Abstract: In this paper, we present a novel set of (q, γ)-Bernstein basis functions that are parameterized by q and and defined over an interval. We give detailed proofs for several key properties of these functions, including the partition of unity, recurrence relations, degree elevation, and the Marsden identity. Additionally, we introduce and validate the (q, γ)-De Casteljau algorithm, providing comprehensive examples to illustrate its implementation. These results are analyzed to highlight the theoretical and practical implications of these (q, γ)-Bernstein basis functions in various fields, such as polynomial approximation, numerical methods, and Computer Aided Geometric Design (CAGD). Furthermore, we discuss potential extensions and applications of these functions, considering their impact on future research and developments in the domain. By exploring these aspects, we aim to offer a robust framework for understanding and utilizing (q, γ)-Bernstein basis functions.

Keywords: (q, γ)-Bernstein polynomials, (q, γ)-Bernstein basis, Marsden's identity, (q, γ)-Bezier curves, De Casteljau, interval

Pages: 660-674