Abstract

The main object of this paper is to construct a new generating function of the () Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the () Bernstein-type polynomials. We also give relations between the () Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second-kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the () Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of () Bernstein-type polynomials, moments of some distributions in Statistics.

1. Introduction

In [1], Bernstein introduced the Bernstein polynomials. Since that time, many authors have studied these polynomials and other related subjects (cf, [125]), and see also the references cited in each of these earlier works. The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. These polynomials have been used not only for approximations of functions in various areas in Mathematics, but also for the other fields such as smoothing in statistics, numerical analysis and constructing Bezier curve which have many interesting applications in computer graphics (cf, [1, 5, 7, 1320, 25] and see also the references cited in each of these earlier works).

The (-) Bernstein polynomials have been investigated and studied by many authors without generating function. So far, we have not found any generating function of (-) Bernstein polynomials in the literature. Therefore, we will consider the following question

How can one construct generating function of (-) Bernstein-type polynomials?

The aim of this paper is to give answer this question and to construct generating function of the (-) Bernstein-type polynomials which is given in Section 3. By using this generating function, we not only give recurrence relation and derivative of the (-) Bernstein-type polynomials, but also find relations between higher-order Bernoulli polynomials, the Stirling numbers of the second-kind, and the Hermite polynomials. In Section 5, by applying Mellin transformation to the generating function of the (-) Bernstein-type polynomials, we define interpolation function, which interpolates the (-) Bernstein-type polynomials at negative integers.

The Bernstein polynomials play a crucial role in approximation theory and the other branches of Mathematics and Physics. Thus in this section we give definition and some properties of these polynomials.

Let be a function on . The classical Bernstein polynomials of degree are defined by

where is called the Bernstein operator and

are called the Bernstein basis polynomials (or the Bernstein polynomials of degree ). There are nth degree Bernstein polynomials. For mathematical convenience, we set if or (cf, [1, 5, 7, 9, 14, 1820]).

If is a continuous function, then the sequence of Bernstein polynomials converges uniformly to on (cf, [10]).

A recursive definition of the th th Bernstein polynomials can be written as

For proof of the above relation see [9].

For , derivatives of the th degree Bernstein polynomials are polynomials of degree :

(cf, [1, 5, 7, 9, 14, 18, 19]). On the other hand, in Section 3, using our new generating function, we give the other proof of (2.4).

Observe that the Bernstein polynomial of degree , , uses only the sampled values of at , , . For , ,

is the density function of beta distribution .

Let be a binomial random variable. Then

(cf, [7]).

The classical higher-order Bernoulli polynomials are defined by means of the following generating function:

The higher-order Bernoulli polynomials play an important role in the finite differences and in (analytic) number theory. So, the coefficients in all the usual central-difference formulae for interpolation, numerical differentiation, and integration and differences in terms of derivatives can be expressed in terms of these polynomials (cf, [2, 11, 12, 24]). These polynomials are related to the many branches of Mathematics. By substituting into the above, we have

where is usual Bernoulli polynomials (cf, [22]).

The usual Stirling numbers of the second-kind with parameters are denoted by , that is, the number of partitions of the set into nonempty set. For any , it is well known that the Stirling numbers of the second-kind are defined by means of the generating function (cf, [3, 21, 23])

These numbers play an important role in many branches of Mathematics, for example, combinatorics, number theory, discrete probability distributions for finding higher-order moments. In [8], Joarder and Mahmood demonstrated the application of the Stirling numbers of the second-kind in calculating moments of some discrete distributions, which are binomial distribution, geometric distribution, and negative binomial distribution.

The Hermite polynomials are defined by the following generating function.

For , ,

which gives the Cauchy-type integral

where is a circle around the origin and the integration is in positive direction (cf, [12]). The Hermite polynomials play a crucial role in certain limits of the classical orthogonal polynomials. These polynomials are related to the higher-order Bernoulli polynomials, Gegenbauer polynomials, Laguerre polynomials, the Tricomi-Carlitz polynomials and Buchholz polynomials, (cf, [12]). These polynomials also play a crucial role not only in Mathematics but also in Physics and in the other sciences. In Section 4 we give relation between the Hermite polynomials and (-) Bernstein-type polynomials.

3. Generating Function of the Bernstein-Type Polynomials

Let be a sequence of Bernstein polynomials. The aim of this section is to construct generating function of the sequence . It is well known that most of generating functions are obtained from the recurrence formulae. However, we do not use the recurrence formula of the Bernstein polynomials for constructing generating function of them.

We now give the following notation:

If , then we assume that .

We define

where , and denotes the second-kind Stirling numbers and

By (3.2), we define the following new generating function of polynomial by

where .

Observe that if in (3.4), we have

Hence

From (3.4), we obtain the following theorem.

Theorem 3.1. Let be a positive integer with . Then one has

By using (3.2) and (3.4), we obtain

The generating function depends on integer parameter , real variable , and complex variable and . Therefore the properties of this function are closely related to these variables and parameter. By using this function, we give many properties of the (-) Bernstein-type polynomials and the other well-known special numbers and polynomials. By applying Mellin transformation to this function, in Section 5, we construct interpolation function of the (-) Bernstein-type polynomials.

By the umbral calculus convention in (3.8), then we obtain

By using the above, we obtain all recurrence formulae of as follows:

where each occurrence of are given by (symbolically ).

By (3.9),

we obtain the following corollary.

Corollary 3.2. Let be a positive integer with . Then one has

Remark 3.3. By Corollary 3.2, for all with , we see that The polynomials are so-called -Bernstein-type polynomials. It is easily seen that which give us (2.2).

By using derivative operator

in (3.2), we obtain

Consequently, we have

or

Observe that by using our generating function we give different proof of (2.4).

Let be a function on . The (-) Bernstein-type polynomial of degree is defined by

where . is called the (-) Bernstein-type operator and , , defined in (3.7), are called the (-) Bernstein-type (basis) polynomials.

4. New Identities on Bernstein-Type Polynomials, Hermite Polynomials, and the Stirling Numbers of the Second-Kind

Theorem 4.1. Let be a positive integer with . Then one has where and denote the classical higher-order Bernoulli polynomials and the Stirling numbers of the second-kind, respectively.

Proof. By using (2.7), (2.9), and (3.4), we obtain By using Cauchy product in the above, we have From the above, we have By comparing coefficients of in both sides of the above equation, we arrive at the desired result.

Remark 4.2. In [1820], Phillips gave many results concerning the -integers, and an account of the properties of -Bernstein polynomials. He gave many applications of these polynomials on approximation theory. In [6], Gould gave a different relation between the Bernstein polynomials, generalized Bernoulli polynomials, and the second-kind Stirling numbers. Oruç and Tuncer [15] gave relation between the -Bernstein polynomials and the second-kind -Stirling numbers. In [13], Nowak studied approximation properties for generalized -Bernstein polynomials and also obtained Stancu operators or Phillips polynomials.

From (4.4), we get the following corollary.

Corollary 4.3. Let be a positive integer with . Then one has

Theorem 4.4. Let be a positive integer with . Then one has

Proof. By (2.10), we have By Cauchy product in the above, we obtain By substituting into (4.8), we have By comparing coefficients of in the both sides of the above equation, we arrive at the desired result.

5. Interpolation Function of the (-) Bernstein-Type Polynomials

The classical Bernoulli numbers interpolate by Riemann’ zeta function, which has a profound effect on number theory and complex analysis. Thus, we construct interpolation function of the (-) Bernstein-type polynomials.

For , and , by applying the Mellin transformation to (3.2), we get

By using the above equation, we defined interpolation function of the polynomials as follows.

Definition 5.1. Let and . We define

By using (5.2), we obtain

where and .

By (5.2), we have as . Thus one has

By substituting into the above, we have

We now evaluate the th -derivatives of as follows:

where .

By substituting into (5.2), we obtain

By substituting (3.7) into the above, we arrive at the following theorem, which relates the polynomials and the function .

Theorem 5.2. Let be a positive integer with and . Then we have

Remark 5.3. Consider the following. Therefore, for , the function interpolates the classical Bernstein polynomials of degree at negative integers.

By substituting into (5.6), we obtain the following corollary.

Corollary 5.4. Let be a positive integer with and . Then one has

6. Further Remarks and Observation

The Bernstein polynomials are used for important applications in many branches of Mathematics and the other sciences, for instance, approximation theory, probability theory, statistic theory, number theory, the solution of the differential equations, numerical analysis, constructing Bezier curve, -analysis, operator theory, and applications in computer graphics. Thus we look for the applications of our new functions and the (-) Bernstein-type polynomials.

Due to Oruç and Tuncer [15], the -Bernstein polynomials share the well-known shape-preserving properties of the classical Bernstein polynomials. When the function is convex, then

where

As a consequence of this one can show that the approximation to convex function by the -Bernstein polynomials is one sided, with for all . behaves in very nice way when one varies the parameter . In [2], the authors gave some applications on the approximation theory related to Bernoulli and Euler polynomials.

We conclude this section by the following questions.

()How can one demonstrate approximation by (-) Bernstein-type polynomials ?()Is it possible to define uniform expansions of the (-) Bernstein-type polynomials ?()Is it possible to give applications of the (-) Bernstein-type polynomials in calculating moments of some distributions in Statistics ?()How can one give relations between the (-) Bernstein-type polynomials and the Milnor algebras.

Acknowledgments

The first author is supported by the research fund of Akdeniz University. The authors would like to thank the referee for his/her comments. The authors also would like to thank Professor George M. Phillips for his comments and his some references.