  The main aim of this paper is to provide a unified approach to deriving
identities for the Bernstein polynomials using a novel generating function. We
derive various functional equations and differential equations using this
generating function. Using these equations, we give new proofs both for a
recursive definition of the Bernstein basis functions and for derivatives of
the nth degree Bernstein polynomials. We also find some new identities and
properties for the Bernstein basis functions. Furthermore, we discuss analytic
representations for the generalized Bernstein polynomials through the binomial
or Newton distribution and Poisson distribution with mean and variance. Using
this novel generating function, we also derive an identity which represents a
pointwise orthogonality relation for the Bernstein basis functions. Finally, by
using the mean and the variance, we generalize Szasz-Mirakjan type basis
functions.
