Approximation and Asymptotic Properties of Szász-Type Operators Generated by Negative-Order Euler Polynomials

In this paper, we introduce and study a Szász-type family of positive linear operators generated by Euler polynomials of negative order on [0,∞). The construction is based on an explicit finite representation of these polynomials with non-negative terms, which ensures the positivity of the corresponding kernel. We prove the basic properties of the operators and show that they can be represented as finite convex combinations of shifted classical Szász operators. We also provide a probabilistic representation of the kernel as a finite mixture of Poisson distributions, which clarifies the role of the parameter k and the resulting moment structure. The corresponding algebraic and central moment identities are derived and used to establish convergence on compact intervals and to obtain quantitative estimates in terms of the modulus of continuity, Lipschitz-type classes, and Peetre’s K-functional. Furthermore, Voronovskaya-type asymptotic results are obtained, including a quantitative form and a second-order asymptotic formula. Numerical tables and a graphical illustration are presented for selected test functions and parameter values, and the results are consistent with the theoretical convergence behaviour. The paper shows that Euler polynomials of negative order provide a positive and structurally tractable framework for constructing Szász-type approximation operators on the positive real axis.