Fractional Supersymmetric Hermite Polynomials

: We provide a realization of fractional supersymmetry quantum mechanics of order r , where the Hamiltonian and the supercharges involve the fractional Dunkl transform as a Klein type operator. We construct several classes of functions satisfying certain orthogonality relations. These functions can be expressed in terms of the associated Laguerre orthogonal polynomials and have shown that their zeros are the eigenvalues of the Hermitian supercharge. We call them the supersymmetric generalized Hermite polynomials.


Introduction
Supersymmetry relates bosons and fermions on the basis of Z 2 -graded superalgebras [1,2], where the fermionic set is realized in terms of matrices of finite dimension or in terms of Grassmann variables [3]. The supersymmetric quantum mechanics (SUSYQM), introduced by Witten [2], may be generated by three operators Q − , Q + and H satisfying Superalgebra (1) corresponds to the case N = 2 supersymmetry. The usual construction of Witten's supersymmetric quantum mechanics with the superalgebra (1) is performed by introduction of fermion degrees of freedom (realized in a matrix form, or in terms of Grassmann variables) which commute with bosonic degrees of freedom. Another realization of supersymmetric quantum mechanics, called minimally bosonized supersymmetric quantum [1,4,5], is built by taking the supercharge as the following Dunkl-type operator: where v(x) is a superpotential. The fractional supersymmetric quantum mechanics of order r (FSUYQM) are an extension of the ordinary supersymmetric quantum mechanics for which the Z 2 -graded superalgebras are replaced by a Z r -graded superalgberas [3,6,7]. The framework of the fractional supersymmetric quantum mechanics has been shown to be quite fruitful. Amongst many works, we may quote the deformed Heisenberg algebra introduced in connection with parafermionic and parabosonic systems [3,4], the C λ -extended oscillator algebra developed in the framework of parasupersymmetric quantum mechanics [8], and the generalized Weyl-Heisenberg algebra W k related to Z k -graded supersymmetric quantum mechanics [3]. Note that the construction of fractional supersymmetric quantum mechanics without employment They satisfy the (anti)commutation relations The generators 1, A ± , R, and relations (7) give us a realization of the R-deformed Heisenberg algebra [1,10]. In [9,13], the authors show that the R-deformed algebra is intimately related to parabosons, parafermions [13] and to the osp(1|2) osp(2|2) superalgebras. From now, we assume that ν > 0. The adjoint Y * ν of the Dunkl operators Y ν with domain S(R) (the space S(R) being dense in L 2 (R, |x| 2ν dx)) is −Y ν and therefore the operator H ν is self-adjoint, its spectrum is discrete, and the wave functions corresponding to the well-known eigenvalues are given by where [x] denotes the greatest integer function and H  n (x) as follows: It is well known that for ν > 0, these polynomials satisfy the orthogonality relations : We define the generalized Klein operator K as a special case of the fractional Dunkl transform F α ν corresponding to α = 2π r . That is, It is well known that the wave functions ψ (ν) n (x) form an orthonormal basis of L 2 (R, |x| 2ν dx) and are also eigenfunctions of the Fourier-Dunkl transform [11,12,15]. In particular, the generalized Klein operator K acts on the wave functions ψ (ν) n (x) as: Let us consider the Z r -grading structure on the space L 2 (R, |x| 2ν dx) as where L 2 j (R, |x| 2ν dx) is a linear subspace of L 2 (R, |x| 2ν dx) generated by the generalized wave functions {ψ nr+j (x) : n = 0, 1, 2, · · · }.
For j = 0, 1, · · · , r − 1, we denote by Π j , the orthogonal projection from L 2 (R, |x| 2ν dx) onto its subspace L 2 j (R, |x| 2ν dx). The action of Π j on L 2 (R, |x| 2ν dx) can be taken to be It is clear that they form a system of resolution of the identity: Note that the orthogonal projection Π j is related to the Klein operator K by

Fractional Supersymmetric Dunkl Harmonic Oscillator
In this section, we shall present a construction of the fractional supersymmetric quantum mechanics of order r (r = 2, 3, . . . ) by using the generalized Klein's operator defined in Equation (11). Following Khare [6,7], a fractional supersymmetric quantum mechanics model of arbitrary order r can be developed by generalizing the fundamental Equations (1) to the forms We introduce the supercharges Q − and Q + as : and the fractional supersymmetric Dunkl harmonic oscillator H ν by where and recall that [ . ] denotes the greatest integer function. Obviously, the operators Q ± and H ν with common domain S(R) are densely defined in the Hilbert space L 2 (R, |x| 2ν dx) and have the Hermitian conjugation relations Furthermore, they satisfy the intertwining relations valid for s = 0, · · · , r − 1: Proposition 1. The supercharges Q ± are nilpotent operators of order r.
Proof. By making use of the following relations we can easily show by induction that Since Q + = Q * − , we also have Q r + = 0.

Associated Generalized Hermite Polynomials
Starting form the following recurrence relations for the generalized Hermite polynomials {H given in [15][16][17], one defines, for each real number c, the system of polynomials H n (x, c) by the recurrence relation: with initial conditions Now, assume that c > 0, c + 2ν > −1.
By Favard's theorem [16], it follows that the family of polynomials {H  n (x, c)} as the associated generalized Hermite polynomials. As shown in ([18] Theorem 5.6.1)(see also [19][20][21]), there are two different systems of associated Laguerre polynomials denoted by L n (x, c). They satisfy the recurrence relations: and Recall the Tricomi function Ψ(a, c; x) given by By [18], the polynomials L n (x, c) satisfy the orthogonality relations when one of the following conditions is satisfied: The monic polynomial version of H ν n (x, c) is given by n (x, c), n = 0, 1, · · · , and satisfies It is easy to see that the polynomial (−1) n H n (x, c).

Thus, by induction, we write them in the form
where S n (x), Q n (x) are monic polynomials of degree n. where Proof. It is directly verified that the polynomials S n (x), Q n (x) given in (39) are orthogonal as they satisfy the recurrence relations From Equation (32), we see that the polynomials S n (x) satisfy the same recurrence relation as (−1) A similar analysis shows that In view of Equations (41) and (42), the explicit form of the associated generalized Hermite polynomials is given by From Equations (36) and (37), we deduce that the system H ν n (x, c) satisfies the orthogonality relations with ζ n = 2 4k (1 + c/2) k (ν + c/2 + 1/2) k , if n = 2k, 2 4k+2 (1 + c/2) k (ν + c/2 + 3/2) k , if n = 2k + 1.

Supersymmetric Generalized Hermite Polynomials
In the sequel, we assume that r is an even integer and we consider the Hermitian supercharge operator Q, defined on S(R), by From Equation (14), we have so it has a self-adjoint extension on L 2 (R, |x| 2ν dx). Furthermore, it acts on the basis ψ ν n as where a (n) s := (nr + s + ν(1 − (−1) s ))/2, s = 1, · · · , r − 1.
On the other hand, by (46), we see that the operator Q leaves invariant the finite dimensional subspace of L 2 (R, |x| 2ν dx) generated by ψ ν nr+s , s = 0, 1, · · · , r − 1. Hence, Q can be represented in this basis by the following r × r tridiagonal Jacobi matrix A It is well known that, if the coefficients of the subdiagonal of some Jacobi Matrix are different from zero, then all the eigenvalues of this matrix are real and nondegenerate [16]. We introduce the normalized eigenvectors φ s of the supercharge Q Qφ s = x s φ s , s = 0, · · · , r − 1 (47) that can be expanded in the basis ψ nr+k , k = 0, 1, · · · , r − 1, as where the coefficients p k obey the three-term recurrence relation [22] a (n) k p k−1 (x) + a (n) k+1 p k+1 (x) = xp k (x), p −1 (x) = 0, p 0 (x s ) = 1, Hence, they become orthogonal polynomials. We denote by P k (x), the monic orthogonal polynomial related to p k (x) by P k (x) = h k p k (x), where h k = a (n) k · · · a (n) 1 and satisfying xP k (x) = P k+1 (x) + 1 2 (k + nr + ν(1 − (−1) k )) P k−1 (x), k = 0, · · · , r − 1, From the three terms recurrence relations (51), the polynomials P k (x) can be identified with the associated generalized Hermite polynomial H (ν) k (x, c), namely, k (x, nr).
It is well known from the theory of orthogonal polynomials that the eigenvalues of the Jacobi matrix  r (x, nr) [16,22]. The weights w s defined in (56) are given by the following formula