Concept
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to "prove" them. These techniques were introduced 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. In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing. 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: 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). See also Faulhaber's formula. In differential calculus, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point.