Bilinear Summation Formulas from Quantum Algebra Representations

Groenevelt, Wolter

doi:10.1007/s11139-004-0145-1

Cited by 19 publications

(40 citation statements)

References 28 publications

(52 reference statements)

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“…The Racah coefficients for the tensor product π + k 1 ⊗ π + k 2 ⊗ π − k 3 can now be determined from v m+r n (j; k 1 , k 2 )F r−j m (τ ; k 1 + k 2 + j, k 3 ) = U (j, ρ; k 1 , k 2 , k 3 , τ )F r−n m (ρ; k 2 , k 3 )G r n (τ ; k 1 , ρ, k 2 − k 3 ) dν(ρ; k 1 , k 2 , k 3 , τ ), or equivalently from [29] showed that they occur as spherical functions for SU q (2), see also [31] and [21]. In [33] the Askey-Wilson polynomials occur as matrix elements for representations of U q (su (1, 1)), in [25], [13] and [8] they occur as Clebsch-Gordan coefficients.…”

Section: (84)mentioning

confidence: 99%

Wilson Function Transforms Related to Racah Coefficients

Groenevelt

2006

Acta Appl Math

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Abstract. The irreducible * -representations of the Lie algebra su(1, 1) consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch-Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for Uq(su(1, 1)), which turn out to be Askey-Wilson functions and Askey-Wilson polynomials. IntroductionIn this paper we study Racah coefficients, or 6j-symbols, for representations of the Lie algebra su(1, 1). The Racah coefficients for certain tensor product representations of su(1, 1) lead to unitary integral transforms with a very-well-poised 7 F 6 -function, called a Wilson function, as a kernel. These Wilson functions and the corresponding integral transforms were recently introduced by the author in [12] with the applications in this paper in mind. We also consider a q-analogue of su(1, 1), namely the quantized universal enveloping algebra U q (su (1, 1)). From the Racah coefficients for certain tensor product representations of U q (su (1, 1)) we obtain a new interpretation of the Askey-Wilson functions transform introduced by Koelink and Stokman [23].As an application we obtain several identities for special functions involving (Askey-) Wilson functions and polynomials.Racah coefficients for su(2) play an important role in the theory of angular momentum in quantum physics [7]. They were first studied in the 1940's by Racah [32], who also obtained an explicit expression as a finite single sum for these coefficients. Only much later it was realized that the Racah coefficients are multiples of polynomials of hypergeometric type, so that the orthogonality relations for the Racah coefficients lead to discrete orthogonality relations for certain polynomials. These polynomials are nowadays called the Racah polynomials, which can be defined bywhere one of the lower parameters is equal to −N , N ∈ Z ≥0 , and 0 ≤ n ≤ N . These are polynomials of degree n in the variable x(x + γ + δ + 1), and they are orthogonal on the set {0, 1, . . . , N };

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Section: (84)mentioning

confidence: 99%

Wilson Function Transforms Related to Racah Coefficients

Groenevelt

2006

Acta Appl Math

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“…compatible with the assignmenth A,1 defined by (22) and (25) are introduced for the infinite components of K 1,1 bȳ…”

Section: Infinite Border Stripsmentioning

confidence: 99%

“…. constitute an infinite-dimensional U q sl(2) -moduleV (y) corresponding to the principal unitary series representation π P ρ,0 of U q su(1, 1) with cos θ = 1 2 q 2iρ + q −2iρ [21], [22]. Each representation π P ρ,0 is irreducible [23].…”

Section: The Mixed Casementioning

confidence: 99%

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A -vertex model: Creation algebras and quasi-particles I

Gade¹

2005

Nuclear Physics B

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Abstract. The infinite configuration space of an integrable vertex model based on Uq gl(2|2) 1 is studied at q = 0. Allowing four particular boundary conditions, the infinite configurations are mapped onto the semi-standard supertableaux of pairs of infinite border strips. By means of this map, a weight-preserving one-to-one correspondence between the infinite configurations and the normal forms of a pair of creation algebras is established for one boundary condition. A pair of type-II vertex operators associated with an infinite-dimensional Uq gl(2|2) -moduleV and its dualV * is introduced. Their existence is conjectured relying on a free boson realization. The realization allows to determine the commutation relation satisfied by two vertex operators related to the same Uq gl(2|2) -module. Explicit expressions are provided for the relevant R-matrix elements. The formal q → 0 limit of these commutation relations leads to the defining relations of the creation algebras. Based on these findings it is conjectured that the type II vertex operators associated withV andV * give rise to part of the eigenstates of the row-to-row transfer matrix of the model. A partial discussion of the R-matrix elements introduced onV ⊗V * is given.

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“…for the polynomials (12). We call the polynomials d n (µ(m); a, b|q) dual little q-Jacobi polynomials.…”

Section: Dual Little Q-jacobi Polynomialsmentioning

confidence: 99%