- Open Access
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.
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