Askey-Wilson polynomials: an affine Hecke algebra approach

By M. Noumi, Jasper V. Stokman. 2000

81 citations recorded by the source corpus.

We study Askey-Wilson type polynomials using representation theory of the double affine Hecke algebra. In particular, we prove bi-orthogonality relations for non-symmetric and anti-symmetric Askey-Wilson polynomials with respect to a complex measure. We give duality properties of the non-symmetric Askey-Wilson polynomials, and we show how the non-symmetric Askey-Wilson polynomials can be created from Sahi's intertwiners. The diagonal terms associated to the bi-orthogonality relations (which replace the notion of quadratic norm evaluations for orthogonal polynomials) are expressed in terms of residues of the complex weight function using intertwining properties of the non-symmetric Askey-Wilson transform under the action of the double affine Hecke algebra. We evaluate the constant term, which is essentially the well-known Askey-Wilson integral, using shift operators. We furthermore show how these results reduce to well-known properties of the symmetric Askey-Wilson polynomials, as were originally derived by Askey and Wilson using basic hypergeometric series theory.

Subfields: Algebra and Number Theory, Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/36c412fe2f04f5f924c19329aad63e87e66b4b22

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Citations in the corpus

The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case (2006, methodology): “raspherical polynomials. For the DAHA of type (C∨ 1 ,C 1) (four parameters) the non-symmetric Askey–Wilson polynomials are obtained. These were earlier treated by Sahi [12] and by Noumi & Stokman [9]. See also Sahi’s recent paper [13]. Comparison of Zhedanov’s AW(3) with the DAHA of type of type”
Double affine Hecke algebras of rank 1 and affine cubic surfaces (2003, methodology): “This structure turned out to be the rank 1 double affine Hecke algebra (DAHA) introduced by Cherednik [5], or, more precisely, its generalization H = H(t1, t2, t3, t4; q) to the nonreduced root system C ∨ C1, studied by Sahi, Noumi and Stokman [ 14 , 16, 17].”
Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces (2004, methodology): “In the D4 case, it is the double affine Hecke algebra of typeC ∨ C1 of Sahi, Noumi and Stokman [Sa, St, NoSt ].”
Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations (2001, result): “We would like to acknowledge that the results of this section can be readily obtained from any of the following works: Granovskii, Lutzenko, and Zhedanov [21], Grunbaum and Haine [27], Noumi and Stokman [43].”
Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra ? (2007, background): “pe(C∨ 1,C 1). Inthebasic(orpolynomial)representationofthisDAHA(denoted byH˜)onthespaceofLaurentpolynomialsinz,thenonsymmetricAskey–Wilson polynomials occuraseigenfunctionsofasuitableelementY ofH˜ (see[13,9],[8,Ch.6]). Itturnsout(see[7]) that a central extension AWg(3,Q 0) of AW(3,Q 0) can beembedded”
Cherednik and Hecke algebras of varieties with a finite group action (2004, methodology): “The fact that this algebra is flat was proved by Cherednik, Sahi, Noumi, Stokman (see [Ch],[Sa],[NoSt],[St]) using a different approach (q-deformed Dunkl operators).”
Askey--Wilson polynomials, quadratic harnesses and martingales (2008, background): “On the other hand, sometimes new examples arise when processes run only on a subinterval of (0,∞).”