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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
Posted:
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.
References
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G. Gasper and M. Rahman, Basic hypergeometric series. With a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications 35, Cambridge Univ. Press, 1990. MR 1052153 (91d:33034)
- 2.
M.E.H. Ismail, The Askey-Wilson operator and summation theorems, M. Ismail, M.Z. Nashed, A. Zayed, A. Ghaleb (Eds.), Mathematical Analysis, Wavelets and Signal Processing, Contemp. Math. 190 (1995), 171-178. MR 1354852 (96j:33011)
- 3.
M.E.H. Ismail and D. Stanton, q-Taylor theorems, polynomial expansions, and interpolation of entire functions, J. Approx. Th. 123 (2003), 125-146. MR 1985020 (2004g:30040)
- 4.
M.E.H. Ismail and D. Stanton, Applications of q-Taylor theorems, J. Comp. Appl. Math. 153 (2003), 259-272. MR 1985698 (2004f:33035)
- 5.
J.M. Marco and J. Parcet, A new approach to the theory of classical hypergeometric polynomials. Trans. Amer. Math. Soc. 358 (2006), 183-214. MR 2171229
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R.V. Wallisser, On entire functions assuming integer values in a geometric sequence, Théorie des nombres (Quebec, PQ, 1987), 981-989, de Gruyter, Berlin, 1989. MR 1024616 (90j:11067)
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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:
http://dx.doi.org/10.1090/S0002-9939-06-08239-6
PII:
S 0002-9939(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
Posted:
January 31, 2006
Communicated by:
Christopher D. Sogge
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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