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q-Bernstein-Type Polynomials for Functions of Two Variables with Their Generating and Interpolation Functions

Mehmet ACIKGOZ, Erdoğan ŞEN, Serkan ARACI*

Atatürk Street, Hatay, Turkey

Abstract: The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of two variables. By using these type polynomials, we derive recurrence formulas and some new interesting identities related to the second kind Stirling numbers and generalized Bernoulli polynomials. Moreover, we give the generating function and interpolation function of these modified q-Bernstein polynomials of two variables and also give the derivatives of these polynomials and their generating function.

Keywords: generating function, Bernstein polynomial of two variables, Bernstein operator of two variables, shift difference operator, q-difference operator, second kind Stirling numbers, generalized Bernoulli polynomials, Mellin transformation, interpolation function

Received October 15, 2013; Revised October 30, 2013; Accepted November 04, 2013

Citation: Mehmet ACIKGOZ, Erdoğan ŞEN, Serkan ARACI, q-Bernstein-Type Polynomials for Functions of Two Variables with Their Generating and Interpolation Functions. Turkish Journal of Analysis and Number Theory, 1 (1). 36-42. January 2013. doi:10.12691/tjant-1-1-8

1. Introduction, Definitions and Notations

In approximation theory, the Bernstein polynomials, named after their creater S. N. Bernstein in 1912, have been studied by many researchers for a long time. But nothing about generating function of Bernstein polynomials were known in the literature. Recently, Simsek and Acikgoz, ([18]), constructed a new generating function of (q-) Bernstein type polynomials based on the q-analysis. They gave some new relations related to (q-) Bernstein type polynomials, Hermite polynomials, Bernoulli polynomials of higher order and the second kind Stirling numbers. By applying Mellin transformation to this generating function they defined the interpolation function of (q-) Bernstein type polynomials. They gave some relations and identities on these polynomials. They constructed the generating function for classical Bernstein polynomials, and for Bernstein polynomials for functions of two variables and gave their properties (see [8,9,10], for details). In [1-7], T. Kim also gave a novel definition of q-Bernstein polynomials and derived not only new but also interesting properties of q-Bernstein polynomials. Actually, we are motivated to write this paper from Kim's arithemtic works.

Throughout this paper, we use some notations like and where denotes the set of natural numbers, and .

Let denotes the set of continuous functions on . For

[picture] | (1.1)

where Here is called the Bernstein operator of two variables of order for . For , the Bernstein polynomial of two variables of degree is defined by

[picture] | (1.2)

where and . Thus, throughout this work, we will assume that and . Then, we easily see the following

[picture] | (1.3)

and they form a partition of unity; that is;

[picture] | (1.4)

and by using the definition of Bernstein polynomials for functions of two variables, it is not difficult to prove the property given above as

[picture] | (1.5)

Some Bernstein polynomials of two variables are given below:

Also, for or , because or There are -th degree Bernstein polynomials (see [10,13] for details).

Some researchers have used the Bernstein polynomials of two variables in approximation theory (See [12,13]). But no result was known anything about the generating function of these polynomials. Note that for , we have

From the above, we obtain the generating function for as follows:

[picture] | (1.6)

where . We notice that,

for (for details, see [9]).

Let . Then, q-integer of by and ( See [2-18] for details). Note that . [2] motivated us to write this paper and we have extended the results given in that paper to modified q-Bernstein polynomials of two variables.

2. The Modified q-Bernstein Polynomials for Functions of Two Variables

For and , the q-Bernstein polynomials of degree are defined by

[picture] | (2.1)

For , consider the q-extension of (1.6) as follows:

[picture] | (2.2)

where . Note that

Definition 1. The modified q-Bernstein polynomials for functions of two variables is defined by means of the following generating function:

[picture] | (2.3)

where .

By comparing the coefficients of (2.2) and (2.3), we obtain a formula for modified q-Bernstein polynomials of two variables given in the following theorem:

Theorem 1. For , then, we have

[picture] | (2.4)

Theorem 2. (Recurrence Formula for ) For , we have

Proof. By using the definition of Bernstein polynomials for functions of two variables defined by (2.4), we have

Theorem 3. For , we get

[picture] | (2.5)

and

Proof. Let be a continuous function of two variables on . Then the modified q-Bernstein operator of order for is defined by

[picture] | (2.6)

where , From Theorem 1 and the definition of modified q-Bernstein operator given by (2.6) for , we have

From Theorem 1, we have

The modified q-Bernstein polynomials of two variables are symmetric polynomials:

by replacing by and by .

Theorem 4. For , and for , then, we procure

[picture] | (2.7)

where is a circle around the origin and integration is in the positive direction.

Proof. By using the definition of the modified q-Bernstein polynomials of two variables and the basic theory of complex analysis including Laurent series that

[picture] | (2.8)

By using (2.7) and (2.8), we obtain

and

[picture] | (2.9)

We also obtain from (2.5) and (2.9) that

[picture] | (2.10)

Therefore we see that from (2.8) and (2.10) that

Theorem 5. (The Derivative Formula for ) For , then, we derive the following

Proof. Using the definition of modified q-Bernstein polynomials for functions of two variables and the property (1.3), we have

and after some calculations, the proof is complete.

Therefore, we can write the modified q-Bernstein polynomials for functions of two variables as a linear combination of polynomials of higher order as follows:

Theorem 6. For , we have

Proof. It follows after expanding the series and some algebraic operations.

Theorem 7. For , we have

Proof. To prove this theorem, we start with the right hand side:

Theorem 8. For , we obtain

Proof. From the definition of modified q-Bernstein polynomials of two variables and binomial theorem with , we have

Theorem 9. The following identity

is true.

Proof. We easily see that from the property of the modified q-Bernstein polynomials of two variables that

and that

Continuing this way, we have

and after some algebraic operations, we obtain the desired result.

We see that from the theorem above, it is possible to write as a linear combination of the two variables modified q-Bernstein polynomials.

For , the Bernoulli polynomials of degree are defined by

and are called the n-th Bernoulli numbers of order . It is well known that the second kind Stirling numbers are defined by

[picture] | (2.11)

for (see [2]). By using the above relations we can give the following theorem:

Theorem 10. For , we have

Proof. By using the generating function of modified q-Bernstein polynomials of two variables, we have

by using the Cauchy product. By comparing last two relations, we have the desired result.

Let be the shift difference operator defined by . By using the iterative method we have

[picture] | (2.12)

for .

By comparing the coefficients on both sides above, we have

[picture] | (2.13)

for . By using the equations (2.11) and (2.12), we obtain the following relation

[picture] | (2.14)

which is the relation of the q-Bernstein polynomials of two variables in terms of Bernoulli polynomials of order and second Stirling numbers with shift difference operator.

Let be the shift operator. Then the q-difference operator is defined by

[picture] | (2.15)

where is and identity operator ( See [2]).

For and , we have

[picture] | (2.16)

where is called the Gaussian binomial coefficients, which are defined by

[picture] | (2.17)

Theorem 11. For , we have

Proof. Let be the generating function of the q-extension of the second kind Stirling numbers as follows:

From the above, we have

where It is easy to see that

[picture] | (2.18)

by similar way

[picture] | (2.19)

We have above equality. Then, we obtain the desired result in Theorem from the equations (2.18), (2.19) and Theorem 7.

3. Interpolation Function of Modified q-Bernstein Polynomial for Functions of Two Variables

For , and , , by applying the Mellin transformation to generating function of Bernstein polynomials of two variables, we get

[picture] | (3.1)

By using the equation (3.1), we define the interpolation function of the polynomials as follows:

Definition 2. Let and , we define

[picture] | (3.2)

By using (3.2), we have as Thus one has

By substituting and into the above, we have .

We now evaluate the th s-derivatives of as follows:

where and

References

[1]   | Kim, T., On the q-extension of Higher-Order Euler polynomials, Proceedings of the Jangjeon Mathematical Society, 15 (2012), no. 3, pp. 293-302.
[2]   | Kim, T., Jang, L.-C., and Yi, H., Note on the modified q-Bernstein polynomials, Discrete Dyanmics in Nature and Society, Volume 2010 (2010), Article ID 706483, 12 pages.
[3]   | Kim, T., A note q-Bernstein polynomials, Russ. J. Math. Phys. 18(2011), pp. 41-50.
[4]   | Kim, T., Choi, J. and Kim, Y. H., Some identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers, Adv. Stud. Contemp. Math. 20(2010), pp. 335-341.
[5]   | Kim, T., Choi, J. and Kim, Y. H., q-Bernstein Polynomials Associated with q-Stirling Numbers and Carlitz's q-Bernoulli Numbers, Abstract and Applied Analysis, Article ID 150975, 11 pages.
[6]   | Kim, T., Choi, J., Kim, Y. H. and Ryoo, C. S., On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials, J. Inequal. Appl. 2010 (2010), Art ID 864247, 12 pages.
[7]   | Kim, T., Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Zp, Russ. J. Math. Phys., 16 (2009), no.4, pp. 484-491.
[8]   | Acikgoz, M., and Araci, S., On the generating function of the Bernstein polynomials, Numerical Analysis and Applied Mathematics, International conference 2010, pp. 1141-1143.
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[10]   | Acikgoz, M., and Araci, S., A study on the integral of the product of several type Bernstein polynomials, IST Transaction of Applied Mathematics Modelling and Simulation,2010, vol. 1, no. 1(2), ISSN 1913-8342, pp. 10-14.
[11]   | Acikgoz, M., and Simsek, M., On multiple interpolation functions of the Nörlund-type q-Euler polynomials, Abstr. Appl. Anal. 2009, Art. ID 382574, 14 pages.
[12]   | Buyukyazici, İ., and İbikli, E., Bernstein polynomials of two variable functions, Graduate School of Natural and Applied Sciences, Department of Mathematics, 1999, 49 pages, Ankara, Turkey.
[13]   | Buyukyazici, İ., and İbikli, E., The approximation properties of generalized Bernstein polynomials of two variables, Applied Math. and Comput. 156 (2004) 367-380.
[14]   | Ryoo, C. S., A note on the weighted -Euler numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), pp. 47-54.
[15]   | Oruc, H., and Phillips, G. M., A generalization of the Bernstein polynomials, Proceedings of the Edinburgh Mathematical society (1999) 42, 403-413.
[16]   | Ostrovska, S., On the q-Bernstein polynomials, Adv. Stud. Contemp. Math. 11 (2) (2005), pp. 193-204.
[17]   | Phillips, G. M., A survey of results on the q-Bernstein polynomials, IMA Journal of Numerical Analysis Advance Access published online on June 23, (2009), pp. 1-12.
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