In this paper, we characterize various sequences of the Appell polynomials by means of the formula of representation of powers viewed from a linear operator perspective. First we construct some Gauss-Weierstrass integral representation operators for various sequence of Appell polynomials such as Euler-Frobenius polynomials, Degenerate Bernoulli polynomials, Bell-Touchard polynomials and for the generalized Tempesta-Bernoulli polynomials. Secondly, we introduce the Appell polynomials as well as the Hurwitz zeta function as eigenfunctions (eigenvectors) of certain well chosen discrete version of Gauss-Weierstrass integral operators. Our approach is inspired by the Gauss-Weierstrass transform for Hermite polynomials.