In mathematics, before the 1970s, the term umbral calculus was understood to mean what is sometimes called Blissard's symbolic method, sometimes attributed to James Joseph Sylvester or to Edouard Lucas. That method is a notational device for deriving identities involving indexed sequences of numbers by pretending that the indices are exponents. Construed literally, it is absurd, and yet it is successful; identities derived via the umbral calculus can also be derived by more complicated methods that can be taken literally without logical difficulty. An example involves the Bernoulli polynomials
In the 1930s and 1940s, Eric Temple Bell tried unsuccessfully to make this kind of argument logically rigorous. The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively. Another combinatorialist, GianCarlo Rota, pointed out that the mystery vanishes if one considers the linear functional L on polynomials in y defined by
Umbral calculus in a more modern sense of the term
When polynomial sequences replace sequences of numbers as images of y^{n} 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.
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