Umbral calculus

The term umbral calculus has two related but distinct meanings.

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. These techniques were introduced in 1861 by John Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively. The use of shadowy techniques was put on a solid mathematical footing starting in the 1970s, and the resulting mathematical theory is also referred to as "umbral calculus".

In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing, however his attempt in making this kind of argument logically rigorous was unsuccessful.

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. Currently, umbral calculus refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences, but may encompass systematic correspondence techniques of the calculus of finite differences.

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.

An example involves the Bernoulli polynomials. Consider, for example, the ordinary binomial expansion (which contains a binomial coefficient):

and the remarkably similar-looking relation on the Bernoulli polynomials: