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

Abstract

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 $\Gamma$ denote a $Q$-polynomial distance-regular graph that contains a Delsarte clique $C$. Assume that $\Gamma$ has $q$-Racah type. Fix a vertex $x\in C$. We partition the vertex set of $\Gamma$ 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 $(C_1^{\vee },C_1)$. 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.

38 pages, 3 figures

References in corpus (3)

• Distance-regular graphs
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Some algebra related to $P$-and $Q$-polynomial association schemes

Cited by in corpus (3)

• Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
• Grassmann graphs, degenerate DAHA, and non-symmetric dual $q$-Hahn polynomials
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle