This chapter introduces a Szász-Durrmeyer type approximation operator generated by Gauss-Appell polynomials. The construction is motivated by the link between Gauss hypergeometric functions, umbral calculus, and Appell polynomial families. After fixing the auxiliary variable as a parameter, the generating function of the Gauss-Appell polynomials is used to define a positive linear operator on the semi-infinite interval. The basic algebraic structure of the operator is examined through its first moments and central moments. These identities make it possible to establish the main approximation properties in a weighted setting. In particular, a Korovkin-type convergence theorem is obtained, showing that the operators approximate continuous functions uniformly on compact subsets. Quantitative estimates are then derived by means of the usual modulus of continuity, Peetre’s K-functional, and the second-order modulus of smoothness. A Voronovskaya type asymptotic formula is also proved, which describes the limiting behaviour of the approximation error and clarifies the role of the first two central moments. Finally, a numerical example supported by graphs and pointwise error tables illustrates the convergence behaviour of the proposed operators. The results show that increasing the main parameter improves the approximation, while the additional parameters provide useful local control over the accuracy.