The $\alpha$-Bernstein operators were initially introduced in the paper by Chen, X., Tan, J., Liu, Z., Xie, J. (2017) titled "Approximation of Functions by a New Family of Generalized Bernstein Operators" (Journal of Mathematical Analysis and Applications, 450(1), 244-261). Since their introduction, these operators have served as a source of inspiration for numerous research endeavors. In this study, we propose a novel technique, founded on a recursive relation, for constructing Bernstein-like bases. A special case of this new representation yields a novel portrayal of Chen's operators. This innovative representation enables the discovery of additional properties of $\alpha$-Bernstein operators and facilitates alternative and more straightforward proofs for certain theorems.

翻译：α-Bernstein算子最初由Chen, X., Tan, J., Liu, Z., Xie, J. (2017)在题为"Approximation of Functions by a New Family of Generalized Bernstein Operators"（《Journal of Mathematical Analysis and Applications》，450(1), 244-261）的论文中引入。自其提出以来，这些算子已成为众多研究工作的灵感来源。在本研究中，我们提出了一种基于递推关系的新技术，用于构造Bernstein型基。该新表示的一个特例为Chen的算子提供了一种新颖的刻画。这一创新性表示使得我们能够发现α-Bernstein算子的更多性质，并为某些定理提供了替代且更简洁的证明。