Formulas and identities involving the Askey–Wilson operator

We derive two new versions of Cooper's formula for the iterated Askey–Wilson operator. Using the second version of Cooper's formula and the Leibniz rule for the iterated Askey–Wilson operator, we derive several formulas involving this operator. We also give new proofs of Rogers' summation formula for ϕ56 series, Watson's transformation, and we establish a Rodriguez type operational formula for the Askey–Wilson polynomials. In addition we establish two integration by parts formulas for integrals involving the iterated Askey–Wilson operator. Using the first of these integration by parts formulas, we derive a two parameter generating function for the Askey–Wilson polynomials. A generalization of the Leibniz rule for the iterated Askey–Wilson operator is also given and used to derive a multi-sum identity.