The method is a notational procedure used 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 properly derived by more complicated methods that can be taken literally without logical difficulty.
In the 1930s and 1940s, Eric Temple Bell tried unsuccessfully to make this kind of argument logically rigorous. The combinatorialistJohn Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively.
Then, using the definition of the Bernoulli polynomials and the definition and linearity of L, one can write
This enables one to replace occurrences of by , that is, move the n from a subscript to a superscript (the key operation of umbral calculus). For instance, we can now prove that:
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.
^*Blissard, John (1861). "Theory of generic equations". The Quarterly Journal of Pure and Applied Mathematics. 4: 279–305.
^E. T. Bell, "The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life", The American Mathematical Monthly45:7 (1938), pp. 414–421.
^Rota, G. C.; Kahaner, D.; Odlyzko, A. (1973). "On the foundations of combinatorial theory. VIII. Finite operator calculus". Journal of Mathematical Analysis and Applications. 42 (3): 684. doi:10.1016/0022-247X(73)90172-8.
G.-C. Rota, D. Kahaner, and A. Odlyzko, "Finite Operator Calculus," Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.
Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, vol. 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185. Reprinted by Dover, 2005.