ON PARAMETRIZATION OF THE q-BERNSTEIN BASIS FUNCTIONS AND THEIR APPLICATIONS

In order to investigate the fundamental properties of q-Bernstein basis functions, we give generating functions for these basis functions and their functional and differential equations. In [16], [15] and [17], we construct a novel collection of generating functions to derive many known and some new identities for the classical Bernstein basis functions. The main purpose of this paper is to construct analogous generating functions for the q-Bernstein basis functions. By using an approach similar to that of our methods in [16] as well as some properties of interpolation functions, we can derive some known and some new identities, relations and formulas for the q-Bernstein basis functions, including the partition of unity property, formulas for representing the monomials, recurrence relations, formulas for derivatives, subdivision identities and integral representations. Furthermore, we give plots of not only our new basis functions, but also their generating functions. Also, we simulate q-Bezier type curves for some selected q values and control points.