Generating functions of bernstein type polynomials and their properties

The main purpose of this thesis is to give Bernstein polynomials and various properties with related to these polynomials in analytic number theory depend on (p,q)-analysis. This thesis consists of seven chapters. First chapter is introduction and literature review. In the second chapter, some basic definitions, notations and theorems which will be used in the other chapters are given. In the third chapter, the results for -Bernstein polynomials under (p,q)-integral operator and their relations between (p,q)-Gamma and (p,q)-Beta functions are obtained. In the fourth chapter, the generating function of (p,q)-Bernstein polynomials are given by using (p,q)-calculus and some properties for these polynomials are investigated by aid of generating function. In the fifth chapter, new two type (p,q)-Bernstein polynomials are given and some results and generating function by aid of these polynomials are obtained. In the sixth chapter, (p,q)-Beta polynomials are defined and some properties for these polynomials are investigated. In the seventh chapter, the conclusions in this thesis are presented.