Taylor series for the Askey-Wilson operator and classical summation formulas

López, Bernardo; Marco, José; Parcet, Javier

doi:10.1090/S0002-9939-06-08239-6

Taylor series for the Askey-Wilson operator and classical summation formulas
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by Bernardo López, José Manuel Marco and Javier Parcet PDF

Proc. Amer. Math. Soc. 134 (2006), 2259-2270 Request permission

Abstract:

An analogue of Taylor’s formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As an application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context.

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