  One of the main purposes of this article is to give functional equations and
differential equations between Bernstein basis functions and generating
functions of B-spline curves. Using these equations, very useful formulas
containing the relationships among the uniform B-spline curves, the Bernstein
basis functions, and other special numbers and polynomials are derived. By
applying p-adic integrals to these polynomials, many novel formulas are also
derived. Furthermore, by applying the partial differential derivative operator
and Laplace transformation to these generating functions, with aid of higher
derivative differential, not only recurrence relations and the higher
derivative formula for B-spline curves, but also infinite series
representations are given.
