On a Family of Parameter-Dependent Bernstein-Type Operators with Multiple Shape Parameters: Incorporating Symmetric Basis Structures

In this paper, we introduce and investigate a novel family of parameter-dependent operators incorporating multiple shape parameters ${\left\{{\lambda }_k\right\}}_{k=1}^n$, whose underlying basis functions include these parameters and exhibit a symmetry property. This parameter-dependent formulation provides a unified and flexible framework for constructing positive linear operators with enhanced approximation and shape-preserving capabilities. We establish fundamental properties of the proposed operators, including nonnegativity, linearity, end-point interpolation, monotonicity preservation and partition of unity; derive their central moments; and determine direct approximation theorems and Voronovskaja-type results. Finally, numerical experiments and graphical illustrations demonstrate the improved performance and adaptability of the proposed scheme compared with existing Bernstein-type variants. The presented framework unifies several classical and generalized operator families while providing additional shape control for practical applications in computer-aided geometric design and function approximation.