Page "Umbral calculus" Paragraph 34

When polynomial sequences replace sequences of numbers as images of y < sup > n </ sup > 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.