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

Authors:
Bernardo López, José Manuel Marco and Javier Parcet

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2259-2270

MSC (2000):
Primary 33D15

Published electronically:
January 31, 2006

MathSciNet review:
2213698

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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.

**1.**George Gasper and Mizan Rahman,*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153****2.**Mourad E. H. Ismail,*The Askey-Wilson operator and summation theorems*, Mathematical analysis, wavelets, and signal processing (Cairo, 1994), Contemp. Math., vol. 190, Amer. Math. Soc., Providence, RI, 1995, pp. 171–178. MR**1354852**, 10.1090/conm/190/02300**3.**Mourad E. H. Ismail and Dennis Stanton,*𝑞-Taylor theorems, polynomial expansions, and interpolation of entire functions*, J. Approx. Theory**123**(2003), no. 1, 125–146. MR**1985020**, 10.1016/S0021-9045(03)00076-5**4.**Mourad E. H. Ismail and Dennis Stanton,*Applications of 𝑞-Taylor theorems*, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 259–272. MR**1985698**, 10.1016/S0377-0427(02)00644-1**5.**José Manuel Marco and Javier Parcet,*A new approach to the theory of classical hypergeometric polynomials*, Trans. Amer. Math. Soc.**358**(2006), no. 1, 183–214 (electronic). MR**2171229**, 10.1090/S0002-9947-04-03620-7**6.**R. V. Wallisser,*On entire functions assuming integer values in a geometric sequence*, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 981–989. MR**1024616**

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Additional Information

**Bernardo López**

Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, 28049, Madrid, Spain

Email:
bernardo.lopez@uam.es

**José Manuel Marco**

Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, 28049, Madrid, Spain

**Javier Parcet**

Affiliation:
Centre de Recerca Matemàtica, Universidad Autónoma de Barcelona, Apartat 50, 08193, Bellaterra, Barcelona, Spain

Email:
jparcet@crm.es

DOI:
https://doi.org/10.1090/S0002-9939-06-08239-6

Keywords:
$q$-Taylor series,
Askey-Wilson operator,
basic hypergeometric function.

Received by editor(s):
May 17, 2004

Received by editor(s) in revised form:
February 24, 2005

Published electronically:
January 31, 2006

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.