Multivariable <i>q</i>‐Hahn polynomials as coupling coefficients for quantum algebra representations

Rosengren, Hjalmar

doi:10.1155/s0161171201012017

Cited by 10 publications

(15 citation statements)

References 32 publications

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“…The functions R (α) m (n) are connection coefficients between families of multivariate orthogonal polynomials. Indeed, it follows from (23), (31) and (45) that 1…”

Section: Connection Coefficients Between Multivariate Polynomialsmentioning

confidence: 99%

“…The coefficients Ψ (α) n (y) can be viewed as nested Clebsch-Gordan coefficients for the positive discrete series of irreducible representations of su q (1,1). A different approach to obtain multivariate q-Hahn polynomials was outlined by Rosengren in [31]. Note that…”

Section: First Basismentioning

confidence: 99%

“…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.…”

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confidence: 99%

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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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“…The functions R (α) m (n) are connection coefficients between families of multivariate orthogonal polynomials. Indeed, it follows from (23), (31) and (45) that 1…”

Section: Connection Coefficients Between Multivariate Polynomialsmentioning

confidence: 99%

Section: First Basismentioning

confidence: 99%

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confidence: 99%

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

2018

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“…We shall take up the problem of special 2-variable orthogonal polynomials that are limiting cases of R τ m,n (x, y) in a subsequent paper. We also need to examine the relationship, if any, of the polynomials obtainable from (5.3) with a host of 2-variable orthogonal polynomials that have been found by various authors in the past 30 years or so, see, for example, [13,25,8,22,16,26,5,9,30,19,15].…”

Section: A Product Formulamentioning

confidence: 99%

A q-analogue of the 9-j symbols and their orthogonality

Rahman

2009

Journal of Approximation Theory

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We introduce a q-analogue of Wigner's 9-j symbols following the notational scheme used by Wilson in identifying the 6-j symbols with Racah polynomials, which eventually led Askey and Wilson to obtain a q-analogue of them, namely, the q-Racah polynomials. Most importantly, we prove the orthogonality of our analogues in complete generality, as well as derive an explicit polynomial expression for these new functions.

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“…However, there seems to be at least one more extension that, to our knowledge, has not yet been investigated. The seed of this extension lies in Rosengren's [8] multivariable extension of the q-Hahn polynomials as well as in Rahman's [7] 2-variable discrete biorthogonal system. In sections 5 and 6 we shall prove the following 2-variable extension of the q-Racah polynomial orthogonality [2, (7.2.18)]:…”

Section: Introductionmentioning

confidence: 99%