Little and Big $q-$Jacobi Polynomials and the Askey-Wilson algebra

TL;DR

This paper explores how little and big q-Jacobi polynomials serve as basis vectors in representations of the Askey-Wilson algebra, revealing their algebraic structure and embedding within quantum groups.

Contribution

It establishes the connection between q-Jacobi polynomials and the Askey-Wilson algebra through representation theory and algebraic embeddings.

Findings

q-Jacobi polynomials form basis vectors for Askey-Wilson algebra representations

Operators diagonalizing these polynomials are identified within the algebra

Embedding of the Askey-Wilson algebra into U_q(sl(2)) is realized through these operators

Abstract

The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra generated by twisted primitive elements of $\mathfrak U_q(sl(2))$. The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding of the Askey-Wilson algebra into $\mathfrak U_q(sl(2))$.