It has been known from  and  that for every probability distribution with finite moments of all orders, there exits a family of monic orthogonal polynomials and a paire of sequences and satisfying the three-term recurrence relation (or the tri-diagonal Jacobi relation)
The sequences () and () are called the Szego-Jacobi parameters of.
The starting point of the quantum probabilistic approach to the theory of orthogonal polynomials (OP) is an operator interpretation of the tri-diagonal Jacobi relation (3) in terms of Creation, Annihilation and Preservation (CAP) operators. This allows to associate, in a canonical way, to any random variable with all moments commutation relations that generalize the Heisenberg commutation relations (corresponding to the Gauss-Poisson class). From the mathematical point of view, this approach has led to some new results in the theory of OP.
In order to give this operator interpretation, we shall recall the notion of the interacting Fock probability space associated with the measure (See  for more details).
Consider an infinite-dimensional separable Hilbert space, in which a complete orthonormal basis is chosen. Let denote the dense subspace spanned by the complete orthonormal basis.
Given the sequence, we associate linear operators given by:
Its known that are mutually adjoint and the linear subspace spanned by the set is invariant under the action of.
The quadruple is called the interacting Fock probability space associated with. The operators and are called the creation operator and the annihilation operators respectively. The linear operator given by
is called the number operator. More generally, with the sequence, we associate the preservation operator by the prescription
Let be the space of classes of complex valued, square integrable functions w.r.t. In the following, we simply denote it by and we assume that the sub-space spanned by the polynomial functions is dense in. So that is an Hilbertian basis of. In such case, we consider the isomorphism U from to whose its restriction on given by:
where. Then the U is unitary and we have
This means that the field operator is the -image of the position operator on providing, in this way, a new interpretation of the recursion relation driving by OP in term of CAP operators. Since the random variable with distribution can be identified, up to stochastic equivalence, with the position operator q on, the previous new formulation of the tri-diagonal Jacobi relation in term of the CAP operators is called the quantum decomposition of the classical random variable. In fact we have seen that
This shows that any classical random variable has a built in non commutative structure which is intrinsic and canonical, and not artificially put by hands, that is a sum of three non commuting random variables.
This result motivated the apparition of a series of papers  -  dealing in the same context and provided many applications in the theory of quantum probability. In the paper  , a similar result was obtained but for the family of random variables having an infinitely divisible distribution (I.D-distribution in the following) and having only the moment of the second order. Here, similarity means that the quantum decomposition can be obtained also for this family of random variables.
Based on the notion of the positive definite kernel and using the Lévy-Khintchine function established for the I.D-distributions, the paper  constructed a natural isomorphism U from the Fock space over the -space w.r.t the Lévy measure to the space. Then the -image of the position operator q is the field operator
where is the function. See papers  and  in which the operator Q was widely studied.
In this approach, the construction was not based on the orthogonal polynomials sequence associated with. But it required only the infinite divisibility property, where the Lévy-Khinchine function have played an important role. Then one can ask about the analytic form of the relation (4), or equivalently the counterpart of the three-term recurrence relation. The only obscure point is the existence of such an analogue of the sequence of the orthogonal polynomials. Since the hypothesis on moments is not satisfied, such a sequence of orthogonal polynomial does exist. But the isomorphism U provided us a such chaos-decomposition of the space. For this reason we ask the question if there exist a such analogue for the family of orthogonal polynomial, if it is the case it must be a total family of orthogonal functions in the space satisfying a recursion relation similar to the well known for OP.
This paper is organized as follows:
In Section 2, we recall some known facts about the bosonic Fock space and the quantum decomposition of classical random variables without moments, having I.D-distributions, obtained in   and  . In Section 3, we compute the action of the generalized field operator on the nth particle vectors (). The main result of this paper will be given in Section 4, so that we compute the action of the position operator on the orthogonal functions. This provide such a generalization of the tri-diagonal recursion relation for OP. Finally, the explicit form of theses functions will be given.
2.1. The Bosonic Fock Space
Let be a separable Hilbert space. Let us denote (resp.) the tensor product of n-copies of (resp.) and let be the unique unitary operator such that
where is a permutation of n-variables.
Let, were is the vacuum vector, let
be the orthogonal projection.
Let us denote
Then. Moreover, the set is linearly independent dense in.
The bosonic creation and annihilation operators are defined, on the total set
where denotes omission of the corresponding variable. The preservation operator associated with the self adjoint operator T on is given by:
2.2. The Quantum Decomposition of Classical Random Variables with I.D-Distributions
In this section, we recall briefly, what has been obtained in the paper  around quantum decomposition of random variables with I.D-distributions and having a finite second order moment.
Let us consider a random variable X with I.D-probability distribution having a finite second order moment. It is known (see  ), that the Fourier transform of given by
where is given by
such that and is the the Lévy measure of. The function is called the Lévy-Khintchine function or the characteristic exponent associated with.
Since the second order moment of is finite, the same result will be true for, i.e,:
We suppose also that the gaussian part of is null (i.e.,). Under these conditions, we have the following results:
The family of the trigonometric functions is total in and the family of the functions
is total in.
Then by applying the Araki-Woods-Parthasarathy-Schmidt isomorphism in  for the infinitely divisible positive definite kernel
we have proved the following theorem (See  for more details and descriptions).
Theorem 2.1. The unique linear operator U given on the exponential vectors by:
is an unitary isomorphism from the Fock space to.
Definition 1. Let q be the multiplication (position) operator in:
Define the operator Q on by
where U is the isomorphism defined by (12). Since is a finite measure on, the operator q is self-adjoint (see  Proposition 1, chapter VIII. 3) and
The operator Q is called the generalized field operator.
It follows from condition (10) that the total set is in the domain of Q. Moreover, one has the following theorem:
Theorem 2.2. Let be the function given by
Then the generalized field operator Q takes the form
where, the expectation of X, and are the creation, annihilation and preservation operators in the Fock space given by the prescriptions as in (5)-(7).
3. The Generalized Field Operator
We denote by the set of all sequences of non negatives integers with finite number of nonzero entries. In the sequel (resp.) will be interpreted as subset of the set (resp.). Throughout the remain of this paper we shall use the following notations:
For and ,
The support of such element is defined by
When is a sequence of elements of an Hilbert space and, we denote
In particular if and, so that takes the form
From  , we recall the following identity which is the analogue of the multinomial Newton formula
which take place whenever the series is convergent.
If is separable and is an Hilbertian basis of it, then the set
is an orthonormal basis of the Hilbert space, with the convention.
Let be the canonic basis of. For, we denote
Note that if, then can be defined as in (16), however it is not an element of, because its kth-entry. In this case, we adapt by convention that
Finally, we recall that
3.2. Computation of the Action of the Generalized Field Operator on the Basis (Fn)n
In the remain, we take and we assume that second order moment of is finite. Let be the function given by, where
Since the set is total in (See (11)), then is also total. Then by the Gram-Schmidt procedure, we construct an Hilbertian basis of it, that is denoted by
Lemma 3.1. If the 4th-moment of is finite then for all.
Proof. We have
Then. Since for all, then it is sufficient to prove that.
where we have used the condition (10).
Proposition 3.1. Let be the orthogonal basis of given by
where is the basis given by (20). Then we have
Remark 1. Note that the relation (22) still true in the case when with convention that.
Proof. From (5), we have
This prove (21).
From (5), we have
Here, we have two cases:
If, then (24), becomes
If, then for all. Therefore (24) gives
But in view of (17), we have which gives that the relation (25) sill true. Hence (22) is proved.
Now, it remains to justify (23). From (7), we get
Since, then it can be written as follows:
Using the fact that is bounded, the Equation (26) becomes
But we have for,
Then (27) becomes
This ends the proof. ,
Corollary 3.1.1 The action of the generalized field operator Q on the basis is given as follows:
Proof. A straightforward computations. ,
4. Orthogonal Functions and Generalization of the Three-Term Recurrence Relation
In this section, we give the action of the multiplication operator q on the functions
Then we deduce the generalization of the three-term recurrence relation in term of the orthogonal functions.
Since U is unitary from to and is an orthogonal basis of, the family is an orthogonal basis of.
Theorem 4.1 Let and let be the diagonal operator from to itself given by
Then for all, we have
Remark 2. Since U is unitary and the basis is orthogonal, then is an orthogonal basis of. Moreover, the chaos decomposition of the Fock space induces the following chaos-decomposition of the space
Now comparing the relation (30) with (3), the only difference is the apparition of a corrective expression in (30) which is in the nth chaos. In the case when it is null, (30) will be exactly the well-known tri-diagonal recurrence relation (3). In this sense the relation (30) can be interpreted as a generalization of the three term recurrence relation. Here, the monic orthogonal polynomial sequence is replaced by a double-entries sequence of orthogonal functions parameterized by and. In addition to the infinite divisibility property, this generalization require only the existence of the second and fourth order moments of.
Proof. From relation (29), we deduce that
Proposition 4.2. We assume that is continuous w.r.t the Lebesgue measure with Radon-Nikodym derivative. Then for all and, one has
Proof. Since is an Hilbertian basis of and,
where the series converge in. It follows, from the multinomial Newton formula (14), that
This implies that
From the definition of U, we get
which is the decomposition of in the basis. Then
On the other hand, we have
This implies that
where denotes the Fourier transform on. Note that the function belongs to the space. It follows that
which is equivalent to
The infinite-divisibility of the distribution gives rise to the Kolmogorov isomorphism U, which was the principal bridge between the Fock space and transforming, in such canonical way, the quantum decomposition identity to the tri-diagonal recurrence relation.
The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number 3378 during the academic year 1436 AH/2015 AD.
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