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The main aim of this paper is to provide a unified approach to derivingidentities for the Bernstein polynomials using a novel generating function. Wederive various functional equations and differential equations using thisgenerating function. Using these equations, we give new proofs both for arecursive definition of the Bernstein basis functions and for derivatives ofthe nth degree Bernstein polynomials. We also find some new identities andproperties for the Bernstein basis functions. Furthermore, we discuss analyticrepresentations for the generalized Bernstein polynomials through the binomialor Newton distribution and Poisson distribution with mean and variance. Usingthis novel generating function, we also derive an identity which represents apointwise orthogonality relation for the Bernstein basis functions. Finally, byusing the mean and the variance, we generalize Szasz-Mirakjan type basisfunctions.

Generating functions for the Bernstein polynomials: A unified approach to deriving identities for the Bernstein basis functions