Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions.


Introduction
The connection between group theory, special functions and orthogonal polynomials is an area that has been of significant interest for many years now and has yielded many beautiful results fundamental in theory and in application. In this work, we give a Lie algebraic description of Tratnik's extension of the Racah polynomials [18]. This description relies on the connection between these polynomials and representations of the quadratic algebra associated with the 'generic' superintegrable system on the three sphere [11] and its realization in terms of positive discrete series representations of su(1, 1) obtained by Genest and Vinet [4]. The system is given by the following superintegrable Hamiltonian and is referred to a generic since all non-degenerate, second-order superintegrable systems on conformally flat pseudo-Euclidean spaces of three dimensions can be obtained through appropriate limits from this system [1]. It admits 6 linearly independent constants of the motion and hence is considered superintegrable [16].
Remarkably, Genest and Vinet [4] have shown that this system can be realized as the tensor product of four copies of positive discrete series representations of su(1, 1). They have used this representation to identify the 9j symbols of the algebra and derive identities for the symbols using the separated solutions of the Hamiltonian (1) in cylindrical coordinates, which are given in terms of Jacobi polynomials. The 9j symbols are given by rational functions multiplied by the vacuum coefficients.
In this paper, we discuss the same tensor product representation but instead focus on bases which separate in spherical coordinates and their associated coupling schemes. We show that bivariate Racah polynomials can be obtained as expansion coefficients between two spherical coordinate systems, or equivalently, two different coupling schemes. The bispectrality of the polynomials is obtained naturally from the model. We show also that there is another set of commuting difference operators for the bivariate Racah polynomials which fits naturally into the scheme. When this difference operator is added to the algebra, it closed to form a quadratic algebra on 9 generators, which we will call QR(9) as it is an extension of the Racah algebra QR(3) from the univariate case.
The remainder of the paper is organized as follows. Section 2 contains the necessary background material on the two-sphere case and the realization of univariate Racah polynomials as expansion co-efficients between different bases for three copies of su(1, 1). Section 3 contains the main results of the paper, namely the connection between the bivariate Racah polynomials and the coupling schemes of four copies of su(1, 1). Section 4 contains the results concerning the additional difference operator and the algebra QR(9) as well as an interpretation of the bispectrality properties of the Racah polynomials in terms of the tensor product representation. Section 5 briefly gives some conclusions and future outlook.

Background
2.1 Positive discrete series representations of su(1, 1) Consider the following operator realization of the positive discrete series representation of su(1, 1) satisfying the su(1, 1) commutation relations As is well known, the tensor product of multiple copies of such representations will itself be a positive discrete series representation of su(1, 1). For example, the tensor product of two such representations would have generators where the upper indices indicate the factor in the tensor product. A subscript is also appended to the variable to distinguish each factor in the representation, i.e.
and similarly for K (i) ± . The tensor product can be decomposed into irreducible components. In the case of the tensor product of two representations, the irreducible components are indexed by an integer N such that the total Casimir − } takes the values ν 12 = ν 1 + ν 2 + N on each irreducible component. In the case of two or more tensor products, the decomposition is not canonical. For example for three components, it is possible to decompose first by coupling V (1) and V (2) into components of V (12) with each irreducible component indexed by the parameter ν 12 and then adding the third component. The basis in this coupling scheme would be |ν 12 , ν with action of operators Q (12) |ν 12 , ν = ν 12 (ν 12 − 1)|ν 12 , ν , Coupling in the other direction, by reducing first V (2) ⊗ V (3) into irreducible components indexed by ν 23 and then into components indexed by the total Casimir operator gives the following alternate basis Q (23) |ν 23 , ν = ν 23 (ν 23 − 1)|ν 12 , ν , ν 23 = ν 2 + ν 3 + n Here Q is the total Casimir operator and is given by .

Three products of su(1, 1) and the Racah algebra
The decomposition of three products of su(1, 1) and its connection with the Racah algebra was discovered by Genest, Vinet and Zhedanov [6]. The algebra and its connection with Racah polynomials will be fundamental for the following sections, so we recall the results here. Consider the following interbasis expansion coefficients P ν23,ν12 = ν 23 , ν|ν 12 , ν , and define the following operators: Having k 1 act on the right gives but also k 1 will also act on |ν 23 , ν as for some expansion coefficients C µ . In order to determine these expansion coefficients and to hence use the recurrence relation (7) we will identify the algebra generated by k 1 and k 2 with the quadratic Racah algebra QR(3) [8,19]. Indeed, computing the commutator of k 1 and k 2 lead to a third operator k 3 defined as where the operator R can be expressed as the sum of permutations of (123) Here ǫ ijk is the sign of the permutation (ijk). This formula holds in general, namely for the operators R ijk are defined via To complete the algebra relations, it remains only to compute the commutators [k 1 , k 3 ] and [k 3 , k 2 ], which are determined via and So that the quadratic algebra relations are with d = 1 2 (Q (1) + Q (2) + Q (3) + Q (123) ), Consider now the Racah polynomials defined by which are polynomials of degree n in λ(x) = x(x + γ + δ + 1). For consistency with the following sections, we shall parametrize the polynomials as The polynomials satisfy the following eigenvalue equation with The operator L(x, β, N ) is of the form given in [7]. Note that the function κ is related to the values of the Casimir operators as follows Defining the operators gives a realization of the algebra QR(3) (11) with the following parameters Note that k R 1 is essentially multiplications by the argument λ(x) = x(x + β 1 − 1) translated. Finally, we are in a position to determine the expansion coefficients P ν23,ν12 as Racah polynomials and identify the operators k 1 and k 2 (5, 6) with k R 1 and k R 2 (19). The identification is accomplished by taking The identification is only determined up to a possible choice of conjugation, namely To determine the necessary gauge, it is important to use the requirement that the basis be normalized. Therefore, we chose The action of the operator Q (23) can then be determined as follows: and so Thus finally, we see the action of Q (23) on the basis |ν 12 , ν is given by where T x |ν 12 , ν = |ν 12 + 1, ν and similarly for T −1 3 Four products of su(1, 1) and bivariate Racah Polynomials In this section we consider four products of su(1, 1) and determine their relation to the bivariate polynomials obtained by Tratnik [18]. As above, we obtain the polynomials through interbasis expansion coefficients and identify them by comparing the algebra of the operators with the algebra generated by the recurrence operators for the polynomials. We take as our initial bases the vectors obtained by coupling first V (1) with V (2) to obtain components V (12) and then coupling this with V (3) to obtain components V (123) and then finally with V (4) to obtain a complete basis indexed by ν 12 , ν 123 and ν satisfying Q (12) |ν 12 , ν 123 , ν = ν 12 (ν 12 − 1)|ν 12 , ν 123 , ν , We will build up the action of each of the intermediate Casimir operators Q (ij) on this basis using the quadratic algebra generated by these 6 operators plus their four linearly independent commutator, the R ijk 's (8), for example R 123 = [Q (12) , Q (23) ] = [Q (13) , Q (12) ] = [Q (23) , Q (13) ].
The remaining algebra relations are obtained through direct, though tedious computations using the definitions of R ijk and Q (ij) . They are as follows Notice that for a fixed ijk, the quadratic algebra relations (28) can be expressed via Q (ijk) as and hence replicate the Racah algebra relations on components with fixed values of the Casimir Q (ijk) by taking k 1 = −Q (ij) /2 and k 2 = −Q (jk) /2. This subalgebra will be utilized to obtain the action of the operator Q (23) on the basis (26). (12) , Q (23) } and univariate Racah Polynomials

The basis for {Q
Let us now consider the basis for V composed of eigenvectors for the commuting operators Q (23) and Q (123) , namely |ν 23 , ν 123 , ν as well as the action of Q (23) on the basis |ν 12 , ν 123 , ν . In order to determine that action of Q (23) on this basis, we note that Q (23) also commutes with Q (123) and so the action of Q (23) will be just to shift the index ν 12 . Indeed the action will be the same as the product of three representations and so the operator Q (23) will be exactly the eigenvalue shift operator of the Racah polynomials give in (25) except that the parameter ν 123 is no longer the total Casimir and is instead given by ν 123 = ν 1 + ν 2 + ν 3 + x 2 . The action of Q (23) on the basis is given by (25) with parameters chosen as in (23) except the integer N is replaced by x 2 . The expansion coefficients for these two bases are then with ω x1 σ n1 = ν 23 (0), ν 123 , ν|ν 12 (x 1 ), ν 123 , ν ν 23 (n 1 ), ν 123 , ν|ν 12 (0), ν 123 , ν , where the following shorthand is used to show the dependence on the integers x 1 and n 1 : The operators are then Q (12) = κ(x 1 , β 1 ),
However, the action of Q (234) , as well as Q (34) , on the basis |ν 12 , ν 123 , ν , or equivalently as shift operators in the variable x 1 and x 2 , have not yet been determined. Matching the eigenvalues of K 4 leads to the identification where with the remaining β given by The final operator to be determined is K 5 , associated with the action of Q (34) . Since Q (34) commutes with Q (12) , its action on the basis will be to shift the variable x 2 . We hypothesize that it is of the form K 5 P (n; x; β; N ) ≡ − 1 2 ν 23 , ν 234 , ν|Q (34) |ν 12 , ν 123 , ν , with K 5 P (n; x; β; N ) ≡ − 1 2 (ω x σ n ) 1/2 Ω 1 (x, β, N )(ω x σ n ) −1/2 P (n; x; β; N ), The exact form ofB(x) andẼ(x) are determined from the quadratic algebra relations, and are given bỹ Thus, we have realized each of the six linearly independent quadratic Casimirs Q (ij) in terms of the following operators Note that the algebra relations do not close without the addition of the shift operator representing Q (34) .

Some consequences of the model
In this section, we derive several results concerning the bivariate Racah polynomials which arise as a result of the above theory.

The algebra QR(9)
By construction, the operators (38-43) give simply another basis for the quadratic Casimir operators and so the algebra generated by these operators is identical to the algebra defined by (27-28). However, we would like to see this as an extension of the algebra QR(3) and so we chose the following basis Here we have dropped the normalization and weight factors since they have no bearing on the algebra relations. Note that whereas the algebra generate by the operators Q (ij) had 6 linearly independent generators, we have restricted to components with a fixed N and so Q, representing the total Casimir operator, will not be considered a generator of the model. This is analogous to the univariate case and the algebra QR(3).
As we shall see, there will be several copies of the the algebra QR(3) in this algebra. The first being the one generated by K 1 and K 2 since these are simply the operators which act on the first factor of R 2 (n; x; β, N ). In order to obtain this algebra, we include the commutator of K 1 and K 2 calling it L 1 . It would be K 3 in QR(3). The copy of the algebra QR(3) is then given by Notice that both K 1 and K 2 commute with K 3 so d 1 , e 11 , and e 12 act as constant for the sub-algebra generated by K 1 , K 2 and L 2 .
We continue taking commutators and see but [K 1 , K 4 ] = 0 and so we call it L 2 . The algebra generated by these operators closes again to form a copy of QR(3) but only if the operator K 5 is included in the list of generators. The algebra relations become The operator K 5 is a shift operator in the variable x 2 only so we expect it to generate, with K 3 another copy of QR(3). We shall see in the next section that this is due to the fact that it is possible to diagonalize K 5 with univariate Racah polynomial in the variable x 2 . Defining gives Again, note that K 1 commutes with both K 3 and K 5 . The final copy of QR(3) is generated by K 2 and K 5 and it is The operators L 1 , L 2 , L 3 and L 4 exhaust the set of linearly independent commutators of the K j 's. The only remaining non-zero commutator is The remaining algebra relations defining QR (9) can, as with the previous ones, be directly determined by (27-28) but we give a set for example. They are To summarize, as in the univariate case, the algebra generated by the eigenvalue operators and multiplication by the variables of the bivariate Racah polynomials can be closed to form a quadratic algebra. But unlike the univariate case, an additional operator not arising as a commutator of two basis operators must be adjoined to the algebra. This operator is a shift operator in the second variable, x 2 . With this additional operator, the algebra closes and has 9 generators, the operators K 1 through K 5 and their 4 linearly independent commutators. The algebra contains at least 4 subalgebras isomorphic to QR(3).
Finally, we emphasize that this algebra is isomorphic to the algebra generated by the operators Q (ij) whose algebra is much more compactly expressed via (27-28). The form of the algebra in terms of the K i is given here to emphasize that it is a natural extension of the algebra QR (3).
Indeed, making the identifications the two commuting operators can be realized as A set of simultaneous eigenvectors for these operators are the normalized, Racah polynomials R 2 (m;ỹ;γ; N ) satisfying the eigenvalues equations To return to x 1 and x 2 we transform to the dual variables, or rather as these were set up already as the dual variables, we transform back to the variables y 1 , y 2 and m 1 , m 2 . In particular, the polynomials can be represented as R 2 (m;ỹ;γ; N ) = R 2 (m; y; γ; N ), with y 1 = x 2 + N + 2ν 1 + 2ν 2 + 2ν 3 − 1, The eigenvalue equations for the polynomials in y, defined via (64), and m are Thus, we see that the operator Ω x 1 (x, β, N ) is another operator that commutes with Λ x 2 (x, β, N ) and the set of these operators are diagonalized by the bivariate Racah polynomials R 2 (m, y, γ, N ).

Conclusions
In this article, we have seen how the bivariate Racah polynomials are related to different su(1, 1) coupling schemes, in analogy with the univariate case [6]. The bivariate Racah polynomials are seen to be expansion coefficients between sets of eigenfunctions for the Hamiltonian (1) in different choices of spherical coordinates. This Hamiltonian arises as the Casimir operator of the tensor product of four copies of su(1, 1) [4] and its conserved quantities are seen to be represented in terms of intermediate Casimir operators for different coupling schemes.
As consequences of the model, we have seen that the algebra generated by multiplication by the arguments of the polynomials along with the eigenvalue operators for the bivariate Racah polynomials do not close to form a quadratic algebra, as in the univariate case, unless an additional operator is included, beyond the operators arising directly as a commutator of the generators. This additional operator is a shift operator in the second variable, x 2 and commutes with shift operator Λ x 2 (or equivalently L x 2 ). Including this additional operator, the algebra is generated by 5 operators plus their 4 linearly independent commutators. This algebra is isomorphic to the algebra generated by the intermediate Casimir operators for the tensor product of four su(1, 1) representations. However, taking the basis as in (44-46) makes explicit that the algebra is an extension of the Racah algebra QR(3) and indeed contains at least four subalgebras isomorphic to QR(3). We have also seen how the symmetry of the Hamiltonian and the representation in terms of the tensor product allows for an natural definition of the dual variables and bispectrality properties of the polynomials. Finally, from the model and the algebra relations we have identified an additional operator Ω x 1 that commutes with Λ x 2 but is a shift operator in the second variable only. This additional operator is necessary for the quadratic algebra to close. The eigenfunctions of {Ω x 1 , Λ x 2 } are also bivariate Racah polynomials but of different arguments and with a different choice of parameters.
Let us finish by mentioning several other possible consequences of this model that would be interesting to investigate. The first is to utilize more explicitly the separated eigenfunctions of the Hamiltonian (1) to derive integral formulas and identities for the Racah polynomials, analogous to the results of [4] for the 9j coefficients of the same model. A similar investigation has also been completed for the univariate case [15].
Another interesting avenue of research would be to consider the limits of these Racah polynomials in terms of the contractions of algebra. Again, in the univariate case, it has been shown that the limits of the Racah polynomials are generated by contractions of the quadratic algebras [12] and furthermore that all such contractions are in fact generated by Lie algebra contractions [10]. For the bivariate case, some results have already been obtained. Equivalence classes of second-order superintegrable systems in conformally flat space have been identified [2] as have the contractions between them [1]. It remains to identify the corresponding limits of the polynomials and the action on the algebras. As an example, the multivariate Hahn polynomials have been represented in a similar manner [5] corresponding to the singular harmonic oscillator in 3D, which can be obtained from the Hamiltonian (1) by an appropriate limit. The connection between these contraction limits, their quadratic algebras and different coupling schemes should provide a fruitful context for classifying multivariate orthogonal polynomials and their limits.
Finally, we mention that this analysis would also be interesting to extend to the q-case and to consider the appropriate algebra for the bivariate q-Racah polynomials [9] and the connections with Leonard triples [3], the universal enveloping algebra U q (sl 2 ) [17] and Hecke algebras [13,14].