Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2013, 1(1), 36-42
DOI: 10.12691/tjant-1-1-8
Open AccessSpecial Issue

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

Mehmet ACIKGOZ1, Erdoğan ŞEN2 and Serkan ARACI3,

1University of Gaziantep, Faculty of Arts and Science, Department of Mathematics, Gaziantep, Turkey

2Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, Tekirdağ, Turkey

3Atatürk Street, Hatay, Turkey

Pub. Date: November 04, 2013

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

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

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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.
 
[9]  Acikgoz, M., and Araci, S., New generating function of Bernstein type polynomials for two variables, Numerical Analysis and Applied Mathematics, International conference 2010, pp. 1133-1136.
 
[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.
 
[18]  Simsek, Y., and Acikgoz, M., A new generating function of q-Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, volume 2010, Article ID 769095, 12 pages.