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GENERATING FUNCTIONS FOR THE BERNSTEIN TYPE POLYNOMIALS: A NEW APPROACH TO DERIVING IDENTITIES AND APPLICATIONS FOR THE POLYNOMIALS

The main aim of this paper is to construct generating functions for theBernstein type polynomials. Using these generating functions, variousfunctional equations and differential equations can be derived. Newproofs both for a recursive definition of the Bernstein type basis functions and for derivatives of the nth degree Bernstein type polynomialscan be given using these equations. This paper presents a novel methodfor deriving various new identities and properties for the Bernstein typebasis functions by using not only these generating functions but alsothese equations. By applying the Fourier transform and the Laplacetransform to the generating functions, we derive interesting series representations for the Bernstein type basis functions. Furthermore, wediscuss analytic representations for the generalized Bernstein polynomials through the binomial or Newton distribution and Poisson distribution with mean and variance. By using the mean and the variance,we generalize Szasz-Mirakjan type basis functions.