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

TL;DR

This paper introduces nonsymmetric Laurent polynomials derived from $Q$-polynomial distance-regular graphs with $q$-Racah type, establishing their orthogonality and relation to nonsymmetric Askey-Wilson polynomials.

Contribution

It extends the theory of $q$-Racah polynomials by constructing nonsymmetric Laurent polynomials from graph structures and linking them to the universal double affine Hecke algebra.

Findings

Orthogonality relations for the Laurent polynomials are established.

The Laurent polynomials are shown to relate to nonsymmetric Askey-Wilson polynomials.

The module structure of the graph's characteristic vectors under $$-Hecke algebra is characterized.

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 $(C^{\vee}_1, C_1)$. 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.