Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs

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• 作者：Jae-Ho Lee ^{jhlee@ims.is.tohoku.ac.jp}
• 关键词：Askey&ndash ; Wilson polynomial ; Nonsymmetric Askey&ndash ; Wilson polynomial ; q-Racah polynomial ; Nonsymmetric q-Racah polynomial ; DAHA of rank one ; Distance-regular graph ; Q-polynomial
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：147
• 期：Complete
• 页码：75-118
• 全文大小：756 K
• 卷排序：147

文摘

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∈Cx∈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 Hˆq of type (C1∨,C1). From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the Hˆq-module and the theory of Leonard systems. Changing Hˆq by Hˆq−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.