Abstract. We construct a commutative algebra A z , generated by d algebraically independent q-difference operators acting on variables z 1 , z 2 , . . . , z d , which is diagonalized by the multivariable Askey-Wilson polynomials P n (z) considered by Gasper and Rahman (2005). Iterating Sears' 4 φ 3 transformation formula, we show that the polynomials P n (z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A n , generated by d algebraically independent difference operators acting on the discrete variables n 1 , n 2 , . . . , n d , which is also diagonalized by P n (z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.
In 1971, Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra sl d+1 (C). Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d 2 + d + 1 terms. This, combined with the duality, establishes their bispectrality. We illustrate our results with several explicit examples.
We consider the generic quantum superintegrable system on the d-sphere with potential V (y) = d+1 k=1 b k y 2 k , where b k are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in d variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in [9]. The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik's multivariable Racah polynomials.
The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.
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 andH. We display an antiautomorphism † of sl 3 (C) that fixes each element of H and each element ofH. 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 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 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 andH are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of H andH on V as a rank 2 generalization of a Leonard pair.
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