of atoms is given in section 3. Finally, in section 4 we give the explicit forms of the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials. 2 The multi-dimensional Favard Lemma In this section we recall the basic properties of the polynomial algebra in d commut- ing indeterminates and we give the multi-dimensional Favard Lemma. We refer the interested reader to [1] for more details. 2.1 The polynomial algebra in d commuting indeterminates Let d N∗ and let ∈ = C[(X ) ] P j 1≤j≤d be the complex polynomial algebra in the commuting indeterminates (Xj)1≤j≤d with the -structure uniquely determined by the prescription that the Xj are self-adjoint. For all ∗ d v =(v1,...,vd) C denote ∈ d Xv := vjXj Xj=1 A monomial of degree n N is by definition any product of the form ∈ d nj M := Xj Yj=1 where, for any 1 j d, n N and n + ... + n = n. ≤ ≤ j ∈ 1 d Denote by n] the vector subspace of generated by the set of monomials of degree less or equal thanP n. It is clear that P = N P ∪n∈ Pn] 2 Definition 1 For n N we say that a subspace is monic of degree n if ∈ Pn ⊂ Pn] = +˙ Pn] Pn−1] Pn (with the convention −1] = 0 and where +˙ means a vector space direct sum) and n has a linear basis Pwith the{ property} that for each b , the highest order term ofPb Bn ∈ Bn is a non-zero multiple of a monomial of degree n. Such a basis is called a perturbation of the monomial basis of order n in the coordinates (Xj)1≤j≤d. Note that any state ϕ on defines a pre-scalar product P ., . : C h iϕ P×P → (a, b) a, b = ϕ(a∗b) 7→ h iϕ with 1 , 1 = 1. h P P iϕ Lemma 2.1 Let ϕ be a state on and denote , = , ϕ be the associated pre-scalar product. Then there existsP a gradation h · · i h · · i = ( , , ) (1) P Pn,ϕ h · · in,ϕ Mn∈N called a ϕ-orthogonal polynomial decomposition of , with the following properties: P (i) (1) is orthogonal for the unique pre-scalar product , on defined by the conditions: h · · i P , = , , n N h · · i|Pn,ϕ h · · in,ϕ ∀ ∈ , m = n Pm,ϕ ⊥ Pn,ϕ ∀ 6 (ii) (1) is compatible with the filtration ( ) in the sense that Pn] n n = , n N, Pn] Ph,ϕ ∀ ∈ Mh=0 (iii) for each n N the space is monic. ∈ Pn,ϕ Conversely, let be given: (j) a vector space direct sum decomposition of P · = (2) P Pn Xn∈N such that = C.1 , and for each n N, is monic of degree n, P0 P ∈ Pn 3 (jj) for all n N a pre-scalar product , n on n with the property that 1P has norm 1 and∈ the unique pre-scalar producth · · i , Pon defined by the conditions: h · · i P , = , , n N h · · i|Pn h · · in ∀ ∈ , m = n Pm ⊥ Pn ∀ 6 satisfies 1 , 1 = 1 and multiplication by the coordinates X (1 j d) are h P P i j ≤ ≤ , -symmetric linear operators on . h · · i P Then there exists a state ϕ on such that the decomposition (2) is the orthogonal polynomial decomposition of withP respect to ϕ. P 2.2 The symmetric Jacobi relations and the CAP operators In the following we fix a state ϕ on and we follow the notations of Lemma 2.1 with the exception that we omit the indexPϕ. We write ., . for the pre-scalar product ., . , h i h iϕ k for the space k,ϕ and Pk] : k] the ., . -orthogonal projector in the pre-Hilbert spaceP sense (seeP [1] for more details).P → P Put h i P = P P n n] − n−1] ∗ N It is obvious that Pn = Pn and PnPm = δnmPn for all n, m . It is proved in [1] that for any 1 j d and any n N∈, one has ≤ ≤ ∈ XjPn = Pn+1XjPn + PnXjPn + Pn−1XjPn (3) with the convention that P−1] = 0. The identity (3) is called the symmetric Jacobi relation. N ε Now for each 1 j d and n we define the operators aj|n, ε +, 0, , with ≤ ≤ ∈ d ∈{ −} respect to a basis e =(ej)1≤j≤d of C as follows: a+ = a+ := P X P : j|n ej |n n+1 j n n n+1 Pn P −→ P 0 0 aj|n = aej |n := PnXjPn : n n (4) Pn P −→ P a− = a− := P X P : j|n ej |n n−1 j n n n−1 Pn P −→ P d Notation: If v =(v1,...,vd) C , where v1,...,vd are the coordinates of v in the basis e, we denote ∈ ε ε av|n := vjaj|n 1X≤j≤d 4 Note that in this context, the sum = (5) P Pn Mn∈N is orthogonal and meant in the weak sense, i.e. for each element Q there is a finite set I N such that ∈ P ⊂ Q = p , p (6) n n ∈ Pn Xn∈I Theorem 2.2 On , for any 1 j d, the following operators are well defined P ≤ ≤ + + aj := aj|n Xn∈N 0 0 aj := aj|n Xn∈N − − aj := aj|n Xn∈N and one has + 0 − Xj = aj + aj + aj (7) in the sense that both sides of (7) are well defined on and the equality holds. P Identity (7) is called a quantum decomposition of the variable Xj. Proposition 2.3 For any 1 j d and n N, one has ≤ ≤ ∈ + ∗ − + ∗ − (aj|n) = aj|n+1 ; (aj ) = aj 0 ∗ 0 0 ∗ 0 (aj|n) = aj|n ; (aj ) = aj Moreover, for each j, k 1,...,d , one has ∈{ } + + [aj , ak ]=0 2.3 3-diagonal decompositions of and multi-dimensional Favard Lemma P Definition 2 For n N a 3–diagonal decomposition of ∈ Pn] n−1 n + 0 n ( k , , k)k=0 , a·|k , a·|k P h· ·i k=0 k=0 is defined by: 5 (i) a vector space direct sum decomposition of such that Pn] · = ; k 0, 1, , n (8) Pk] Ph ∀ ∈{ ··· } h∈{X0,...,k} where each is monic. Pk (ii) for each k 0, 1, , n a pre-scalar product , on . ∈{ ··· } h · · ik Pk (iii) two families of linear maps v Cd a+ ( , ) , k 0, 1, , n 1 ∈ 7−→ v|k ∈L Pk Pk+1 ∈{ ··· − } v Cd a0 ( , ) , k 0, 1, , n ∈ 7−→ v|k ∈L Pk Pk ∈{ ··· } such that: Rd + - forall v , av|k maps the ( k, ., . k)-zero norm subspace into the ( k+1, ., . k+1)- zero norm∈ subspace; P h i P h i - for all v Rd, a0 is a self-adjoint operator on the pre-Hilbert space ( , , ), ∈ v|k Pk h· ·ik thus in particular it maps ( , , )-zero norm subspace into itself; Pk h · · ik - denoting (when no confusion is possible) the adjoint of a linear map from ( k−1 , , k−1) to ( , ∗ , ) for any k 0, 1, , n , and defining P h · · i Pk h · · ik ∈{ ··· } a− := (a+ )∗ ; a+ := 0 ; k 0, 1, , n 1 , v Cd v|k v|k−1 v|−1 ∈{ ··· − } ∈ the following identity is satisfied: + 0 − Rd Xv = av|k + av|k + av|k ; k 0, 1, , n 1 , v Pk ∈{ ··· − } ∈ Remarks: For the following remarks we refer to [1].