The Rahman polynomials and the Lie algebra sl_3(C)

We interpret the Rahman polynomials in terms of the Lie algebra $sl_3(C)$. Using the parameters of the polynomials we define two Cartan subalgebras for $sl_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism $\dagger$ of $sl_3(C)$ that fixes each element of $H$ and each element of $\tilde{H}$. We consider a certain finite-dimensional irreducible $sl_3(C)$-module $V$ consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form $<,>$ on $V$ such that $<\beta \xi,\zeta>=<\xi,\beta^\dagger \zeta>$ for all $\beta \in sl_3(C)$ and $\xi,\zeta \in V$. We display two bases for $V$; one diagonalizes $H$ and the other diagonalizes $\tilde{H}$. Both bases are orthogonal with respect to $<,>$. We show that when $<,>$ is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we recover the previously known orthogonality relation for the Rahman polynomials. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by the dual polynomials. These recurrence relations show that the Rahman polynomials are bispectral. In our theory the roles of $H$ and $\tilde{H}$ are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of $H$ and $\tilde{H}$ on $V$ as a rank 2 generalization of a Leonard pair.

We will use the fact that for nonnegative integers m, n (−m) n = 0 if n > m.
We credit the following result to Rahman and Hoare [5]; however a similar and more general theorem was given earlier in [11,Theorem 1.1].
In this paper we interpret the Rahman polynomials in terms of the Lie algebra sl 3 (C).
Our results are summarized as follows. Using the parameters {p i } 4 i=1 we define two Cartan subalgebras for sl 3 (C), denoted H andH. We display an antiautomorphism † of sl 3 (C) that fixes each element of H and each element ofH. We consider an irreducible sl 3 (C)-module V consisting of the homogeneous polynomials in three variables that have total degree N. We display a nondegenerate symmetric bilinear form , on V such that βξ, ζ = ξ, β † ζ for all β ∈ sl 3 (C) and ξ, ζ ∈ V . We display two bases for V ; one diagonalizes H and the other diagonalizesH. Both bases are orthogonal with respect to , . We show that when , is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we obtain an elementary proof of Theorem 1.1. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by their duals. These recurrence relations show that the Rahman polynomials are bispectral, a feature hinted at by Grünbaum [4]. We view the actions of H andH on V as a rank 2 generalization of a Leonard pair [12]; this is discussed along with some open problems at the end of the paper.

The Lie algebra sl 3 (C)
We will be discussing the Lie algebra sl 3 (C); for background information see [1]. Let Mat 3 (C) denote the C-algebra consisting of the 3 × 3 matrices that have all entries in C. We index the rows and columns by 0, 1, 2. For 0 ≤ i, j ≤ 2 let e ij denote the matrix in Mat 3 (C) that has (i, j)-entry 1 and all other entries 0. The Lie algebra sl 3 (C) is the set of matrices in Mat 3 (C) that have trace 0, together with the Lie bracket [β, γ] = βγ − γβ. We will consider two Cartan subalgebras of sl 3 (C), denoted H andH. The subalgebra H consists of the diagonal matrices in sl 3 (C). Define Then ϕ, φ form a basis for H. We will describeH shortly. Define A brief calculation shows νW UW U t = I, where I denotes the identity matrix and U t denotes the transpose of U. Define and note that ϑθ = ν. Define R =θW U t and note that R −1 = ϑW U. We have Note that RW R t =θϑ −1W . For 0 ≤ i, j ≤ 2 defineẽ ij = Re ij R −1 . DefineH = RHR −1 and note thatH is a Cartan subalgebra of sl 3 (C). Defineφ = RϕR −1 andφ = RφR −1 . Note thatφ,φ is a basis forH. We havẽ By an antiautomorphism of sl 3 (C) we mean a C-linear bijection θ : There exists an antiautomorphism † of sl 3 (C) such that β † =W β tW −1 for all β ∈ sl 3 (C). Note that (β † ) † = β for all β ∈ sl 3 (C). We have β e 01 e 12 e 02 e 10 e 21 e 20 ϕ φ β † e 10η1 /η 0 e 21η2 /η 1 e 20η2 /η 0 e 01η0 /η 1 e 12η1 /η 2 e 02η0 /η 2 ϕ φ (4) Note that † fixes each element of H and each element ofH. We now show that H,H together generate Then

An sl (C)-module
We now define a certain sl 3 (C)-module. Let x, y, z denote mutually commuting indeterminates. Let C[x, y, z] denote the C-algebra consisting of the polynomials in x, y, z that have all coefficients in C. We abbreviate A = C[x, y, z]. By a derivation of A we mean a C-linear map ∂ : A → A such that ∂(ξζ) = ∂(ξ)ζ + ξ∂(ζ) for all ξ, ζ ∈ A. By [1, p. 207] the space A is an sl 3 (C)-module on which each element of sl 3 (C) acts as a derivation and ξ e 01 .ξ e 12 .ξ e 02 .ξ e 10 .ξ e 21 .ξ e 20 .ξ ϕ.ξ φ.ξ Let V denote the subspace of A consisting of the homogeneous polynomials that have total degree N. The following is a basis for V : The action of sl 3 (C) on this basis is described as follows.
By the above data V is an sl 3 (C)-submodule of A, and this submodule is irreducible [9, p. 97]. Let I denote the set consisting of the 3-tuples of nonnegative integers whose sum is N. For λ = (r, s, t) ∈ I let V λ denote the subspace of V spanned by x r y s z t . Then V = λ∈I V λ (direct sum) and this is the weight space decomposition of V with respect to H. By construction dim(V λ ) = 1 for all λ ∈ I.
We now consider the weight space decomposition of V with respect toH. To describe this decomposition we make a change of variables. Recall the matrix R and definẽ Thus R is the transition matrix from x, y, z tox,ỹ,z. Using R =θW U t we obtaiñ Using R −1 = ϑW U we obtain The action of sl 3 (C) onx,ỹ,z is described as follows.
The action of sl 3 (C) on this basis is described as follows.
For each λ = (r, s, t) ∈ I letṼ λ denote the subspace of V spanned byx rỹszt . Observe that V = λ∈IṼ λ (direct sum) and this is the weight space decomposition of V with respect tõ H. By construction dim(Ṽ λ ) = 1 for all λ ∈ I.
We comment on how H andH act on the weight spaces of the other one. A pair of elements (r, s, t) and (r ′ , s ′ , t ′ ) in I will be called adjacent whenever (r −r ′ , s−s ′ , t−t ′ ) is a permutation of (1, −1, 0). Then H andH act on the weight spaces of the other one as follows. For all λ ∈ I,H

A bilinear form
In this section we introduce a symmetric bilinear form , on V . As we will see, both We define , as follows. With respect to , the vectors (6) are mutually orthogonal and We are using the notation ξ 2 = ξ, ξ . The form , is symmetric, nondegenerate, and satisfies (12). Using (4) and the tables below (6) we obtain Line (13) follows from (15) and since † fixes each element ofH. The following is the basis for V that is dual to (6) with respect to , .
The vectorx N is equal to N!ν N times the sum of the vectors (16); this is verified using (7) and the trinomial theorem.
Evaluating (18) using these comments we obtain We have verified (17) and the lemma is proved. 2 The following is the basis for V that is dual to (10) with respect to , .
The vectorx N is equal to N!ν N times the sum of the vectors (19).

The Rahman polynomials and sl 3 (C)
In this section the Rahman polynomials are related to the sl 3 (C)-module V .
We would like to acknowledge that the following result is similar to [11, Section 2], although the setup is somewhat different.
Theorem 5.1 For a basis vectorx ρỹσzτ from (10), For a basis vector x ρ y σ z τ from (6), Proof: We first prove (20). Since ρ + σ + τ = N, Using (8) and the trinomial theorem, Similarly using (9), Evaluating (22) using the above comments and (1), we obtaiñ In the expression (23) consider the first factor. Using (7) and the trinomial theorem, Therefore (23) is equal to which is equal to After changing variables s = b + i + j, t = c + k + ℓ and using (1) the above sum becomes 0≤r,s,t r+s+t=N Therefore (23) is equal to (24). Upon replacing (23) by (24) the expressionx ρỹσzτ becomes a sum which is equal to the right-hand side of (20). This proves (20) and the proof of (21) is similar. 2 Theorem 5.2 For nonnegative integers s, t whose sum is at most N, both P (s, t,φ + N/3 I,φ + N/3 I)x N = x r y s z t , (25) where r = N − s − t.
Proof: By (20),x In the above equation take the inner product of each side with x r y s z t , and use the fact that the bases (6), (16) are dual with respect to , . 2 Proof of Theorem 1.1. We first verify (2). Abbreviate ρ = N − σ − τ . By (20) we havẽ In the above equation take the inner product of each side withx rỹszt . In the resulting equation simplify the left-hand side using Lemma 4.1 and simplify the right-hand side using Theorem 5.3. This gives (2). Line (3) is similarly verified. 6 Some seven-term recurrence relations In this section we display two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by their duals. To obtain these relations we use the sl 3 (C)-module V from Section 3. We would like to acknowledge here the work of Grünbaum [4,Section 6], which hints at the existence of these relations and contains a wealth of related data for the case N = 5.
Theorem 6.1 Fix nonnegative integers s, t whose sum is at most N, and nonnegative integers σ, τ whose sum is at most N. Then (i)-(iv) hold below.
(i) (s − N/3)P (s, t, σ, τ ) is a weighted sum with the following terms and coefficients: is a weighted sum with the following terms and coefficients: (iii) (σ − N/3)P (s, t, σ, τ ) is a weighted sum with the following terms and coefficients: is a weighted sum with the following terms and coefficients: In the above tables r = N − s − t and ρ = N − σ − τ .
The left-hand side of (27) is equal to N!ν N times (s − N/3)P (s, t, σ, τ ). To see this, first observe ϕ.x r y s z t = (s − N/3)x r y s z t from the tables below (6), and then use Theorem 5.3. The right-hand side of (27) is equal to N!ν N times the weighted sum in the theorem statement. To obtain this fact, reduce the right-hand side of (27) using the following three steps: (a) eliminate ϕ using the long formula in Section 2; (b) evaluate the result using the tables below (10); (c) apply Theorem 5.3. The result follows from the above comments.
(ii)-(iv) Similar to the proof of (i) above. 2 We interpret Theorem 6.1 as follows. Parts (i), (ii) indicate that the Rahman polynomials are common eigenvectors for a pair of difference operators, while parts (iii), (iv) indicate that the dual Rahman polynomials are common eigenvectors for analogous difference operators.
Following [2] we can say that the Rahman polynomials solve a bispectral problem. This feature of the Rahman polynomials was hinted at earlier by Grünbaum [4].

Directions for future research
Motivated by the results in this paper we now pose some problems. To state the problems we adopt a general point of view. Let F denote a field and let V denote a vector space over F with finite positive dimension. Let End(V ) denote the F-algebra consisting of the F-linear maps from V to V . Fix integers M ≥ 1 and N ≥ 0. Let I = I(M, N) denote the set consisting of the (M +1)-tuples of nonnegative integers whose sum is N. A pair of elements (r 0 , r 1 , . . . , r M ) and (r ′ 0 , r ′ 1 , . . . , r ′ M ) in I will be called adjacent whenever (r 0 − r ′ 0 , r 1 − r ′ 1 , . . . , r M − r ′ M ) is a permutation of (1, −1, 0, 0, . . . , 0). Problem 7.1 Find all the pairs H,H that satisfy the following conditions.
(i) H is an M-dimensional subspace of End(V ) whose elements are diagonalizable and mutually commute.
(ii)H is an M-dimensional subspace of End(V ) whose elements are diagonalizable and mutually commute.
(iii) There exists a bijection λ → V λ from I to the set of common eigenspaces of H such that for all λ ∈ I,H (iv) There exists a bijection λ →Ṽ λ from I to the set of common eigenspaces ofH such that for all λ ∈ I, (v) There does not exist a subspace W of V such that HW ⊆ W ,HW ⊆ W , W = 0, W = V .
Note 7.3 For M = 1 a solution to Problem 7.1 is essentially the same thing as a Leonard pair [12]. The Leonard pairs have been studied extensively and are well understood; see [13] and the references therein. The Leonard pairs correspond to a class of orthogonal polynomials in one variable. This class coincides with the terminating branch of the Askey scheme [10] and consists of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito, and orphan polynomials [13,Section 35]. We expect that for general M the solutions to Problem 7.1 correspond to a class of M-variable orthogonal polynomials that resembles the above class. In particular the M-variable polynomials from [3, Appendix A.3], [6], [11] are likely members of this class. For the ambitious reader we now pose some harder problems. Note 7.6 For M = 1 a solution to Problem 7.5 is essentially the same thing as a tridiagonal pair [8]; for a discussion of tridiagonal pairs see [7] and the references therein. The tridiagonal pairs over an algebraically closed field are classified [7, Corollary 18.1].
Problem 7.7 Above Problem 7.1 we defined an adjacency relation on a set I. Given its features a Lie theorist will recognize that the solutions to Problems 7.1, 7.5 belong to the root system A M . Investigate the analogous objects that belong to other root systems.

Acknowledgement
The second author would like to thank Hajime Tanaka for several illuminating conversations on the general subject of this paper, and in particular for pointing out reference [11]. We believe that [11] is important and deserves to be better known among researchers in special functions.