## Dualities in the \$q\$-Askey scheme and degenerated DAHA. (arXiv:1803.02775v1 [math.CA])

The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials \$R_n[z]\$ which are eigenfunctions of a second-order \$q\$-difference operator \$L\$, and of a second-order difference operator in the variable \$n\$ with eigenvalue \$z + z^{-1}=2x\$. Then \$L\$ and multiplication by \$z+z^{-1}\$ generate the Askey-Wilson (Zhedanov) algebra. A nice property of the Askey-Wilson polynomials is that the variables \$z\$ and \$n\$ occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures: the Askey-Wilson algebra and the double affine Hecke algebra (DAHA). In this paper we follow the degeneration of the Askey-Wilson polynomials until two arrows down and in four diferent situations: for the orthogonal polynomials themselves, for the degenerate Askey-Wilson algebras, for the non-symmetric polynomials and for the (degenerate) DAHA and its re查看全文

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 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials \$R_n[z]\$ which are eigenfunctions of a second-order \$q\$-difference operator \$L\$, and of a second-order difference operator in the variable \$n\$ with eigenvalue \$z + z^{-1}=2x\$. Then \$L\$ and multiplication by \$z+z^{-1}\$ generate the Askey-Wilson (Zhedanov) algebra. A nice property of the Askey-Wilson polynomials is that the variables \$z\$ and \$n\$ occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures: the Askey-Wilson algebra and the double affine Hecke algebra (DAHA). In this paper we follow the degeneration of the Askey-Wilson polynomials until two arrows down and in four diferent situations: for the orthogonal polynomials themselves, for the degenerate Askey-Wilson algebras, for the non-symmetric polynomials and for the (degenerate) DAHA and its re