Bernstein polynomials and their reflections in analytic numbers theory

This thesis is formed from thirteenth sections. First section is introduction.In the second, third, fourth and fifth sections, respectively, Bernstein operator and Bernstein polynomials, Bernoulli numbers and polynomials, Euler numbers and polynomials and Genocchi numbers and polynomials and their properties are given.In the sixth section, incoming work areas of many mathematicians and other scientists are mentioned from q-analysis (quantum calculus).In the seventh section, basic information of p-adic analysis, Volkenborn integral and properties and fermionic p-adic integral and properties are examined under three main headings.In the eighth, nineth, tenth, eleventh and twelfth sections, the links between Legendre polynomials, Hermite polynomials, Genocchi polynomials, Euler polynomials and Bernoulli polynomials with Bernstein polynomials are given, respectively. Additionally, the results obtained throughout the work are given.