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