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Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties
Advances in Difference Equations volume 2013, Article number: 55 (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.
In this paper, we introduce a two-parameter generating function, which generates not only the Bernstein basis polynomials, but also the Bleimann-Butzer-Hahn basis functions. The generating function that we propose is given by
where , , . Here, where I is a subinterval of ℝ such that the expansion in (1) is valid. The following two cases will be important for us.
The case , . In this case, we let and we see that
generates the unifying Bernstein basis polynomials which were introduced and investigated in . We should note further that gives
Note that the Bernstein operators are given by
and by the Korovkin theorem, it is known that for all , where denotes the space of continuous functions defined on , and the notation ‘⇉’ denotes the uniform convergence with respect to the usual supremum norm on . Very recently, interesting properties of Bernstein polynomials were discussed in [7, 9–11] and .
The case , . In this case, we let and define
We will see that this generating function produces the generalized Bleimann-Butzer-Hahn basis functions . Furthermore, the special case
generates the well-known Bleimann-Butzer-Hahn basis functions:
The Bleimann-Butzer-Hahn operators were introduced in  and defined by
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 .
Theorem 1 If , we have
Direct calculations give
Comparing (1) and (2), we get the result. □
Corollary 2 By taking , in Theorem 1, we obtain the explicit representation of the unifying Bernstein basis polynomials :
Furthermore, is the well-known Bernstein basis.
Corollary 3 Taking , in Theorem 1, we get the explicit representation of the generalized Bleimann-Butzer-Hahn 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
Recently , the first author introduced the unification of the Apostol-Bernoulli, Euler and Genocchi polynomials by
For the convergence of the series in (3), we refer to [, p.2453].
Some of the well-known polynomials included by are listed below.
Remark 4 Having and in (3), we get
Note that are the generalized Apostol-Bernoulli polynomials defined through the following generating relation:
where α and λ are arbitrary real or complex parameters and . Note that when , the order α should be restricted to nonnegative integer values. These polynomials were introduced by Luo and Srivastava  and investigated in [16, 17] and . The Apostol-Bernoulli polynomials and numbers are obtained by the generalized Apostol-Bernoulli polynomials, respectively, as follows:
Taking in the above relations, we obtain the classical Bernoulli polynomials and Bernoulli numbers .
Remark 5 Letting and in (3), we get
Note that when , the order α should be restricted to nonnegative integer values. The Apostol-Genocchi polynomials and numbers are respectively given by
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 .
Remark 6 Setting and in (3), we get
Recall that the Apostol-Euler polynomials are generalized by Luo  and given by the generating relation
for arbitrary real or complex parameters α and λ and . The Apostol-Euler polynomials and numbers are given respectively by
When , the above relations give the classical Euler polynomials and Euler numbers .
Now, recall that the Stirling numbers of the second kind are denoted by and defined by (see [, p.58 (15)])
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.
Theorem 7 The following representation:
holds true between the unified Bernstein and Bleimann-Butzer-Hahn basis and Apostol-type polynomials.
Proof We get, using (1), that
On the other hand, since
we can write from (4) that
Now, using (3) in the above relation, we get
Whence the result. □
Now, we list some important corollaries of the above theorem.
Corollary 8 Since and , we obtain the following :
Furthermore, for , we have the following known relation:
Corollary 9 Since and , we get
Corollary 10 Since and , we obtain
Furthermore, when , we have the following:
Corollary 11 Since and , we get
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.
Theorem 12 For the unified family, we have the following implicit summation formulae:
Proof Writing () in (1), we get
Letting , we get
Equating real and imaginary parts, we get (5).
Now, taking and () in (1), we obtain
Therefore, we get
which is precisely (6).
Finally, for , ,
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.
Corollary 13 For the unified Bernstein family, we have the following implicit summation formulae:
On the other hand, taking in (8) and (9), we get the following relations for the Bernstein basis:
Since the case , gives the unified Bleimann-Butzer-Hahn family, we immediately obtain the following corollary.
Corollary 14 For the unified Bleimann-Butzer-Hahn family, we have the following implicit summation formulae:
Taking in (10) and (11), we get the following relations for the Bleimann-Butzer-Hahn basis:
Finally, we obtain a summation formula for the unified Bernstein and Bleimann-Butzer-Hahn basis as follows.
Theorem 15 For all ; , the following implicit summation formula holds true:
Proof Letting in (1) and then using the fact that
Multiplying both sides by and then expanding the function , we get, after using (12) twice, that
Now, using (12) with the index pairs and , we get
Since the left-hand side is equal by (13) to
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.
Corollary 16 For all , the following implicit summation formula:
holds true for the unified Bernstein family. Taking in (16), we get the following relation for the Bernstein basis:
Since the case , gives the unified Bleimann-Butzer-Hahn family, we have the following result.
Corollary 17 For all , the following implicit summation formula:
holds true for the unified Bleimann-Butzer-Hahn family. Upon taking in (17), we get the following relation for the Bleimann-Butzer-Hahn basis:
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Dedicated to Professor Hari M Srivastava.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Özarslan, M.A., Bozer, M. Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties. Adv Differ Equ 2013, 55 (2013). https://doi.org/10.1186/1687-1847-2013-55
- generating function
- Bernstein polynomials
- Bernoulli polynomials
- Euler polynomials
- Genocchi polynomials
- Stirling numbers of the second kind