Askey-Wilson polynomials: an affine Hecke algebra approach
Askey-Wilson polynomials: an affine Hecke algebra approach
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发表时间：
2000-01
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通讯作者：
M. Noumi;J. Stokman
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作者：
M. Noumi;J. Stokman
利用双仿射Hecke代数的表示理论研究Askey-Wilson型多项式。特别地，我们证明了非对称和反对称的Askey-Wilson多项式关于复测度的双正交关系。我们给出了非对称的Askey-Wilson多项式的对偶性质，并且我们展示了如何从Sahi的交织器中创建非对称的Askey-Wilson多项式。与双正交关系相关的对角项（它取代了正交多项式的二次范数评估的概念）使用双仿射Hecke作用下非对称Askey-Wilson变换的交织性质以复权重函数的留数表示。代数。我们评估的常数项，这基本上是著名的Askey-Wilson积分，使用移位算子。我们还展示了这些结果如何减少到众所周知的对称Askey-Wilson多项式的属性，因为最初是由Askey和Wilson使用基本的超几何级数理论。
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.