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 and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas, who used the technique extensively.
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 : 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 variableb is an "umbra". See alsoFaulhaber's formula.
Similar relationships were also observed in the theory offinite differences. The umbral version of the Taylor series is given by a similar expression involving the k-th forward differences of a polynomial functionf, where is the Pochhammer symbol used here for the falling sequential product. A similar relationship holds for the backward differences and rising factorial. This series is also known as the Newton series or Newton's forward difference expansion. The analogy to Taylor's expansion is utilized in the calculus of finite differences.
Bell and Riordan
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.
The modern umbral calculus
Another combinatorialist, Gian-Carlo Rota, pointed out that the mystery vanishes if one considers the linear functionalL on polynomials in z defined by 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. 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 mappingL, 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.
