Compositional calculus

AbstractThis paper presents an extension of the umbral calculus to infinitely many variables, in which the role of functional composition of formal power series in one variable is replaced by that of plethysm of formal power series in infinitely many variables. The basic notion involved is that of sequences of polynomials of plethystic type (in infinitely many variables). We obtain a plethystic analog of the transfer formula, which yields a plethystic analog of Abel polynomials. We also find combinatorial interpretations of the plethystic Abel polynomial and the plethystic inverse Abel polynomial