Formulas and identities involving the Askey-Wilson operator

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• 作者：Mourad E.H. Ismail^a ; ^b ; ¹ ; ^{mourad.eh.ismail@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Plamen Simeonov^c ; ^{simeonovp@uhd.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：33C45 ; 33D45 ; 33Dxx ; 30E20
• 刊名：Advances in Applied Mathematics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：76
• 期：Complete
• 页码：68-96
• 全文大小：510 K

文摘

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 ${}_6\varphi _5$ 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.