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.References
- 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
- 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, DOI 10.1090/conm/190/02300
- 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, DOI 10.1016/S0021-9045(03)00076-5
- 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, DOI 10.1016/S0377-0427(02)00644-1
- 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. MR 2171229, DOI 10.1090/S0002-9947-04-03620-7
- 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
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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2259-2270
- MSC (2000): Primary 33D15
- DOI: https://doi.org/10.1090/S0002-9939-06-08239-6
- MathSciNet review: 2213698