05690930

j

05690930 Simsek, Yilmaz A\c{c}{\i}kg\"oz, Mehmet A new generating function of ($q$-) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010, Article ID 769095, 12 p. (2010). 2010 Hindawi Publishing Corporation, New York, NY EN generating function recurrence relations derivatives Hermite polynomials Bernoulli polynomials of higher order Stirling numbers of the second kind Mellin transform interpolation moments of distributions

doi:10.1155/2010/769095

Summary: 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 relations and derivatives of the ($q$-) Bernstein-type polynomials. We also give relations between the ($q$-) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and Stirling numbers of the second kind. 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, and moments of some distributions in Statistics.