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The Dunkl oscillator in the plane II : representations of the symmetry algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger-Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra u(2). Two of the symmetry generators, J_3 and J_2, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J_3 is diagonal and the operator J_2 acts in a tridiagonal fashion. In the circular basis, the operator J_2 is block upper-triangular with all blocks 2x2 and the operator J_3 acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J_2 in the circular basis are generated by the Heun polynomials and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J_2 are generated by little -1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J_2 is diagonal is then considered. In this basis, the defining relations of the Schwinger-Dunkl algebra imply that J_3 acts in a block tridiagonal fashion with all blocks 2x2. The matrix elements of J_3 in this basis are given explicitly.


Introduction
This is the second part of this series concerned with the analysis of the isotropic Dunkl oscillator model. In part I, the model has been shown to be superintegrable, the wavefunctions have been obtained in Cartesian and polar coordinates and the overlap coefficients have been found [5]. In the present work, the representations of the symmetry algebra of the model, called the Schwinger-Dunkl algebra (see below), are investigated. As shall be seen, this study entails remarkable connections with special functions such as the Heun, little −1 Jacobi and para-Krawtchouk polynomials.

Superintegrability
One recalls that a quantum system defined by a Hamiltonian H in d dimensions is maximally superintegrable if it admits 2d − 1 algebraically independent symmetry generators S i that commute with the Hamiltonian where one of the symmetries is the Hamiltonian itself, e.g. S 1 ≡ H. Moreover, a superintegrable system is said to be of order ℓ if ℓ is the maximal order of the symmetries S i in the momentum variables.

The Dunkl oscillator model
The isotropic Dunkl oscillator model [2,5,11] in the plane is possibly the simplest twodimensional system described by a Hamiltonian involving reflections. It is second-order superintegrable and is defined by the Hamiltonian [5] where D µx i x i is the Dunkl derivative [4,14] D µx i with I denoting the identity operator and R x i , x i ∈ {x, y}, standing for the reflection operator with respect to the plane x i = 0. Hence the reflections R x , R y that appear in the Hamiltonian (1) have the action R x f (x, y) = f (−x, y), R y f (x, y) = f (x, −y), and thus evidently R 2 x i = I. In connection with the nomenclature of the standard harmonic oscillator, the model is called isotropic because the quadratic potential is SO (2) invariant. For the full Hamiltonian (1) to have this symmetry requires of course that µ x = µ y .
The Schrödinger equation associated to H is separable in both Cartesian and polar coordinates. The spectrum of energies E is given by where n x , n y are non-negative integers. The wavefunctions are well defined for the values µ x , µ y ∈ (− 1 2 , ∞); the case µ x = µ y = 0 corresponds to the standard quantum harmonic oscillator. It is easily seen from (2) that the energy level E N exhibits a N + 1-fold degeneracy.

Symmetries of the Dunkl oscillator
The symmetries of the Dunkl oscillator Hamiltonian (1) can be obtained by the Schwinger construction using the parabosonic creation/annihilation operators [5,8,10]. We consider the operators [14,15] It is verified that the operators A x i ± satisfy the following commutation relations: where {x, y} = xy + yx denotes the anticommutator. In addition to the commutation relations (4), one has In terms of the operators (3), the Hamiltonian (1) takes the form where is the Hamiltonian of the one-dimensional Dunkl oscillator. The symmetry generators of the Dunkl oscillator model are as follows. Consider the operator It is directly verified that [H, J 3 ] = 0. Since J 3 can be written as using (6), it is clear that this symmetry corresponds to the separability of the Schrödinger equation in Cartesian coordinates [5]. A second symmetry generator is given by It is again directly verified that [J 2 , H] = 0. In terms of Dunkl derivatives, this operator has the expression and it has been shown [5] that J 2 is the symmetry corresponding to the separation of variables in polar coordinates. A third symmetry J 1 , algebraically dependent of J 2 , J 3 , is obtained by taking J 1 = −i[J 2 , J 3 ]. This additional symmetry generator reads In addition to J i , i = 1, . . . , 3, it is directly checked that the reflections R x , R y also commute with H.

The main object: the Schwinger-Dunkl algebra sd(2)
The symmetry algebra of the Dunkl oscillator, called the Schwinger-Dunkl algebra, is denoted sd (2) and defined by the commutation relations where R 2 x i = I, x i ∈ {x, y}, and the Hamiltonian H is a central element. The algebra sd(2) admits the Casimir operator [7] which commutes with all the generators. In the present realization, the Casimir operator C takes the value Note that the involution P = R x R y also commutes with all the generators and thus can be viewed as a second Casimir operator. Furthermore, it is easily seen that when µ x = µ y = 0, the Schwinger-Dunkl algebra sd(2) reduces to the Lie algebra u(2), which is the symmetry algebra of the standard isotropic 2D quantum oscillator in the plane. The irreducible representations of sd(2) can be used to account for the degeneracies in the spectrum of H. In finite-dimensional representations of degree N +1, the action of the symmetry generators J 1 , J 2 , J 3 , R x and R y indicate how the degenerate eigenstates of H corresponding to the energy value E N transform into one another under the action of the symmetries. In the following, three bases for the finite-dimensional irreducible representations of sd(2) will be constructed and the explicit formulas for the action of the symmetry generators on each basis will be derived.

Outline
Here is the outline of the paper. In Section 2, we construct the Cartesian basis in which the symmetry generator J 3 is diagonal and J 2 acts in a tridiagonal fashion. In Section 3, we introduce the circular creation/annihilation operators and study the associated circular basis in which J 2 is block upper-triangular and J 3 is tridiagonal. We show that the interbasis expansion coefficients involve the Krawtchouk polynomials and we derive the spectrum of J 2 algebraically. In Section 4, we obtain the eigenvectors of J 2 in the circular basis for odd-dimensional representations and show that these eigenvectors are generated by the Heun polynomials and that their components are para-Krawtchouk polynomials. The fully isotropic case is shown to involve the little −1 Jacobi polynomials. In Section 5, the eigenvectors of J 2 in the circular basis for evendimensional representations are studied. In Section 6, we examine the basis in which J 2 is diagonal and show that J 3 acts in a six-diagonal fashion on this basis. We conclude with an outlook.

The Cartesian basis
In this section the Cartesian basis for the finite-dimensional representations of sd(2) is constructed using the realization (7), (8), (9) of the algebra generators in terms of the creation/annihilation operators (3). The representation spaces spanned by the Cartesian basis correspond to the spaces of degenerate wavefunctions with energies E N ,N ∈ N, separated in Cartesian coordinates, although a different normalization is used for the basis vectors. The action of the sd(2) generators on the wavefunctions were obtained by a direct computation in [5] using the expressions of the symmetries in terms of Dunkl derivatives. Here the actions of the generators and the spectra of the Hamiltonian H and the symmetry generator J 3 are obtained in a purely algebraic manner.
The Cartesian basis vectors are labeled by two non-negative integers n x , n y and are denoted by |n x , n y . These basis vectors are defined by where |0 x , 0 y is the "vacuum" vector. The vacuum vector has the defining properties The action of the reflection operators and the creation/annihilation operators on the Cartesian basis vectors can be derived from the above definitions and the commutation/anticommutation relations (4) and (5). From the anticommutation relations and the vacuum parity conditions (12b), it directly follows that R x |n x , n y = (−1) nx |n x , n y , R y |n x , n y = (−1) ny |n x , n y .
By the definition of the basis vectors (11), one has also A x + |n x , n y = |n x + 1, n y , A y + |n x , n y = |n x , n y + 1 .
To derive the action of the operators A x i − on the Cartesian basis, one needs the commutator identity which is easily proven by induction. It is convenient to introduce the µ-numbers [14] [n] µ = n + µ(1 − (−1) n ).
Using the identity (15) and the formulas (11), (12a) and (12b), one finds and similarly for A y − . Using the results (13), (14) and (17) along with the expressions of the symmetry generators J i , i = 1, 2, 3, and H, in terms of the operators A x i ± given in (7), (8) and (9), one finds that the action of the symmetries on the Cartesian basis is given by and the action of J 1 can be obtained directly by commuting J 2 and J 3 . The central element H has the action H|n x , n y = (n x + n y + µ x + µ y + 1)|n x , n y .
Hence the spectra of the symmetry generator J 3 and of the full Hamiltonian H of the Dunkl oscillator have been recovered in a purely algebraic manner. As is expected, the symmetry operators J 1 , . . . J 3 and the involutions R x , R y transform the set of vectors |n x , n y with a given value of N = n x + n y into one another; these vectors are the degenerate eigenvectors of H with energy E N = N + µ x + µ y + 1.

The circular basis
In this section, the circular basis for the finite-dimensional representations of sd(2) is constructed using the left/right circular operators. The actions of the symmetries on this basis are obtained and the spectrum of the generator J 2 is derived from these actions. The expansion coefficients between the circular and Cartesian bases, which involve the Krawtchouk polynomials, are also examined.
The left/right circular operators for the 2D Dunkl oscillator are introduced following the analogous construction in the standard 2D harmonic oscillator [1]. We define where A x i ± are the creation/annihilation operators of the Dunkl oscillator that obey the commutation relations (4). The inverse relations are easily seen to be The Dunkl oscillator in the plane II

8
The left/right operators obey the commutation relations and the algebraic relations involving the reflections become The circular basis vectors |n L , n R are labeled by the two non-negative integers n L , n R and are defined by where |0 L , 0 R is the circular vacuum vector with the properties Given the definition (22), one has A L + |n L , n R = |n L + 1, n R , A R + |n L , n R = |n L , n R + 1 .
From the relations (21) and the definition (22), it follows that R x |n L , n R = (−1) n L +n R |n R , n L , R y |n L , n R = |n R , n L .
Consider the commutator identities which can be proven straightforwardly by induction. From the definition (22), the vacuum conditions (23a), (23b) and the above identities, the action of A L − on the circular basis elements |n L , n r can be derived by a direct computation. For n L = n R , one has For n L > n R , one finds Finally, for n L < n R , one obtains To obtain the corresponding formulas for the action of A R − , one needs the identities Using the same procedure as for A L − , we obtain the action of A R − . For n L = n R , we have When n L > n R , one finds and for n R > n L , the result is As is seen from the formulas, the operators A L/R − have the effect of sending the circular basis vectors |n L , n R to all circular basis vectors |i L j R with i L + j R = n L + n R − 1.
In terms of the circular operators (20), the symmetry generators and the central element H take the rather symmetric form Using the above formulas and the actions of the circular operators A L/R ± , the matrix elements of the sd(2) generators in the circular basis can be computed; they are given below for J 2 and J 3 . The action of the Hamiltonian H is It is clear that the generators preserve the subspace spanned by the basis vectors {|n L , n R | n L + n R = N}. As is seen from the action of H, this corresponds to the space of degenerate eigenstates of H with energy E N . The properties of representations of the symmetry generators in the circular basis will now be used to derive the transition matrix from the circular basis to the Cartesian basis and to obtain the eigenvalues of J 2 algebraically.

Transition matrix from the circular to the Cartesian basis
We consider the N + 1-dimensional energy eigenspace spanned by the circular basis vectors |n L , n R with n L + n R = N and redefine the basis vectors as follows On this basis, a direct computation shows that the generator J 3 has the action where we have defined Since J 3 is diagonal in the Cartesian basis and tridiagonal in the circular basis, the two bases are related by a similarity transformation involving orthogonal polynomials. Let us consider the decomposition of the Cartesian basis vector v where j ∈ {0, . . . , N}. Acting on both sides of (27) with J 3 and using (19c) and (26), one arrives at the following recurrence relation satisfied by the expansion coefficients C n (j): with C −1 = 0. Upon factoring out the initial value we obtain the recurrence relation where P 0 (j) = 1. It follows from (28) that P n (x) is a polynomial of degree n in x. Upon substituting P n (j) = P n (j)/n!, we obtain the normalized recurrence relation It is directly seen that the polynomials P n (j) are the monic Krawtchouk polynomials K n (x; p, N) [9] with parameter p = 1/2 and variable x evaluated at x = j. We thus have where the constant C 0 (j) can be chosen to ensure the unitarity of the transition matrix by using the orthogonality relation of the Krawtchouk polynomials. Despite the differences that the Dunkl and standard harmonic oscillators exhibit, the relations between the circular and Cartesian bases are identical in both cases.

Matrix representation of J 2 and spectrum
The circular representation space can be used to derive the spectrum of the symmetry operator J 2 . To exhibit the structure of J 2 , we introduce the following notation for the basis vectors: where ⌊x⌋ is the floor function. We adopt the convention that for convenience. We denote by B 2 the circular basis such that n L + n R = N with the ordering As an example, consider the case N = 3. The basis reads and corresponds to the following ordering of the standard circular basis vectors |n L , n R : For N = 4, one has which corresponds to Using the action of the operators A L/R ± and the formulas (25a), (25b), the matrix representation of J 2 in the circular basis B 2 is derived in a straightforward manner. We find that for N even, the N + 1-dimensional square matrix representing J 2 in the basis B 2 is block upper-triangular with all blocks 2 × 2 in addition to a row of 1 × 2 blocks. The matrix reads with m = N/2 and where we have with ω k corresponding to the lower part of Ω k . We have taken In the N odd case, one obtains with m = (N − 1)/2 and where Since in both cases the matrices representing J 2 are block upper-triangular, it follows from elementary linear algebra that the set of eigenvalues of J 2 is the union of the sets of eigenvalues of each diagonal block Γ k or Γ k . By the direct diagonalization of the 2 × 2 diagonal blocks, we obtain that when N is even, the eigenvalues of J 2 are given by where m = N/2 and where the eigenvalue λ − 0 = 0 is non-degenerate. When N is odd, the spectrum of J 2 has the form where m ′ = (N − 1)/2. These eigenvalues are indeed the eigenvalues of J 2 that were obtained in the first part [5] by solving the differential equation arising from the realization of J 2 in terms of Dunkl derivatives; here they have been obtained in a purely algebraic manner. It is seen from the matrices (29), (31) that when µ x = µ y = 0, the matrices representing J 2 are diagonal. This corresponds to the standard result for the harmonic oscillator, where the circular basis is the eigenbasis of the symmetry J 2 .
Given that in the case of the Schwinger-Dunkl algebra sd (2), the circular basis does not diagonalize J 2 directly, it is of interest to inquire about the eigenvectors of J 2 in this basis. This is the subject of the next two sections.

Diagonalization of J 2 : the N even case
This section is devoted to the computation of the eigenvectors of J 2 in the circular basis for the N even case. To perform the calculation, we shall make use of an auxiliary operator Q whose eigenvectors have been related to those of J 2 in the previous paper [5]. The evaluation of the eigenvectors of Q is somewhat involved and consequently it is instructive to first expose the main steps of the computation.
Firstly, the structure of Q will be used to reduce the eigenvalue problem to a system of recurrence relations for the components of the eigenvectors. Secondly, using generating functions, the recurrence system will be transformed into a system of differential equations and the solutions will be expressed in terms of Heun polynomials. Thirdly, using well-known properties of Heun functions, the explicit expressions for the components of the eigenvectors will be obtained in terms of a special case of complementary Bannai-Ito polynomials [6] which correspond to para-Krawtchouk polynomials [18]. Lastly, the relation between the eigenvectors of J 2 and Q obtained in the first paper [5] will be used to write the final expression for the eigenvectors of J 2 in the circular basis.

The operator Q and its simultaneous eigenvalue equation
The operator Q has been used in the first part [5] to obtain the overlap coefficients between the wavefunctions in Cartesian and polar coordinates of the 2D Dunkl oscillator [7]. It is defined in terms of J 2 through the relation Given the action (24) of the reflections operators, it is seen that R x , R y have the following matrix representation in the circular basis B 2 : Using the formula (34) for the reflections and the formulas (29), (30) for the expression of J 2 in the N even case, one obtains from (33) The 1 × 2 blocks δ i correspond to the lower part of the blocks ∆ i . From the block upper-triangular structure, it follows that the eigenvalues ν ± k of Q are given by and we also have ν − 0 = −ζ − 1/2. Let us denote by |k, ± Q the eigenvectors of Q with eigenvalues ν ± k . We wish to evaluate the components of these eigenvectors in the circular basis. We define their expansion in the circular basis by for k = 1, . . . , m and where the vectors |ℓ, ± are vectors of the circular basis B 2 . It is clear from the matrix representation of Q that |0, − Q = |0, − and thus v − 0 (0) = 1. We shall study the simultaneous eigenvalue equation for the operator Q. Since the matrix representing Q is block upper-triangular, the matrix of eigenvectors will have the same structure. We define the matrix of eigenvectors of Q as follows: and where V ℓk is the 1×2 block corresponding to the lower part of V ℓk . The simultaneous eigenvalue equation for the matrix [Q] B 2 can be written as As will be seen, the components (37) of the eigenvectors of Q can be derived from the eigenvalue equation (38) by solving the associated system of recurrence relations.

Recurrence relations
It will prove convenient to consider the two sectors corresponding to the eigenvalues ν + k , ν − k separately. In block form, for ℓ = 1, . . . , m and k = 1, . . . , m, the eigenvalue equation (38) can be written in the form with Φ ℓ and ∆ j given in (35) and where the range of the sum has been determined by the structure of the eigenvector matrix W .

The ν + k eigenvalue sector
We consider the eigenvectors |k, + Q of Q with the expansion and associated to the eigenvalue ν + k = 2k + ζ − 1/2. It is understood that u + 0 (k) does not belong to this decomposition. For ℓ = 1, . . . , m, it directly seen that the eigenvalue equation (39) is equivalent to the following system of recurrence relations: where we have defined and where the explicit dependence of the components u ± ℓ on k has been dropped for notational convenience. The case ℓ = 0 is treated below. The system of recurrence relations (41) is "reversed": the values of u ± i are obtained from the values of u ± j with i < j and j < p. The terminating conditions are at ℓ = k. In this case (41) reduces to In accordance to the system (42), we choose the following terminating conditions

Upon introducing
the system (41) is directly seen to be equivalent to [k]B ℓ = iℓA ℓ .
The above system accounts for the ℓ = 0 case. Indeed, it is seen that B 0 = 0 and hence u − 0 = A 0 /2. These equations can be simplified by factoring out the terminating conditions where α 0 = (1 − i) and β 0 = (1 + i). It is seen that the normalized components A ℓ , B ℓ are real and satisfy the system with the terminating conditions A k = B k = 1. The system (44) can be simplified by introducing the reversed components a ℓ = A k−ℓ and b ℓ = B k−ℓ . Using the index n, the system takes the usual form with the initial conditions a 0 = b 0 = 1. Hence the components u ± ℓ (k) of the eigenvector |k, ± Q of the operator Q have the expression where a n and b n are the unique solutions to the system (45).

The ν − k eigenvalue sector
We consider the eigenvectors |k, − Q corresponding to the eigenvalue ν − k of Q with the circular basis expansion and associated eigenvalue ν − k = −2k − ζ − 1/2. An analysis similar to the preceding one shows that the components v ± ℓ (k) differ from the components u ± ℓ (k) only by their terminating conditions. Again

This yields
where

Generating function and Heun polynomials
We have seen that the evaluation of the components of the eigenvectors of Q in the circular basis depends on the solution of the recurrence system (45). As it turns out, an explicit solution for a n (k) and b n (k) can be obtained using generating functions. We introduce the ordinary generating functions We shall make use of the elementary identities z∂ z A(z) = n 0 n a n z n , Using the above identities, it is easily seen that the system of recurrence relations (45) for the quantities a n , b n is equivalent to the following system of differential equations for the generating functions A(z), B(z): By direct substitution, we find that the generating function A(z) satisfies the secondorder differential equation This corresponds to Heun's differential equation [3,13]. The general form of the Heun differential equation is with α + β + 1 = γ + δ + ǫ. Comparing (51) with (50), we thus write A(z) = Hℓ(a, q; α, β, γ, δ, z) with the parameters The function Hℓ(a, q; α, β, γ, δ) denotes the solution to (51) that corresponds to the exponent 0 at z = 0 and assumes the value 1 at that point. This is obviously the case of A(z). It will be seen that A(z) is in fact a polynomial of degree 2k, and hence that the Heun function (52) is in fact a Heun polynomial. Given the system (49), we also have
Since there is a singularity in the recurrence coefficients for the polynomials P n (ξ), the correspondence between the polynomials P n (ξ) and the CBI polynomials outlined above is valid only for n = 0, . . . , k and hence the Heun polynomial A(z) generates only the first k para-Krawtchouk polynomials. As is easily seen by induction, the recurrence relation (55) generates center-symmetric polynomials P n . Hence for n > k, we have P n (ξ) = P 2k−n (ξ). Putting the preceding results together, we write a n = (−1) n 4 n n! (k + 1 − n) n (2k + ζ − n) n I n (ξ/2; ρ 1 , ρ 2 , r 1 , r 2 ) for n k and a n = a 2k−n , n = k + 1, . . . , 2k for n = k + 1 · · · , 2k. The hypergeometric expression of the CBI polynomials (58) provides an explicit formula for the coefficients a n and the coefficients b n are easily evaluated from the recurrence system (45). Combining those results with the formulas (46) and (47) yields the expansion coefficients of the eigenvectors of the operator Q in the circular basis. Note that these expansion coefficients only involve a j , b j with j = 0, . . . , k and hence only (62) is needed.

Eigenvectors of J 2
To obtain the expansion coefficients of the eigenvectors of J 2 in the circular basis, it is necessary to relate the eigenvectors of J 2 to those of Q. This relation has been obtained in the previous paper [5]. In the present notation, we have where |k, ± J 2 are the eigenvectors of J 2 corresponding to the eigenvalues and where the coefficient ω k is The inverse relation, which allows to express the eigenvectors of J 2 in terms of the known eigenvectors of Q reads |k,

The fully isotropic case
We now consider the case µ x = µ y = µ. This corresponds to a fully isotropic 2D Dunkl oscillator, where two "identical" parabosonic oscillators are combined. Returning to the system of differential equations (49) for the generating functions, one has Solving for A(z), we find The solution corresponding to the initial value a 0 = 1 is given by and we also have Hence in the isotropic case, the generating functions are simply the Jacobi polynomials. It follows from the hypergeometric generating function that Thus it is seen that the in the fully isotropic case µ x = µ y = µ, the formulas for the expansion coefficients of the eigenvectors of Q simplify substantially.

Diagonalization of J 2 : the N odd case
In this section, we obtain the expression for the eigenvectors of J 2 in the circular basis when N = n L + n R is an odd integer. In spirit, the computation is similar to the N even case presented in the previous section. We proceed along the same lines.

The operator Q and its simultaneous eigenvalue equation
The operator Q is defined by Given the action (24) of the reflections operators, it is seen that they have the following matrix representation in the circular basis B 2 : Using the matrix representation of J 2 in the circular basis B 2 given in (31), one finds From the block upper-triangular structure, it follows that the eigenvalues ν ± k of Q are for k = 0, . . . , m. Although a different labeling has been used, it is directly checked that the eigenvalues of Q are the same for the N even and N odd case, except for the additional one. We denote the eigenvectors of Q corresponding to the eigenvalues ν ± k by |k, ± Q and define their expansion in the circular basis by for k = 0, . . . , m and where the vectors |ℓ, ± are the vectors of the circular basis B 2 . We shall once again study the simultaneous eigenvalue equation for the operator Q. We define the matrix of eigenvectors .

The simultaneous eigenvalue equation for the matrix [Q] B 2 reads
(65) As in section 4, the simultaneous equation (64) will be shown to be equivalent to a system of recurrence relation for the components u ℓ (k) ± , v ± ℓ (k) of the eigenvectors of Q in the circular basis.

Recurrence relations
We now construct the recurrence systems for the component of the eigenvectors of the operator Q. For k = 0, . . . , m and ℓ = 0, . . . , m, the simultaneous equation (64) takes the form

The ν − k eigenvalue sector
Let us begin by considering the eigenvectors |k, − Q of Q corresponding to the eigenvalue ν − k and their expansion in the circular basis It is directly seen from (66),(63) and (65) that we have where the explicit dependence on k of the components v ± ℓ (k) has been dropped for notational convenience. The recurrence system (67) is "reversed". The terminating conditions are at ℓ = k. In this case, (67) becomes Choosing v − k = 1, we obtain the terminating conditions

Upon taking
where α 0 = (1 − i) and β 0 = (1 + i), it is easily seen that A ℓ and B ℓ are real and that the system (67) is equivalent to with the terminating conditions A k = 1 and B k = 1. Introducing the reversed components a n = A k−n , b n = B k−n , we obtain the system [k + µ x + 1/2] b n = [k − n + µ x + 1/2] a n + 2µ x (−1) n with the initial conditions a 0 = 1 and b 0 = 1. Taking into account all the preceding transformations, we have where a ℓ , b ℓ are the unique solutions to the recurrence system (68).

The ν + k eigenvalue sector
Let us now consider the eigenvectors |k, + Q corresponding to the eigenvalue ν + k with expansion in the circular basis. Proceeding along the same lines as in the previous computation, we find that the terminating condition are of the form and that the components are given by where

Generating functions and Heun polynomials
As is seen from the formulas (69) and (70), the main part of the components of the eigenvectors of Q in the circular basis is given by the solutions to the recurrence system (68). As in section 4 for the N even case, we bring the ordinary generating functions Upon using the identities (48), we obtain from (68) the associated differential system By direct substitution, we find that the generating functions are expressed in terms of the Heun functions where and where the parameters q B , · · · δ B are obtained from (73) by the transformation µ x ↔ µ y . The form of the parameters involved in the Heun functions show that once again one has a truncation at degree 2k + 1 and hence the Heun functions appearing in (72) are in fact Heun polynomials. The generating functions A(z) and B(z) are polynomials of degree 2k + 1.

Expansion of Heun polynomials in complementary Bannai-Ito polynomials
The expansion of the Heun polynomials can be obtained using the associated three-term recurrence relation. Since the expansion coefficients of B(z) and A(z) are related by the simple relation z ↔ −z and µ x ↔ µ y , we shall focus on the expansion of A(z).
We examine the expansion of the Heun polynomial appearing in (72a) where the parameters are given in (73). Using the recurrence coefficients given in (54), one finds that the expansion coefficients P n are polynomials of degree r in ξ = µ x − µ y that obey the recurrence relation with P −1 = 0, P 0 = 1 and where where we have taken N = 2k. Introducing the monic polynomials P r (ξ) = Pr(ξ) σ 1 ···σr , we obtain where Upon comparing (76) with the recurrence coefficients for the CBI polynomials given in (57), it is directly seen that the polynomials defined by the recurrence (76) correspond to CBI polynomials with the parametrization Hence, when r k, we have where ρ 1 , ρ 2 , r 1 and r 2 are given by (77) and I n (x; ρ 1 , ρ 2 , r 1 , r 2 ) are the complementary Bannai-Ito polynomials. As can be seen by the recurrence relation (75), the expansion coefficients of the Heun function (74) truncate at order 2k + 1 and are center-symmetric. Hence, for r > k, we have Taking into account the relation between A(z) and B(z), the expansion coefficients of the Heun function appearing in (72b), denoted by T n (ξ), are easily seen to be T r (ξ) = 4 r r! (k + 1 − r) r (2k + ζ + r − 1) r I r ((1 − ξ)/2; ρ 1 , ρ 2 , r 1 , r 2 ), r 2k, Collecting all the previous results, we write a n = P n (ξ) + P n−1 (ξ), where P −1 = 0, T −1 = 0 and P n (ξ), T n (ξ) are given by (78), (79), (80) and (81).

Eigenvectors of J 2
To obtain the expansion of the eigenvectors of J 2 in the circular basis, one must relate the eigenvectors of the operator Q to the eigenvectors of J 2 . This relation has been obtained in the previous paper [5]. We have where |k, ± J 2 are the eigenvectors of J 2 corresponding to the eigenvalues and where v k = ξ + 2i (k + µ x + 1/2)(k + µ y + 1/2) 2k + ζ + 1 .
The inverse relation reads |k, Using the relations (83), the results (82), (70) and (69), one has an explicit expression for the expansion of the eigenvectors of J 2 in the circular basis for the case N odd.

The fully isotropic case : −1 Jacobi polynomials
We consider again the case µ x = µ y = µ which corresponds to the fully isotropic Dunkl oscillator, where two independent identical parabosonic oscillators are combined. We return to the system of differential equations for the generating functions of the components of the eigenvectors of Q given in (71). When µ x = µ y = µ, one has It is easily seen from the above formulas that A(z) = B(−z). Hence the generating function A(z) satisfies the differential equation which may be cast in the form of an eigenvalue equation where Rf (z) = f (−z). It is recognized that the operator L is a special case of the defining operator of the little −1 Jacobi polynomials [17]. The little −1 Jacobi polynomials, denoted by P −1 n (x), obey the eigenvalue equation where Comparing the operators Ω and L, it is seen that the generating function A(z) corresponds to a −1 Jacobi polynomial of degree n = 2k + 1 with parameters Using this identification and the explicit formula for the little −1 Jacobi polynomials derived in [17], we obtain This directly yields the following result for the recurrence coefficients a n : Using the symmetry B(z) = A(−z), we also obtain Once again, the components of the eigenvectors of Q drastically simplify in the isotropic case µ x = µ y = µ. Moreover, the preceding computations entail a relation between a special case of Heun polynomials and the little −1 Jacobi polynomials.

Representations of sd(2) in the J 2 eigenbasis
We now investigate the representation space in which the operator J 2 is diagonal. The matrix elements of the generators of the Schwinger-Dunkl algebra will be derived using the defining relations of sd (2), which read where R 2 x = R 2 y = I. It will prove convenient to treat the even and odd dimensional representations separately.

The N odd case
We first consider the case where N is odd. The representation space C is spanned in this case by the basis vectors |k, ± with k ∈ {0, . . . , m} on which the generator J 2 acts in a diagonal fashion where m = (N − 1)/2 and where the eigenvalues of J 2 , derived in section 3, are given by λ ± k = ± (k + µ x + 1/2)(k + µ y + 1/2), k = 0, . . . , m.
Since R x , R y anti-commute with J 2 and given that R x R y is central in the algebra sd(2), we can take R x |k, ± = ǫ|k, ∓ , R y |k, ± = |k, ∓ , where ǫ = ±1. Here we choose ǫ = −1, which corresponds to the representation encountered in the model. In the basis {|0, + , |0, − , · · · , |m, + , |m, − }, the matrices representing the involutions R x , R y have the form which is identical to their action in the circular basis. The Hamiltonian H has the action H|k, ± = (N + µ x + µ y + 1)|k, ± .
The action of the operator J 3 on this representation space can be derived by imposing the commutation relation (6) using (6) to define J 1 . The action of J 3 on the basis |k, ± is taken to be With these definitions, it is easily seen that the commutation relation (6) is equivalent to the following system of relations where the relations (84a),(84b) were obtained by acting on |k, + and the relations (84c),(84d) by acting on |k, − . It follows directly from the solution of the system (84) that J 3 acts in a six-diagonal fashion on the eigenbasis of J 2 . For j = k, we obtain where ξ = µ x − µ y , ζ = µ x + µ y . For j = k + ℓ or j = k − ℓ with ℓ > 1, only the trivial solution occurs, so the matrix representing J 3 in the eigenbasis of J 2 is block tridiagonal with all block 2× 2. Hence it acts in a six-diagonal fashion on the eigenbasis of J 2 . Using the commutation relations and the system (84), it is possible to obtain an expression for the matrix elements of J 3 which involves a set of arbitrary non-zero parameters {β n } for n = 0, . . . , m. After considerable algebra, one finds that the matrix J 3 has the form where the blocks are given by The matrix elements of the central blocks are given by (85). The components of the upper blocks U k have the form The matrix elements of the lower blocks have the form M + k+1k = ξ(k + 1)(2k − N + 1)(k + 1 + ζ)(2k + 2ζ + N + 3) 4(2k + ζ + 2)(2k + ζ + 1) 3 , where E k = 1 2ξ 1 − 4(k + µ x + 1)(k + µ y + 1) + 4{(k + µ x + 1/2) 2 (k + µ y + 1/2) 2 } 1/2 .
We note that these matrix elements are valid for µ x = µ y . In the latter case, the form of the spectrum of J 2 changes and the computation has to be redone from the start.

The N even case
We now consider the N even case. The representation space C is spanned in this case by the basis vectors |0, − and |k, ± with k = 1, . . . , m on which the operator J 2 acts in a diagonal fashion J 2 |k, ± = λ ± k |k, ± , k = 0, . . . , m, where the eigenvalues of J 2 , determined in section 3, are given by the formula λ ± k = ± k(k + ζ), k = 0, . . . , m.
Note that the eigenvalue λ 0 is non-degenerate. In this representation, we choose the following action for the involutions R x , R y : and hence the reflections have the matrix representation R x = R y = diag(1, σ 1 , . . . , σ 1 ), σ 1 = 0 1 1 0 .
The parameters of the sequence {α k } are arbitrary but non-zero; they could be fixed, for example, by examining the action of J 2 on the eigenstates of the 2D Dunkl oscillator in the polar coordinate representation. We have thus obtained the action of the operator J 3 on the eigenstates of J 2 . Recall that J 3 is the symmetry operator associated to the separation of variables in Cartesian coordinates and J 2 is the symmetry associated to the separation of variables in polar coordinates.

Conclusion
We have investigated the finite-dimensional irreducible representations of the Schwinger-Dunkl algebra sd (2), which is the symmetry algebra of the two-dimensional Dunkl oscillator in the plane. The action of the symmetry generators in the representations were obtained in three different bases. In the Cartesian basis, the symmetry generator J 3 associated to separation of variables in Cartesian coordinates is diagonal, and the symmetry J 2 is tridiagonal. In the circular basis, the operator J 3 acts in a threediagonal fashion and J 2 has a block upper-triangular structure with all blocks 2×2. The eigenvalues of J 2 can be evaluated algebraically in the circular basis and the expansion coefficients for the eigenvectors of J 2 in this basis are generated by Heun polynomials and are expressed in terms of the para-Krawtchouk polynomials. Finally, it was shown that in the eigenbasis of J 2 , the operator J 3 acts in a block tridiagonal fashion with all blocks 2 × 2, that is, that J 3 is six-diagonal. It has been seen that the Dunkl oscillator model is superintegrable and closely related to the −1 orthogonal polynomials of the Bannai-Ito scheme. In this connection, the study of the 3D Dunkl oscillator model and the singular 2D Dunkl oscillator could also provide additional insight in the physical interpretation of the orthogonal polynomials of the Bannai-Ito scheme.