## Abstract and Applied Analysis

### A New Generating Function of ($q\text{-}$) Bernstein-Type Polynomials and Their Interpolation Function

#### Abstract

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

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 769095, 12 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620710

Digital Object Identifier
doi:10.1155/2010/769095

Mathematical Reviews number (MathSciNet)
MR2607127

Zentralblatt MATH identifier
1185.33013

#### Citation

Simsek, Yilmaz; Acikgoz, Mehmet. A New Generating Function of ( $q\text{-}$ ) Bernstein-Type Polynomials and Their Interpolation Function. Abstr. Appl. Anal. 2010 (2010), Article ID 769095, 12 pages. doi:10.1155/2010/769095. https://projecteuclid.org/euclid.aaa/1288620710