ABSTRACT
This chapter retraces the recent history of the Umbral Calculus. It studies the classic results concerning polynomial sequences of binomial type. Much of the calculus of finite differences leans upon certain analogies between two linear operators defined on functions ranging over a field of characteristic zero. Sequences of polynomials of binomial type are of frequent occurence in combinatorics, probability, statistics, function theory and representation theory. The chapter extends the domain of every shift-invariant operator to a more general domain originating from the Hardy field called the logarithmic algebra consisting of all expansions of real functions in a neighborhood of infinity in terms of the monomials. It discusses shift-invariant operators and explains the logarithmic version of the binomial theorem using the shift-invariant operators.
