In mathematics, before the 1970s, umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove them.
The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively.
In the 1970s, Steven Roman, Gian-Carlo Rota, and others developed the umbral calculus by means of linear functionals on spaces of polynomials.
The method is a notational procedure used for deriving identities involving indexed sequences of numbers by pretending that the indices are exponents.
Consider, for example, the ordinary binomial expansion (which contains a binomial coefficient): and the remarkably similar-looking relation on the Bernoulli polynomials: Compare also the ordinary derivative to a very similar-looking relation on the Bernoulli polynomials: These similarities allow one to construct umbral proofs, which on the surface cannot be correct, but seem to work anyway.
Thus, for example, by pretending that the subscript n − k is an exponent: and then differentiating, one gets the desired result: In the above, the variable b is an "umbra" (Latin for shadow).
The umbral version of the Taylor series is given by a similar expression involving the k-th forward differences
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.
[3] A small sample of that theory can be found in the article on polynomial sequences of binomial type.
Rota later applied umbral calculus extensively in his paper with Shen to study the various combinatorial properties of the cumulants.