The Selberg Integral and Its Applications

Andrews, George E.; Askey, Richard; Roy, Ranjan

doi:10.1017/cbo9781107325937.009

Cited by 18 publications

(30 citation statements)

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“…Note that Γ ∅ , Γ [1] and Γ [n] are not included in the generating set of Y n , as they are central in A n . It is manifest from the defining relations (21) that Y n is the largest possible Abelian subalgebra of A n .…”

Section: Proposition 2 ([19]mentioning

confidence: 99%

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The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

Bie

Genest

Vinet

2016

Advances in Mathematics

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Abstract. The kernel of the Z n 2 Dirac-Dunkl operator is examined. The symmetry algebra An of the associated Dirac-Dunkl equation on S n−1 is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the DiracDunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of An and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of An is proposed.

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Section: Proposition 2 ([19]mentioning

confidence: 99%

“…In (35) and (36), P (α,β) n (x) are the Jacobi polynomials with P (α,β) −1 (x) = 0 and (a) n = Γ(a+n) Γ(a) the Pochhammer symbol [1]. Proof.…”

Section: Proposition 5 ([2]mentioning

confidence: 99%

The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

Bie

Genest

Vinet

2016

Advances in Mathematics

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“…where 2 F 1 is the classical hypergeometric series [23]. It must be stressed finally that a myriad of analytic models can be generated from those associated to known orthogonal polynomials by a procedure known as "spectral surgery" [10].…”

Section: A Review Of Perfect State Transfer In a XX Spin Chainmentioning

confidence: 99%

Quantum spin chains with fractional revival

2016

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Abstract. A systematic study of fractional revival at two sites in XX quantum spin chains is presented and analytic models with this phenomenon are exhibited. The generic models have two essential parameters and a revival time that does not depend on the length of the chain. They are obtained by combining two basic ways of realizing fractional revival in a spin chain each bringing one parameter. The first proceeds through isospectral deformations of spin chains with perfect state transfer. The second arises from the recurrence coefficients of the para-Krawtchouk polynomials with a bi-lattice orthogonality grid. It corresponds to an analytic model previously identified that can possess perfect state transfer in addition to fractional revival.

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“…Anyway, according to refs. [148,187], this analytic continuation can be embodied in an integrate form …”

Section: D2 Integration Of Legendre Polynomialsmentioning

confidence: 99%