Durrmeyer and Stancu Generalization of Szász-type Operators with Beta Function via U-Charlier-Poisson Polynomials

This paper is dedicated to the comprehensive characterization of a novel generalized Szász–type positive linear operator, a recent addition to the field of constructive approximation. Our approach utilizes a high-precision moment theory to investigate the operator's intrinsic structure and convergence dynamics. Significantly, we introduce and define its powerful integral extensions, specifically the Durrmeyer and the corresponding Durrmeyer-Stancu parametric variants. The central finding involves the explicit calculation of the closed-form first and second algebraic moments for these modified operators. These derived moment relations constitute the fundamental analytic framework used throughout the study. Based on these foundational identities, we establish a range of approximation properties. The analysis confirms the validity of Korovkin's theorem conditions, guaranteeing uniform convergence. Furthermore, we determine the optimal order of convergence by developing sharp direct estimates which functionally depend on the modulus of continuity and the associated Peetre’s $K$-functional. The operators' performance is meticulously quantified across various Lipschitz smoothness classes. To complete the asymptotic description, we leverage the momentum-based structure to formulate precise Voronovskaya-type asymptotic theorems. These theorems provide critical insight into the relationship between the operator's convergence rate and the local differential properties of the approximated function. Collectively, these findings contribute significantly to the existing literature concerning the asymptotic behavior and approximation capacity of generalized linear positive operators.