The umbral calculus of symmetric functions
The author develops an umbral calculus for the symmetric functions in an infinite number of variables. This umbral calculus is analogous to Roman-Rota umbral calculus for polynomials in one variable, see \textit{S. M. Roman} and \textit{G.-C. Rota} [Adv. Math. 27, 95-188 (1978; Zbl 0375.05007)]. Although not explicit in the Roman-Rota treatment of umbral calculus, it is apparent that the underlying notion is the Hopf-algebra structure of the polynomials in one variable. To make the required analogy, the author uses the whole dual of the Hopf-algebra structure of the set of symmetric functions, to classify the algebra, coalgebra, and bialgebra maps in terms of symmetric functions of binomial type and admissable systems of functionals. Many examples of families of binomial type are studied, and the classical definition of Sheffer sequences of polynomials is extended to this context, and analogues of Bernoulli and Euler polynomials are introduced. The author also defines a family of symmetric functions of Schur type.
scientific article; zbMATH DE number 997644