Formulas And Identities Involving The Askey-Wilson Operator

Keywords

Askey-Wilson operator; Askey-Wilson polynomial; Basic hypergeometric series; Cooper's formula; Generating function; Integration by parts; Rodriguez formula

Abstract

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 ℙ56 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.

Publication Date

5-1-2016

Publication Title

Advances in Applied Mathematics

Volume

76

Number of Pages

68-96

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.aam.2016.02.002

Socpus ID

84959279960 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84959279960

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