Little and big q-Jacobi polynomials and the Askey–Wilson algebra

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 $$\mathfrak U_q(sl(2))$$.