Outcomes of the Abel Identity

We discuss some outcomes of an umbral generalization of the Abel identity. First we prove that a concise proof of the Lagrange inversion formula can be deduced from it. Second, we show that the whole class of Sheffer sequences, if manipulated to an umbral level, coincides with the subclass of Abel polynomials. Finally, we apply these techniques to obtain explicit formulae for some classical polynomial sequences, even in non Sheffer cases (Chebyshev and Gegenbauer polynomials).