Unification of the Bernstein-type polynomials and their applications

In this paper, we investigate some new identities related to the unification of the Bernstein-type polynomials, Bernoulli polynomials, Euler numbers and Stirling numbers of the second kind. We also give some remarks and applications of the Bernstein-type polynomials related to solving high even-order differential equations by using the Bernstein-Galerkin method. We also give some applications on these polynomials and differential equations.

MSC: 11B68, 12D10, 14F10, 26C05, 26C10, 30B40, 30C15, 42A38, 44A10.

Generating functions play an important role in the investigation of various useful properties of the sequences and differential equations. These functions are also used to find many properties and formulas for the sequences. In [1], the author constructed certain generating functions for the unification of the classical Bernstein polynomials. Using these generating functions, the author derived several interesting and useful identities for these polynomials. The Bernstein polynomials have been defined by many different ways, for example, by q-series, by complex function and by many algorithms. The Bernstein polynomials are used in approximations of functions as well as in other fields such as smoothing in statistics, in numerical analysis, constructing the Bezier curves. The Bernstein polynomials are also used to solve differential equations.

According to Farouki [2], the Bernstein polynomial basis was introduced 100 years ago (Bernstein, 1912) as a means to constructively prove the ability of polynomials to approximate any continuous function, to any desired accuracy, over a prescribed interval. Their slow convergence rate and the lack of digital computers to efficiently construct them caused the Bernstein polynomials to lie dormant in the theory rather than practice of approximation for the better part of a century. The Bernstein coefficients of a polynomial provide valuable insight into its behavior over a given finite interval, yielding many useful properties and elegant algorithms that are now being increasingly adopted in other application domains.

Recently, the author [1] introduced and investigated the following generating functions which use a unification of the classical Bernstein polynomials:

where , and . The following function is a generating function of the polynomials

where .

An explicit formula of the polynomials is given by the following theorem [1].

Theorem 1.1Let. Letb, nandsbe nonnegative integers. If, then we have

Remark 1.2 If we set in (2), we have

which denotes the classical Bernstein basis function (cf. [1-7]). Consequently, the polynomials are a unification of the Bernstein polynomials.

The remainder of this study is organized as follows.

Section 2: We give many properties of the unification of the Bernstein-type polynomials: partition of unity, alternating sum, subdivision property. We also give many functional equations and differential equations of this generating function. Using these equations, many properties of the unification of the Bernstein-type polynomials can be found. Section 3: Integral representations of the unification of the Bernstein-type polynomials are given. Using these representations, we give an identity. Section 4: By using the Laplace transform, we find some identities of the unification of the Bernstein-type polynomials. Section 5: By using a new generating function, we prove the Marsden identity for the unification of the Bernstein-type polynomials. Section 6: By using generating functions, we give relations between the unification of the Bernstein-type polynomial, the unification of the Bernoulli polynomial of higher order and the Stirling numbers of the second kind. Section 7: By using the unification of the Bernstein-type polynomials and the Bernstein-Galerkin methods, we solve high even-order differential equations. Section 8: We give some remarks on the unification of the Bernstein-type polynomials and Bezier-type curves.

In this section, we investigate some properties of the unification of the Bernstein-type polynomials.

2.1 Partition of unity

The unification of the Bernstein-type polynomials does not have partition of unity. That is, by using (1), we derive the following functional equation:

By using the same method as that in [7] and (2), we arrive at the formula for the polynomials

Remark 2.1 The polynomials have partition of unity. That is,

2.2 Alternating sum

By using (1), we derive the following functional equation which is used to find an alternating sum of the unification of the Bernstein-type polynomials :

By using the same method as that in [7] and (2), we arrive at a formula for the alternating sum of the polynomials , which is given by the following theorem.

Theorem 2.2

2.3 Subdivision property

Here, we give partial differential equations and a functional equation of the generating function for the unification of the Bernstein-type polynomials . By using this functional equation, we derive the subdivision property unification of the Bernstein-type polynomials .

We set

By using the above functional equation and (2), we derive the subdivision property for the polynomials by the following theorem.

Theorem 2.3

or

wheredenotes the classical Bernstein basis function.

Remark 2.4 Substituting into (4), we obtain the subdivision property for the classical Bernstein basis functions:

Using (3), we give the following partial differential equations:

and

By applying these partial differential equations, we obtain the following derivative relations which are related to the subdivision property unification of the Bernstein-type polynomials , respectively:

Theorem 2.5

and

In this section, we derive integral representations of the unification of the Bernstein-type polynomials . We also give an identity which connects the binomial coefficients, gamma and beta functions.

The beta function is a function of two complex variables α and β, defined by

(cf. [[8], p.9, Eq. (60)]). The beta function is related to the gamma function; one has

Replacing α by and β by in the above equation, we get

(cf. [[8], p.9, Eq. (62)]).

Theorem 3.1

or

Proof

where

By using (5), we easily arrive at the desired result. □

Binomial coefficients play an important role in mathematics and mathematical physics, especially in statistics, probability and analytic number theory. Therefore, by using (7) and (8), we derive the following identity related to the binomial coefficients, gamma and beta functions:

In this section, by using the Laplace transform, we give some identities of the unification of the Bernstein-type polynomials .

Using the generating function in (2), we get

Integrating equation (10) (by parts) with respect to t from zero to infinity, we have

If we appropriately use the case of the following Laplace transform of the function :

by substituting (12) into (11), we arrive at the following theorem.

Theorem 4.1Let. Then we have

Remark 4.2 If we set in (13), then we arrive at Theorem 15 in [7].

We modify (2) as follows:

From the above equation, we get

Therefore, we arrive at the following theorem.

Theorem 4.3

In this section, by using generating functions, we prove the Marsden identity for the unification of the Bernstein-type polynomials . This identity is associated with a formula for rational linear transformation of B-splines, which are of interest in computer-aided geometric design and approximation theory.

We set

or

We derive the following functional equation:

From the above functional equation, we get

Therefore

From the above equation, we have

By comparing the coefficients of on the both sides of the above equation, we obtain

Therefore, we arrive at the Marsden identity which is given by the following theorem.

Theorem 5.1

Remark 5.2 By using (14), we also obtain the Marsden identity for the classical Bernstein polynomials as follows:

The so-called unification of the Bernoulli, Euler and Genocchi polynomials were defined by Ozden [9]. The polynomials are defined by means of the following generating function:

where k is an integer parameter, a and b are real parameters and β is a complex parameter. Observe that

(cf. [9,10]).

The above generating function is related to some special polynomials as follows.

Remark 6.1 Substituting into (15), we have the Apostol-Bernoulli polynomials (cf. [11-13]):

substituting , and into (15), we have the Apostol-Euler polynomials:

substituting , and into (15), we have the Apostol-Genocchi polynomials:

substituting into (15), we have

where denotes the classical Bernoulli polynomials and substituting and into (15), we have

where denotes the classical Euler polynomials.

Now, the modification of (15) is given by

The following definition provides a natural generalization and unification of λ-Stirling numbers of the second kind, which is defined by Srivastava [12,13].

Definition 6.2 Let and . The generalized λ-Stirling type numbers of the second kind are defined by means of the following generating function:

Remark 6.3 By setting in (17), we get

where denotes the Stirling numbers of the second kind. It is also well known that

so that

being the Kronecker symbol (cf. [8,12,14,15]).

Theorem 6.4Letb, nandsbe nonnegative integers with. Then we have

whereanddenote the unification Bernoulli polynomial of higher order and-Stirling numbers of the second kind, respectively.

Proof By (2), we have

By using (2), (16) and (17) in the above equation, we have

From the above equation, after some calculation, we find the desired result. □

Theorem 6.5Letbandnbe nonnegative integers with. Then we have

wheredenotes the Euler polynomials of higher order.

Proof By (2), we have

From the above, we have

By using the Cauchy product in the above, after some calculation, we find the desired result. □

We recall from the work of Gould [[14], Vol. 7, Eq. (2.45)] that

where

By substituting (20) into (19), we arrive at the following result.

Corollary 6.6The following identity holds true:

In [4], Doha et al. gave an application of the Bernstein polynomials for solving high even-order differential equations by using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods. The methods do not contain generating functions for proving explicitly the derivatives formula of the Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves. Here, we prove this formula for the unification of the Bernstein-type polynomials by a higher-order partial differential equation and functional equations. We also give some remarks and applications related to these polynomials and the Bernstein-Galerkin method.

We modify (1) as follows:

where and . Let b, k, n and s be nonnegative integers and , then we get

so that, obviously,

By using the same method as in [15], we now give a higher-order partial differential equation for the generating function as follows.

We set

and

We have

By using Leibnitz’s formula for the vth derivative, with respect to x, of the product of the above two functions, we obtain the following higher-order partial differential equation:

By using (1) in the above partial differential equation, we get the following higher order partial differential equation:

By substituting (21) into the above equation, after some calculation, we arrive at the following theorem.

Theorem 7.1Let. Letb, sandvbe nonnegative integers with. Then we have

where

Integrating equation (22) (by parts) with respect to x from 0 to 1 and using Theorem 3.1, we have

for all b and s.

We recall from the work of Doha et al. [4] that if is a differentiable function of degree m and defined on , then a linear combination of the Bernstein polynomials can be written. Therefore, we can easily have

where , otherwise and

By using the same method as in [4], we write

where , otherwise . We now give an application for the solution of high even-order differential equations. We also recall from the work of Doha et al. [4] that for ,

by the following boundary conditions:

(cf. [4]). By using the same method as that of Doha et al. [4], we apply the unification of the Bernstein-type polynomials to the Bernstein-Galerkin approximation for solving (24); that is,

and

By applying the Bernstein-Galerkin approximation (24), we find as follows. For solving this equation, we need the following notations, which we recall from the work of Doha et al. [[4], p.9, Eq. (4.4)].

The inner product on is defined by

By using this inner product, we modify (24) as follows:

where , and

The matrix representation of the above equation is given by

where

By using (23), one can easily find A, (); that is,

and

Remark 7.2 According to Doha et al.[4], it is important to apply the Galerkin-spectral Bernstein approximation for how to choose an appropriate basis for such that the linear system resulting in the Bernstein-Galerkin approximation to (25) is possible. That is,

where for all . The 2m boundary conditions lead to the first m, and the least m expansion coefficients are zero.

Remark 7.3 By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, Doha et al.[4] solved the following boundary value problem:

subject to the boundary conditions , with the exact solution

(cf. see for detail [4,16]).

The unification of the Bernstein-type polynomials is used to construct Bezier-type curves which are used in computer-aided graphics design and related fields and also in the time domain, particularly in animation and interface design (cf. [2,6]).

The Bezier-type curve of degree n can be generalized by the author [1] as follows:

where , denotes the unification of the Bernstein-type polynomials and are the control points.

The unification of the Bernstein-type polynomials might affect the shape of the curves.

The author declares that he has no competing interests.

The author completed the paper himself. The author read and approved the final manuscript.

Dedicated to Professor Hari M Srivastava.

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

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