The Modern Umbral Calculus
Another combinatorialist, Gian-Carlo Rota, pointed out that the mystery vanishes if one considers the linear functional L on polynomials in y defined by
Then one can write
etc. Rota later stated that much confusion resulted from the failure to distinguish between three equivalence relations that occur frequently in this topic, all of which were denoted by "=".
In a paper published in 1964, Rota used umbral methods to establish the recursion formula satisfied by the Bell numbers, which enumerate partitions of finite sets.
In the paper of Roman and Rota cited below, the umbral calculus is characterized as the study of the umbral algebra, defined as the algebra of linear functionals on the vector space of polynomials in a variable x, with a product L1L2 of linear functionals defined by
When polynomial sequences replace sequences of numbers as images of yn under the linear mapping L, then the umbral method is seen to be an essential component of Rota's general theory of special polynomials, and that theory is the umbral calculus by some more modern definitions of the term. A small sample of that theory can be found in the article on polynomial sequences of binomial type. Another is the article titled Sheffer sequence.
Rota later applied umbral calculus extensively in his paper with Shen to study the various combinatorial properties of the cumulants.
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