  In this paper, we explore the effectiveness of almost purely operational
methods in the study of umbral calculus. To accomplish this goal, we
systematically reconstruct the theory operationally, offering new proofs and
results throughout. Our approach is applied to the study of invertible power
series, where we notably offer a concise two-line proof of Lagrange's inversion
theorem and derive formulas for both fractional and regular compositional
iterates, for which the latter is new. In addition, we will develop several new
insights into umbral operators, including their fractional exponents and novel
expressions, in connection with the previously discussed applications. Finally,
we will elaborate on both established and new concepts and examples, such as
pseudoinverses of delta operators and a natural generalization of the Laguerre
polynomials.
