Multivariable Askey–Wilson function and bispectrality

Geronimo, Jeffrey S.; Илиев, Пламен

doi:10.1007/s11139-010-9244-3

Cited by 16 publications

(13 citation statements)

References 23 publications

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“…In the context of q−special functions and the q−Virasoro algebra, a similar extension has been considered too [67,108]. In view of these, it is clear that an extension of the bispectral problem (2.16), (2.17) for N > 1 can be considered similarly, see for instance [109]. These works may establish a relation between the q−Onsager algebra and an algebraic structure extending the q−Virasoro algebra.…”

Section: Perspectivesmentioning

confidence: 99%

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A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

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Abstract. For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q = 1).MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

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Section: Perspectivesmentioning

confidence: 99%

“…Van Diejen and E. Koelink for sending us reprints [17] and [110], respectively. P.B thanks E. Koelink for detailed explanations on the work [110], as well as P. Iliev for communications and sending us a reprint of [109]. We thank P. Terwilliger for important comments on the manuscript and sending us the preprint [43].…”

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confidence: 99%

A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

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“…where h n (x, α, β, N; q) are the q-Hahn polynomials (15). These polynomials are orthogonal with respect to the same measure as the Gasper-Rahman q-Hahn polynomials H (α) n (y) given by (25).…”

Section: Another Family Of Multivariate Q-hahn Polynomialsmentioning

confidence: 99%

“…where h n (x, α, β, N; q) and p n (x, α, β; q) are the q-Hahn and q-Jacobi polynomials defined in (15) and (34). We now set x 1 = (u − v)/2, x 2 = (u + v)/2 as well as t = q u+1 , and we consider the limit of (51) as v → ∞.…”

Section: The D = 3 Case: One-variable Q-racah Polynomialsmentioning

confidence: 99%

Coupling coefficients of su(1,1) and multivariate q-Racah polynomials

2018

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Gasper & Rahman's multivariate q-Racah polynomials are shown to arise as connection coefficients between families of multivariate q-Hahn or q-Jacobi polynomials. The families of q-Hahn polynomials are constructed as nested Clebsch-Gordan coefficients for the positive-discrete series representations of the quantum algebra su q (1,1). This gives an interpretation of the multivariate q-Racah polynomials in terms of 3n j symbols. It is shown that the families of q-Hahn polynomials also arise in wavefunctions of q-deformed quantum Calogero-Gaudin superintegrable systems.Keywords: multivariate q-Racah polynomials, representations of su q (1,1), Clebsch-Gordan coefficients, superintegrable systems.Coupling coefficients of su q (1, 1) & multivariate q-Racah polynomials bases of multivariate Jacobi polynomials on the simplex [5] and used this to compute connection coefficients for families of discrete classical orthogonal polynomials studied in [19], as well as for orthogonal polynomials on balls and spheres. In [18], Iliev has established the connection between the bispectral operators for the multivariate Racah polynomials and the symmetries of the generic superintegrable system on the d-sphere. In [2], De Bie, Genest, van de Vijver and Vinet have unveiled the relationship between this superintegrable model, d-fold tensor product representations of su(1, 1) and the higher rank Racah algebra. See also [8,29,30].Many of these results have yet to be extended to the q-deformed case. The interpretation of the univariate q-Racah polynomials as 6 j coefficients for the quantum algebra su q (1, 1) is well known [37], as is its relation with the Zhedanov algebra and the operators involved in the bispectrality of the one-variable q-Racah polynomials [16,38]. Dunkl has also shown in [4] that these polynomials arise as connection coefficients between bases of two-variable q-Jacobi or q-Hahn polynomials. On the multivariate side, Rosengren has observed that q-Hahn polynomials arise by considering nested Clebsch-Gordan coefficients for su q (1, 1) and derived some explicit formulas [31]. Moreover, Scarabotti has examined similar families of multivariate q-Hahn polynomials associated to binary trees and their connection coefficients [32].Nevertheless, the identification of the multivariate q-Racah polynomials as 3n j coefficients of the quantum algebra su q (1, 1) has not been achieved. Moreover, the connection between q-Racah polynomials, both univariate and multivariate, and superintegrable systems remains to be determined. The present paper addresses these questions. As stated above, it will be shown that Gasper & Rahman's multivariate q-Racah polynomials arise as connection coefficients between bases of multivariate orthogonal q-Hahn or q-Jacobi polynomials. The bases will be constructed using the nested Clebsch-Gordan coefficients for multifold tensor product representations of su q (1, 1), which will provide the exact interpretation of the multivariate q-Racah polynomials in terms of coupling coefficients for that quantum algebr...

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“…We will recover the orthogonality relations with respect to (1.3) below. Iliev [15] (see also [12]) showed that the multivariate Askey-Wilson polynomials are eigenfunctions of d commuting difference operators. Furthermore, the multivariate Askey-Wilson polynomials are also eigenfunctions of commuting difference equations in m, i.e.…”

Section: Introductionmentioning

confidence: 99%

A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

Groenevelt

2019

International Mathematics Research Notices

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We study matrix elements of a change of base between two different bases of representations of the quantum algebra Uq(su(1, 1)). The two bases, which are multivariate versions of Al-Salam-Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman's multivariate Askey-Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey-Wilson polynomials are solutions of a multivariate bispectral q-difference problem.Date: September 13, 2018.

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Multivariable Askey–Wilson function and bispectrality

Cited by 16 publications

References 23 publications

A bispectral q-hypergeometric basis for a class of quantum integrable models

A bispectral q-hypergeometric basis for a class of quantum integrable models

Coupling coefficients of su(1,1) and multivariate q-Racah polynomials

A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

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