Construction a new generating function of Bernstein type polynomials. (English) Zbl 1225.30003

Summary: The main purpose of this paper is to reconstruct generating function of Bernstein type polynomials. Some properties of these generating functions are given. By applying this generating function, not only the derivatives of these polynomials but also recurrence relations of these polynomials are found. Interpolation functions of these polynomials are also constructed by the Mellin transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the Stirling numbers of the second kind, and Bernoulli polynomials of higher order are given. Furthermore, some remarks associated with the Bézier curves are given.


30C10 Polynomials and rational functions of one complex variable
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
30E05 Moment problems and interpolation problems in the complex plane
33B15 Gamma, beta and polygamma functions
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