In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah polynomials and their relatives in the Askey scheme from the duality property of $Q$-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the $q$-Racah polynomials in the above situation. Let $Γ$ denote a $Q$-polynomial distance-regular graph that contains a Delsarte clique $C$. Assume that $Γ$ has $q$-Racah type. Fix a vertex $x \in C$. We partition the vertex set of $Γ$ according to the path-length distance to both $x$ and $C$. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra $\hat{H}_q$ of type $(C^{\vee}_1, C_1)$. From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the $\hat{H}_q$-module and the theory of Leonard systems. Changing $\hat{H}_q$ by $\hat{H}_{q^{-1}}$ we show how our Laurent polynomials are related to the nonsymmetric Askey-Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric $q$-Racah polynomials.
Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for Macdonald-Koornwinder polynomials
我们提出了代表阿斯基-威尔逊多项式的四重级数展开。为了得到这个结果，我们依次使用了几个基本超几何级数的求和和变换公式，包括Verma的域的q-扩张和Wimp展开式，Andrews的Watson的3F 2和的终止q-模拟，Singh的二次变换.作为应用，我们给出了BCn（n in Z>0）型Koornwinder多项式的一个列图显式公式.在参数特殊化的情况下，我们用一行图恢复了Bn，Cn和Dn型Macdonald多项式的Lassalle公式，从而证明了他的结论.
CNRS, IRIF Université Paris Cité（法国国家科学研究中心，IRIF巴黎西岱大学）；Department of Mathematics, University of California Berkeley（加州大学伯克利分校数学系）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Nu Quantum Ltd.（Nu量子有限公司）；Department of Mathematics, Sungkyunkwan University (SKKU)（成均馆大学（SKKU）数学系）
本文为 $q$-Askey 方案和 Askey 方案中的每个正交多项式族建立了以 lecture hall 图上的路径表示的混合矩和系数的组合模型，推广了 Corteel 和 Kim 关于小 $q$-Jacobi 多项式的结果，并首次给出了 Askey-Wilson 多项式参数对称性的组合证明。
