On zeros of the degenerate Apostol-type polynomials and their applications in approximation

n this paper, we consider the degenerate Apostol-type polynomials and compute their zeros numerically and present them graphically. In particular, we explore the application of these polynomials in approximation theory by introducing a family of Szász-type positive linear operators built from degenerate Apostol-type polynomials defined via a generating function. These operators are explicitly given in terms of those polynomials and act on samples of the target function at uniformly spaced nodes. Their basic moment properties show that they preserve constants, reproduce linear behavior up to a bias that is inversely proportional to n, and admit a closed expression for the second moment. Using Korovkin’s theorem, we prove the uniform convergence of these Szaśz-type approximation operators on compact subsets of the nonnegative real line for continuous functions under mild growth conditions. Also, we derive a quantitative error bound in terms of the modulus of continuity. The theory is complemented with numerical examples and computations of absolute error of approximations for the specific choices of parameters, illustrating the approximation behavior and error trends of these operators.