Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere

Transcription

1 Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere Willard Miller, [Joint with E.G. Kalnins (Waikato) and Sarah Post (CRM)] University of Minnesota Special Functions and Orthogonal Polynomials of Lie Groups and their Applications. Decin W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 1 / 33

2 Outline 1 Superintegrable Systems and their Algebras The Generic Superintegrable System on the -Sphere 3 The Generic Superintegrable System on the 3-Sphere Creating the Model The model and basis functions 4 Conclusion and Outlook W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin / 33

3 Superintegrable Systems and their Algebras Superintegrable Systems In n dimensions, we call a classical or quantum Hamiltonian H = n pi + V (x i ), H = + V (x i ) i=0 (maximally, Nth-order) Superintegrable if it admits n 1 symmetry operators, ie. {L i, H} = 0, [L i, H] = 0, i = 0,..., n 1, L 1 = H such that L,, L n 1 are polynomial, degree at most N, in the momenta or as differential operators. We also require that these operators be independent either functionally or algebraically. Superintegrable systems can be solved algebraically as well as analytically and are associated with special functions and exact solvability. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 3 / 33

4 Superintegrable Systems and their Algebras Integrability and Superintegrability An integrable system has n algebraically independent symmetry operators in involution. A superintegrable system has n 1 algebraically independent symmetry operators (the maximum possible). The symmetries of a merely integrable system generate an abelian algebra, those of a superintegrable system generate an algebra that is necessarily nonabelian. Claim: Superintegrability captures what it means for a Hamiltonian system to be explicitly solvable. Some simple but important superintegrable systems: Kepler-Coulomb problem: Kepler s 3 laws, Hohmann transfer for celestial navigation. hydrogen atom: periodic table of the elements classical and quantum harmonic oscillator W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 4 / 33

5 A D Example 1 Superintegrable Systems and their Algebras Here n =, so n 1 = 3. Hamiltonian: H = x + y α (4x + y ) + bx c y. Generating symmetry operators: H, L 1, L where L 1 = x 4α x bx L = 1 {M, y} + y ( b 4 xα ) ( 1 4 c ) x y. Here M = x y y x and {A, B} = AB + BA. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 5 / 33

6 A D Example Superintegrable Systems and their Algebras Add the commutator R = [L 1, L ] to the symmetry algebra. Then [L 1, R] = bh + 16α L bl 1 [L, R] = 8L 1 H 6L 1 H + bl 8α (1 c ) R + 4L L 1H 8L 1H + 16α L + 4bL H b{l 1, L } +16α (3 c )L 1 3α H b (1 c ) = 0 W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 6 / 33

7 Superintegrable Systems and their Algebras A D Example 3 We have a second order superintegrable system whose symmetry algebra closes at level 6. If Ψ is an eigenvector of H with eigenvalue E, i.e., HΨ = EΨ, and L is a symmetry operator then also H(LΨ) = E(LΨ), so the symmetry algebra preserves eigenspaces of H. Thus we can use the irreducible representations of the symmetry algebra to explain the accidental degeneracies of the eigenspaces of H. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 7 / 33

8 A D Example 4 Superintegrable Systems and their Algebras By classifying the finite dimensional irreducible representations of the symmetry algebra we can show that the possible bound state energy levels must take the form E m = 4α(m c) + b 16α where m = 0, 1, and the multiplicity of the energy level E m is m + 1. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 8 / 33

9 Superintegrable Systems and their Algebras General Associated Symmetry Algebras In general, the integrals of the motion generate algebra relations via taking commutators [L i, L j ] = R ij where R ij are (usually new) symmetries of the Hamiltonian. Often, if we include these new symmetries in our algebra, our algebra will close, i.e. [L i, R ij ] = P(L i ), P is a polynomial. Also, since n 1 is maximal number of algebraically independent symmetries, there will be functional relations between the R ij s the L i s. These relations determine our symmetry algebra, usually not a Lie algebra. The eigenspaces of the Hamiltonian (Schrödinger operator) are invariant under the action of the symmetry algebra, so the irreducible representations of the symmetry algebra enable us to determine the possible multiplicities of the eigenspaces and the spectrum. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 9 / 33

10 Superintegrable Systems and their Algebras Function Space Representations How do special functions and orthogonal polynomials arise? By solving the original quantum eigenvalue problem for the Hamiltonian via separation of variables. By finding other function space representations of the symmetry algebra action and solving those. In either case the symmetry algebra structure yields important information about the special functions. The quantum wave functions, along with the integrals, give a reducible representation of the algebra. We look for irreducible representations (so that the Hamiltonian is a constant) and look for operators which reproduce the structure equations. Taking a guide from classical mechanics, where such models are guaranteed to exist, we look for operators on a Hilbert space with n 1 dimensions. For the case of n =, all maximally superintegrable systems are known and their algebras close to form quadratic algebras. Each such algebra has been represented via function space realizations. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 10 / 33

11 Superintegrable Systems and their Algebras Plan of the remainder of the talk To introduce the superintegrable system on the n-sphere, which is in some sense the most general, and which admits a quadratic algebra structure. The representation theory of the -d system coincides with the theory of Wilson polynomials. The representation theory of the 3-d system coincides with the theory of a two variable generalization of the Wilson polynomials introduced by Tratnik [Tratnik 1991a, Tratnik 1991b]. These representations give information about the original system, i.e. eigenvalues of symmetry operators and inter-basis expansion coefficients. The algebra gives the structure of recurrence operators which act on the Wilson polynomials. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 11 / 33

12 The Generic Superintegrable System on the -Sphere The D System:Definition For n = we define the generic sphere system by embedding of the unit -sphere x1 + x + x 3 = 1 in three dimensional flat space. Then the Hamiltonian operator is H = 1 i<j 3 (x i j x j i ) + 3 a k x k=1 k, i xi. The 3 operators that generate the symmetries are L 1 = L 1, L = L 13, L 3 = L 3 where for 1 i < j 4. Here, H = 1 i<j 3 L ij L ji = (x i j x j i ) + a ixj xi L ij + 3 k=1 + a jxi xj, a k = H 0 + V, V = a 1 x 1 + a x + a 3 x3. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 1 / 33

13 The Generic Superintegrable System on the -Sphere The D System:Algebra For the generic -sphere quantum system the structure equations can be put in the symmetric form [Kalnins, Miller, and Post 007] ɛ ijk [L jk, R] = 4{L jk, L ij } 4{L jk, L ik } (8 + 16a j )L ik + (8 + 16a k )L ij + 8(a j a k ), R = 8 3 {L 3, L 13, L 1 } (16a 1 + 1)L 3 (16a + 1)L 13 (16a 3 + 1)L ({L 3, L 13 } + {L 13, L 1 } + {L 1, L 3 }) ( a 1)L ( a )L ( a 3)L (a 1 + a + a 3 ) +48(a 1 a + a a 3 + a 3 a 1 ) + 64a 1 a a 3. Here ɛ ijk is the pure skew-symmetric tensor, R = [L 3, L 13 ] and {A, B} = AB + BA with an analogous definition of {A, B, C} as a symmetrized sum of 6 terms. Also, recall L 3 = H L 1 L a 1 a a 3. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 13 / 33

14 The Generic Superintegrable System on the -Sphere The Wilson Polynomials Before we proceed to the model, we us present a basic overview of some of the characteristics of the Wilson polynomials [Wilson 1980] w n (t ) w n (t, α, β, γ, δ) = (α + β) n (α + γ) n (α + δ) n ( ) n, α + β + γ + δ + n 1, α t, α + t 4F 3 ; 1 α + β, α + γ, α + δ = (α + β) n (α + γ) n (α + δ) n Φ (α,β,γ,δ) n (t ). The polynomial w n (t ) is symmetric in α, β, γ, δ. The Wilson polynomials are eigenfunctions of a divided difference operator given as τ τφ n = n(n + α + β + γ + δ 1)Φ n where τ = 1 t E A F(t) = F(t + A), τ = 1 t (E 1/ E 1/ ), [ (α + t)(β + t)(γ + t)(δ + t)e 1/ (α t)(β t)(γ t)(δ t)e 1/]. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 14 / 33

15 The Generic Superintegrable System on the -Sphere The D System:Representation via Wilson Polynomials Using a 1 = 1 4 b 1, a = 1 4 b, a 3 = 1 4 b 3, and α = b 1+b +1 µ, β = b 1+b +1, γ = b b 1 +1, δ = b 1+b 1 + b + µ +, the algebra relations are realized by H = E and L 1 = 4t + b 1 + b, L 3 = 4τ τ (b + 1)(b 3 + 1) + 1, E 4µ + (b 1 + b + b 3 ) + 5)(4µ + (b 1 + b + b 3 ) b 1 b b 3. The model realizes the algebra relations for arbitrary complex µ and restricts to a finite dimensional irreducible representation when µ = m N. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 15 / 33

16 The Generic Superintegrable System on the 3-Sphere The 3D System:Definition We define the Hamiltonian operator via the embedding of the unit 3-sphere x1 + x + x 3 + x 4 = 1 in four dimensional flat space. H = 1 i<j 4 (x i j x j i ) + 4 a k x k=1 k A basis for the second order constants of the motion is for 1 i < j 4. Here, L ij L ji = (x i j x j i ) + a ixj xi H = 1 i<j 4 L ij + 4 a k. k=1, i xi. + a jxi xj, W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 16 / 33

17 The Generic Superintegrable System on the 3-Sphere The 3D System:Algebra In the following i, j, k, l are pairwise distinct integers such that 1 i, j, k, l 4, and ɛ ijk is the completely skew-symmetric tensor such that ɛ ijk = 1 if i < j < k. Let us define A to be the algebra generated by the L ij for all i, j = 1,.., 4 and I, the identity. The structure relations for the algebra are as follows: There are 4 linearly independent commutators of the second order symmetries (no sum on repeated indices) [Kalnins, Miller and Post 011]: R l = ɛ ijk [L ij, L jk ] The fourth order structure equations are [L ij, R j ] = 4ɛ ilk ({L ik, L jl } {L il, L jk } + L il L ik + L jk L jl ) [L ij, R k ] = 4ɛ ijl ({L ij, L il L jl } + ( + 4a j )L il ( + 4a i )L jl + a i a j ). W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 17 / 33

18 The fifth order structure equations are obtainable directly from the fourth order equations and the Jacobi identity. The sixth order structure equations are R l = 8 3 {L ij, L ik, L jk } (1 + 16a k )L ij (1 + 16a i )L jk (1 + 16a j )L ik ({L ij, L ik + L jk } + {L ik, L jk }) + ( a k)l ij + ( a i)l jk +( a j)l ij + 64a i a j a k + 48(a i a j + a j a k + a k a i ) (a i + a j + a k ), ɛ ikl ɛ jkl {R i, R j } = 4 3 ({L il, L jk, L kl } + {L ik, L jl, L kl } {L ij, L kl, L kl }) {L ik, L jl } {L il, L jk } {L ij, L kl } + 4L kl {L jl + L jk + L il + L ik, L kl } (6 + 8a l ){L ik, L jk } (6 + 8a k ){L il, L jl } 3 3 L kl ( 8 3 8a l)(l jk + L ik ) ( 8 3 8a k)(l jl + L il ) +( a k + 4a l + 3a k a l )L ij 16(a k a l + a k + a l ).

19 Finally, there is also an eighth order functional relation between the 6 integrals which is [ 1 8 L ij L kl 1 9 {L ik, L il, L jk, L jl } 1 36 {L ij, L ik, L kl } i,j,k,l 7 6 {L ij, L ij, L kl } (1 + 3 a l){l ij L ik L jk } + 3 L ijl kl ( a k 3 4 a l a k a l )L ij +( a l){l ik, L jk }+( 4 3 a k a l a ka l )L ij + 3 a ia j a k a l + a i a j a k a ia j ] = 0 Here, {A, B, C, D} is the 4 term symmetrizer of 4 operators and the sum is taken over all pairwise distinct i, j, k, l. For the purposes of the representation, it is useful to redefine the constants as a i = b i 1 4.

20 Subalgebras The Generic Superintegrable System on the 3-Sphere We note that the algebra described above contains several copies of -sphere algebra. Then, we can see that there exist subalgebras A k generated by the set {L ij, I} for i, j k and that these algebras are exactly those associated to the D analog of this system. Furthermore, if we define H k i<j,i,j k L ij ( j k b i 3 4 )I then H k will commute with all the elements of A k and will represent the Hamiltonian for the associated system. For example, A 4 A the algebra generated by L 1, L 13, L 3 and the identity, I has as it center H 4 = L 1 + L 13 + L 3 + (3/4 b 1 b b 3)I which is the Hamiltonian for the associated -sphere system. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 0 / 33

21 The Generic Superintegrable System on the 3-Sphere Creating the Model Creating the Model:A basis for L 13, H 4 As described above, we seek to construct a representation of A by extending the representations obtained for the subalgebras A k. The most important difference for our new representation is that the operator H 4 is in the center of A 4 but not A. Hence, it can no longer be represented as a constant. We use the information about its eigenvalues to make an informed choice for its realization by choosing variables t and s, such that H 4 = 1 4 4s, L 13 = 4t 1 + b 1 + b 3. In this basis, the eigenfunctions d l,m for a finite dimensional representation are given by delta functions d l,m (s, t) = δ(t t l )δ(s s m ), 0 l m M, with the spectrum of s is {( s m ) = (m (b 1 + b + b 3 )/) } and the spectrum of t is {t l = (l + (b 1 + b + 1)/) }. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 1 / 33

22 The Generic Superintegrable System on the 3-Sphere Creating the Model Creating the Model: A basis for L 1, H 4 Using the information from the representation of A 4 we hypothesize that L 1 take the form of an eigenvalue operator for Wilson polynomials in the variable t L 1 = 4τt τ t (b 1 + 1)(b + 1) + 1/ with parameters are given by: α = b s, β = b 1 + b 3 + 1, γ = b 1 b 3 + 1, δ = b + 1 s. The basis functions corresponding to diagonalizing H 4 and L 1 can be taken, essentially, as the Wilson polynomials f n,m (t, s) = w n (t, α, β, γ, δ)δ(s s m ), where s m = m (b 1 + b + b 3 )/ as above. Note that w n (t ) actually depends on m (or s ) through the parameters α, δ. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin / 33

23 The Generic Superintegrable System on the 3-Sphere Creating the Model Creating the Model: A basis for L 13, L 4 A reasonable guess of the form of the operator L 4 is as a difference operator in s, since it commutes with L 13. For L 4 we take L 4 = 4 τ s τ s (b + 1)(b 4 + 1) + 1. Here τ s is a difference operator in s with parameters α = t + b + 1 b 1 + b + b 3, β = M 1, γ = M + b 4 + b 1 + b + b 3 +, δ = t + b + 1. With the operator L 4 thus defined, the unnormalized eigenfunctions of the commuting operators L 13, L 4 in the model take the form g n,k where 0 l M, 0 k M l, and g l,k = δ(t t l )w k (s, α, β, γ, δ). W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 3 / 33

24 To complete the model, we will need parameter dependent raising and lowering operators for the Wilson polynomials, RΦ n = R = 1 y [T 1/ T 1/ ]. n(n + α + β + γ + δ 1) (α + β)(α + γ)(α + δ) Φ(α+1/,β+1/,γ+1/,δ+1/) n 1. L = 1 h (α 1/ + y)(β 1/ + y)(γ 1/ + y)(δ 1/ + y)t 1/ y (α 1/ y)(β 1/ y)(γ 1/ y)(δ 1/ y)t 1/i. LΦ n = (α + β 1)(α + γ 1)(α + δ 1)Φ (α 1/,β 1/,γ 1/,δ 1/) n+1. L αβ = 1 h (α 1/ + y)(β 1/ + y)t 1/ + (α 1/ y)(β 1/ y)t 1/i. y L αβ Φ n = (α + β 1)Φ (α 1/,β 1/,γ+1/,δ+1/) n.

25 We finalize the construction of our model by realizing the operator L 34. The operator L 34 must commute with L 1, so we hypothesize that it is of the form L 34 = A(s)S(L αβ L αγ ) t + B(s)S 1 (R αβ R αγ ) t + C(s)(LR) t + D(s). On the other hand, we can consider the action of L 34 on the basis g l,k. Considering L 34 primarily as an operator on s we hypothesize that it must be of the form L 34 = Ã(t)T (L α β L α γ) s + B(t)T 1 (R α β R α γ ) s + C(t)(LR) s + D(t)s + Ẽ(t) + κl 1. By a long and tedious computation we can verify that the 3rd order structure equations are satisfied for certain choices of functions. We also obtain the quantization for E, for finite dimensional representations, E = (M + 4 b j + 3) ) 1. j=1

26 The functional coefficients for L 34 take the following form : A(s) = (M + b 1 + b + b 3 s + )(M + b 1 + b + b 3 + b 4 + s + 4), s(s + 1) B(s) = (M + b 1 + b + b 3 + s + )(M + b 1 + b + b 3 + b 4 s + 4), s(s 1) C(s) = + (M + b 1 + b + b 3 + 3)(M + b 1 + b + b 3 + b 4 + 3) 4s, 1 D(s) = s ( M + b 1 + b + b 3 + b ) (b 1 + b ) + b 3 + b 4 + b 3 + b 4 + M ((b 1 + b + 1) b 3 )(M + b 1 + b + b 3 + 3)(M + b 1 + b + b 3 + b 4 + 3) (4s 1) Ã(t) = (b 1 b 3 + t + 1)(b b 3 + t), t(t + 1) B(t) = (b 1 b 3 t + 1)(b b 3 t), t(t 1) C(t) = + (b 3 b 1 ) 4t, 1 D(t) =, and κ = 4. The expression for Ẽ(t) takes the form Ẽ(t) = µ 1 + µ /(4t 1) where µ 1, µ are constants, but we will not list it here in detail.

27 We shall now review what we have constructed, up to this point. We realize the algebra A by the following operators H = (M + H 4 = 1 4 4s 4 b j + 3) + 1 I j=1 L 13 = 4t 1 + b 1 + b 3 L 1 = 4τ t τ t (b 1 + 1)(b + 1) + 1/ L 4 = 4 τ s τ s (b + 1)(b 4 + 1) + 1 L 34 = A(s)S(L αβ L αγ ) t + B(s)S 1 (R αβ R αγ ) t + C(s)(LR) t + D(s) The operators L 3, L 14 can be obtained through linear combinations of this basis.

28 We have computed three sets of orthogonal basis vectors corresponding to diagonalizing three sets of commuting operators, {L 13, H 4 }, {L 1, H 4 } and {L 13, L 4 }, respectively, d l,m (s, t) = δ(t t l )δ(s s m ), 0 l m M, (1) f n,m (s, t) = w n (t, α, β, γ, δ)δ(s s m ), 0 n m M, () g l,k (s, t) = w k (s, α, β, γ, δ)δ(t t l ) 0 l k + l M (3) We also have a nonorthogonal basis given by h n,k (s, t) = t n s k, 0 n + k M. Recall that the spectrum of the variables s, t is given by t l = l + b 1 + b 3 + 1, s m = (m b 1 + b + b 3 ), 0 l m M.

29 The Generic Superintegrable System on the 3-Sphere Weight Function Creating the Model It is possible to write an inner product on the finite dimensional representation as f (t, s), g(t, s) = f (t, s)g(t, s)ω(t, s)δ(t t l )δ(s s m )dsdt, with ω(t l, s m) = M!(1 + b 4) M m (M + b 1 + b + b 3 + b 4 + 3) m(m + + b 1 + b + b 3 ) (M m)!(1 + b 4 ) M (M + b 1 + b + b 3 + 3) m( + b 1 + b + b 3 ) (1 + b 1) l (1 + b 1 + b 3 ) l (1 + b ) m l ( + b 1 + b + b 3 ) m+l (l b 1 + b 3 ) l!(m l)!(1 + b 3 ) l ( + b 1 + b 3 ) m+l (1 + b 1 + b 3 )c 0,0 Th normalization of the identity 1, 1 = 1 gives c 0,0 = (3 + b 1 + b + b 4 ) M (3 + b 1 + b + b 3 ) M (1 + b 4 ) M (1 + b 3 ) M, W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 9 / 33

30 The Generic Superintegrable System on the 3-Sphere Creating the Model Relation with Tratnik Polynomials The two-variable extension of the Wilson polynomials defined by Tratnik [Tratnik 1991a, Tratnik 1991b] are given for 0 n 1 n 1 + n M by R (n 1, n ; β i ; x 1, x ; M) = r n1 (β 1 β 0 1, β β 1 1, x 1, x + β 1 ; x 1 ) r n (n 1 + β β 0 1, β 3 β 1, n 1 M 1, n 1 + β + M; n 1 + x ) Recall, the Racah polynomials are related to the Wilson polynomials via ( r n (a, b, c, d, x) = w n ã, b, c, d, (x + ã) ) The weight function for the -variable Tratnik polynomials, R agree with ω(x, y) under the substitution x 0 = 0, x 1 = t b 1 + b 3 + 1, x = s 1 b 1 + b + b 3, x 3 = M, β 0 = b 3, β 1 = b 1 + b 3 + 1, β = b 1 + b + b 3 +, β 3 = b 1 + b + b Further, it can be seen by direct computation that the operators which are diagonalized by the Tratnik polynomials [Geronimo and Iliev 010] can be written in terms of the operators L 1 and L 14 + L 1 + L 4 and so form a basis for the representation of A. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 30 / 33

31 Conclusion and Outlook Recap To introduce the superintegrable system on the n-sphere, which is in some sense the most general, and which admits a quadratic algebra structure. The representation theory of the -d system can be constructed in terms of Wilson polynomials. The representation theory of the 3-d system can be constructed in terms of a two variable generalization of the Wilson polynomials introduced by Tratnik. These representations give information about the original system, i.e. eigenvalues of symmetry operators and inter-basis expansion coefficients. The algebra gives the structure of recurrence operators which act on the Wilson polynomials. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 31 / 33

32 Conclusion and Outlook Future Projects Study the system on the n-sphere Consider limiting processes on the system to get other representations in terms of orthogonal polynomials. Representations of higher-order algebras and relations with higher-order integrals. Thanks for listening. W. Miller (University of Minnesota) Two-Variable Wilson Polynomials Decin 3 / 33

33 M. V. Tratnik. Some multivariable orthogonal polynomials of the Askey tableau-continuous families. J. Math. Phys. 3(8), , M. V. Tratnik. Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 3(9), 33734, Fokas A S and Gel fand I M 1996 Surfaces on Lie groups, on Lie algebras, and their integrability Comm. Math. Phys J. S. Geronimo and P. Iliev. Bispectrality of Multivariable Racah-Wilson Polynomials. Constructive Approximation, 31, , DOI: /s , 010 J. Wilson, Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal., 1980, 11, W. Miller Jr., A note on Wilson polynomials, SIAM J. Math. Anal., 1987, E. G. Kalnins, W. Miller, Jr and S. Post. Wilson polynomials and the generic superintegrable system on the -sphere. J. Phys. A: Math. Theor. 40, , (007). E. G. Kalnins, W. Miller, Jr and S. Post. Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere. Sigma 7, 051, (011).