Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties
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Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties Mehmet Ali Özarslan and Mehmet Bozer* *
Correspondence: [email protected] Eastern Mediterranean University, Mersin 10, Gazimagusa, TRNC, Turkey
Abstract 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 Keywords: generating function; Bernstein polynomials; Bernoulli polynomials; Euler polynomials; Genocchi polynomials; Stirling numbers of the second kind
1 Introduction 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
Ga,b (t, x; k, m) :=
–k xk t k ( + ax)k
m
∞
tn t[ +bx ] (a,b) e +ax = Pn (x; k, m) , (mk)! n! n=
()
where k, m ∈ Z+ := {, , . . .}, a, b ∈ R, t ∈ C. Here, x ∈ I where I is a subinterval of R such that the expansion in () is valid. The following two cases will be important for us. . The case a = , b = –. In this case, we let x ∈ [, ] and we see that m G,– (t, x; k, m) = –k xk t k
∞
tn t[–x] (,–) e = Pn (x; k, m) (mk)! n! n=
generates the unifying Bernstein basis polynomials Pn(,–) (x; k, m) := Bn (mk, x) which were introduced and investigated in []. We should note further that G,– (t, x; , m) gives ∞
G,– (t, x; , m) = [xt]m
tn t[–x] e = Bn (m, x) m! n! n=
which generates the celebrated Bernstein basis polynomials (see [–])
Bn (m, x) := Bnm (x) =
n k x ( – x)n–m . m
© 2013 Özarslan and Bozer; licensee Springer. 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.
Özarslan and Bozer Advances in Difference Equations 2013, 2013:55 http://www.advancesindifferenceequations.com/content/2013/1/55
Page 2 of 14
Note that the Bernstein operators Bn : C[, ] → C[, ] are given by Bn (f ; x) =
n n k m f x ( – x)n–m , n m m=
n ∈ N := {, , . . .}
and by the Korovkin theorem, it is known that Bn (f ; x) ⇒ f (x) for all f ∈ C[, ], where C[, ] denotes the space of continuous functions defined on [, ], and the notation ‘⇒’ denotes the uniform convergence with respect to the usual supremum norm on C[, ]. Very recently, interesting properties of Bernstein polynomials were discussed in [, –] and []. . The case a = , b = . In this case, we let x ∈ [, ∞) and define
–k xk t k G, (t, x; k, m) := ( + x)k =
∞
m
t[ ] e +x (mk)!
Pn(,) (x; k, m)
n=
tn . n!
We will see that this generating function produces the
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