  The main aim of this paper is to provide a novel approach to deriving
identities for the Bernstein polynomials using functional equations. We derive
various functional equations and differential equations using generating
functions. Applying these equations, we give new proofs for some standard
identities for the Bernstein basis functions, including formulas for sums,
alternating sums, recursion, subdivision, degree raising, differentiation and a
formula for the monomials in terms of the Bernstein basis functions. We also
derive many new identities for the Bernstein basis functions based on this
approach. Moreover, by applying the Laplace transform to the generating
functions for the Bernstein basis functions, we obtain some interesting series
representations for the Bernstein basis functions.
