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In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing, perhaps not altogether successfully.
In the 1970s, Steven Roman, Gian-Carlo Rota, and others developed the umbral calculus by means of linear functionals on spaces of polynomials. See in particular "The Umbral Calculus" by Roman and Rota, cited below.
That method is a notational device 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 derived by more complicated methods that can be taken literally without logical difficulty. An example involves the Bernoulli polynomials
(where we denote Bernoulli numbers bn by the lower-case b in order to distinguish them from the Bell numbers Bn). We can derive the identity
by methods that any staid conservative will bless cheerfully, or we can give a simpler argument by proceeding "umbrally", pretending that the subscript n − k is an exponent:
and then differentiate, getting
The variable b is an "umbra" ( Latin for shadow).
In the 1930s and 1940s, Eric Temple Bell tried unsuccessfully to make this kind of argument logically rigorous. The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively.
Another combinatorialist, Gian-Carlo Rota, pointed out that the mystery vanishes if one considers the linear functional L on polynomials in y defined by
Then one can write
etc. Rota later stated that much confusion resulted from the failure to distinguish between three equivalence relationIn mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive symmetric and transitive i. if the relation is written as ~ it holds for all a b and c in X that # (Reflexivity) a ~ a # (Symmetry) if a ~ b then b ~ a # (Transs 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 recursionIn mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in formula satisfied by the Bell numbers, which enumerate partitionsA partition of U into 6 blocks: a Venn diagram representation. In mathematics, a partition of a set X is a division of X into non-overlapping parts or blocks that cover all of X''. Definition A partition of a set X is a set of nonempty subsets of X such t 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 algebraAlgebra In mathematics, an algebra over a field K, or a K-algebra is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. Some authors use the t of linear functionals on the vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for 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.