Abstract and Applied Analysis
Volume 2010 (2010), Article ID 769095, 12 pages
doi:10.1155/2010/769095
Research Article

A New Generating Function of (q-) Bernstein-Type Polynomials and Their Interpolation Function

Yilmaz Simsek1 and Mehmet Acikgoz2

1Department of Mathematics, Faculty of Arts and Science, University of Akdeniz, 07058 Antalya, Turkey
2Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, 27310 Gaziantep, Turkey

Received 19 January 2010; Accepted 1 March 2010

Academic Editor: Lance Littlejohn

Copyright © 2010 Yilmaz Simsek and Mehmet Acikgoz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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