Formulas and identities involving the Askey-Wilson operator. (English) Zbl 1336.33026

Ismail, Mourad E. H.; Simeonov, Plamen

Adv. Appl. Math. 76, 68-96 (2016).

Summary: 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 \({}_6\phi_5\) 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.

MSC:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Keywords:

Askey-Wilson operator; Askey-Wilson polynomial; basic hypergeometric series; Cooper’s formula; generating function; integration by parts; Rodriguez formula; summation theorem; transformation

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