Generating functions for unification of the multidimensional Bernstein polynomials and their applications

The aim of this paper is to construct generating functions for m-dimensional unification of the Bernstein basis functions. We give some properties of these functions. We also give derivative formulas and a recurrence relation of the m-dimensional unification of the Bernstein basis functions with help of their generating functions. By combining the m-dimensional unification of the Bernstein basis functions with m variable functions on simplex and cube, we give m-dimensional unification of the Bernstein operator. Furthermore, by applying integrals method including the Riemann integral, the q-integral, and the p-adic integral to some identities for the (q-) Bernstein basis functions, we derive some combinatorial sums including the Bernoulli numbers and Euler numbers and also the Stirling numbers and the Cauchy numbers (the Bernoulli numbers of the second kind). © 2018 John Wiley & Sons, Ltd.