Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties
© Özarslan and Bozer; licensee Springer 2013
Received: 30 November 2012
Accepted: 31 January 2013
Published: 13 March 2013
In this paper we introduce the unification of Bernstein and Bleimann-Butzer-Hahn basis via the generating function. We give the representation of this unified family in terms of Apostol-type polynomials and Stirling numbers of the second kind. More generating functions of trigonometric type are also obtained to this unification.
MSC:11B65, 11B68, 41A10, 30C15.
- 1.The case , . In this case, we let and we see that
- 2.The case , . In this case, we let and define
Denoting by the space of real-valued bounded continuous functions defined on , they proved that as . On the other hand, the convergence is uniform on each compact subset of , where the norm is the usual supremum norm of . For the review of the results concerning the Bleimann-Butzer-Hahn operators obtained in the period 1980-2009, we refer to .
The following theorem gives the explicit representation of the basis family defined in (1). Note that throughout the paper, we let for .
Comparing (1) and (2), we get the result. □
Furthermore, is the well-known Bernstein basis.
Moreover, is the Bleimann-Butzer-Hahn basis function.
We organize the paper as follows. In Section 2, we obtain the representation of this unified family in terms of Apostol-type polynomials and Stirling numbers of the second kind. In Section 3, we give more trigonometric generating functions for this unification and obtain a certain summation formula. All the special cases are listed at the end of each theorem.
2 Representation in terms of Apostol-type polynomials and Stirling numbers
For the convergence of the series in (3), we refer to [, p.2453].
Some of the well-known polynomials included by are listed below.
Taking in the above relations, we obtain the classical Bernoulli polynomials and Bernoulli numbers .
When , the above relations give the classical Genocchi polynomials and Genocchi numbers .
Although our results do not contain the Apostol-Euler polynomials, for the sake of completeness, we give their definitions as a special case of the polynomial family .
When , the above relations give the classical Euler polynomials and Euler numbers .
The following theorem states an interesting explicit representation of the unified basis in terms of Apostol-type polynomials and relation between Stirling numbers of the second kind.
holds true between the unified Bernstein and Bleimann-Butzer-Hahn basis and Apostol-type polynomials.
Whence the result. □
Now, we list some important corollaries of the above theorem.
3 Generating functions of trigonometric type
In this section, we obtain a trigonometric generating relation for the unified Bernstein and Bleimann-Butzer-Hahn basis. Furthermore, we give a certain summation formula for this unification. We start with the following theorem.
Equating real and imaginary parts, we get (5).
which is precisely (6).
Equating real and imaginary parts we get (7). □
Since we obtain the unified Bernstein family in the case , , we have the following corollary at once.
Since the case , gives the unified Bleimann-Butzer-Hahn family, we immediately obtain the following corollary.
Finally, we obtain a summation formula for the unified Bernstein and Bleimann-Butzer-Hahn basis as follows.
the proof is completed by comparing the coefficients of in (14) and (15). □
In the case , , we obtain the following result for the unified Bernstein family at once.
Since the case , gives the unified Bleimann-Butzer-Hahn family, we have the following result.
Dedicated to Professor Hari M Srivastava.
- Simsek Y: Constructing a new generating function of Bernstein type polynomials. Appl. Math. Comput. 2011, 218: 1072-1076. 10.1016/j.amc.2011.01.074MATHMathSciNetView ArticleGoogle Scholar
- Acikgoz M, Aracı S: On generating function of the Bernstein polynomials. AIP Conf. Proc. 2010, 1281: 1141-1143. Proceedings of the International Conference on Numerical Analysis and Applied MathematicsView ArticleGoogle Scholar
- Bayad A, Kim T: Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials. Russ. J. Math. Phys. 2011, 18(2):133-143. 10.1134/S1061920811020014MATHMathSciNetView ArticleGoogle Scholar
- Bernstein SN: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Commun. Soc. Math. Kharkov 1912-13, 13: 1-2.Google Scholar
- Bleimann G, Butzer PL, Hahn L: A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 1980, 42: 255-262.MATHMathSciNetView ArticleGoogle Scholar
- Busé L, Goldman R: Division algorithms for Bernstein polynomials. Comput. Aided Geom. Des. 2008, 25: 850-865. 10.1016/j.cagd.2007.10.003MATHView ArticleGoogle Scholar
- Kim M-S, Kim T, Lee B, Ryoo C-S: Some identities of Bernoulli numbers and polynomials associated with Bernstein polynomials. Adv. Differ. Equ. 2010., 2010: Article ID 305018Google Scholar
- Kim T, Jang L-J, Yi H: A note on the modified q -Bernstein polynomials. Discrete Dyn. Nat. Soc. 2010. doi:10.1155/2010/706483Google Scholar
- Morin G, Goldman R: On the smooth convergence of subdivision and degree elevation for Bézier curves. Comput. Aided Geom. Des. 2001, 18: 657-666. 10.1016/S0167-8396(01)00059-0MATHMathSciNetView ArticleGoogle Scholar
- Phillips GM CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 14. In Interpolation and Approximation by Polynomials. Springer, New York; 2003.View ArticleGoogle Scholar
- Simsek Y, Acikgoz M: A new generating function of ( q -) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010., 2010: Article ID 769095Google Scholar
- Zorlu S, Aktuglu H, Ozarslan MA: An estimation to the solution of an initial value problem via q -Bernstein polynomials. J. Comput. Anal. Appl. 2010, 12: 637-645.MATHMathSciNetGoogle Scholar
- Ulrich A, Mircea I: The Bleimann-Butzer-Hahn operators old and new results. Appl. Anal. 2011, 90(3-4):483-491. 10.1080/00036810903517639MATHMathSciNetView ArticleGoogle Scholar
- Ozarslan MA: Unified Apostol-Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2011, 62(6):2452-2462. 10.1016/j.camwa.2011.07.031MathSciNetView ArticleGoogle Scholar
- Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 2005, 308(1):290-302. 10.1016/j.jmaa.2005.01.020MATHMathSciNetView ArticleGoogle Scholar
- Luo Q-M: On the Apostol-Bernoulli polynomials. Cent. Eur. J. Math. 2004, 2(4):509-515. 10.2478/BF02475959MATHMathSciNetView ArticleGoogle Scholar
- Luo Q-M, Srivastava HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2006, 51(3-4):631-642. 10.1016/j.camwa.2005.04.018MATHMathSciNetView ArticleGoogle Scholar
- Srivastava HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 2000, 129(1):77-84. 10.1017/S0305004100004412MATHView ArticleGoogle Scholar
- Luo Q-M: Fourier expansions and integral representations for the Genocchi polynomials. J. Integer Seq. 2009., 12: Article ID 09.1.4Google Scholar
- Luo Q-M: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 2011, 48(2):291-309.MATHMathSciNetGoogle Scholar
- Luo Q-M: Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 2006, 10: 917-925.MATHGoogle Scholar
- Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht; 2001.MATHView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.