An Inverse to the Askey-Wilson Operator

Abstract

We study properties of the kernel of a right inverse of the Askey-Wilson divided difference operator on L 2 weighted with the weight function of the continuous q-Jacobi polynomials. This operator is embedded in a one-parameter family of integral operators, denoted by D -t q whose kernel is related to the Poisson kernel. It is shown that as t → 1 - , the t-commutator (D q D -t q - tD -t q D q )f tends to the constant term in the orthogonal expansion of f in continuous q-Jacobi polynomials.