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 (1861) 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 in its penumbra systematic correspondence techniques of the calculus of finite differences.
The 19th-century umbral calculus
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.
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.
Umbral Taylor series
Similar relationships were also observed in the theory of finite differences. The umbral version of the Taylor series is given by a similar expression involving the k 'th forward differences of a polynomial function f,
is the Pochhammer symbol used here for the falling sequential product. A similar relationship holds for the backward differences and rising factorial.
Bell and Riordan
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.
The modern umbral calculus
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 (the key operation of umbral calculus). For instance, we can now prove that
by expanding the right-hand side as
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 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. 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.
- E. T. Bell, "The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life", The American Mathematical Monthly 45:7 (1938), pp. 414–421.
- Rota, G. C.; Kahaner, D.; Odlyzko, A. (1973). "On the foundations of combinatorial theory. VIII. Finite operator calculus". Journal of Mathematical Analysis and Applications. 42 (3): 684. doi:10.1016/0022-247X(73)90172-8.
- G.-C. Rota and J. Shen, "On the Combinatorics of Cumulants", Journal of Combinatorial Theory, Series A, 91:283–304, 2000.
- Bell, E. T. (1938), "The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life", The American Mathematical Monthly, Mathematical Association of America, 45 (7): 414–421, ISSN 0002-9890, JSTOR 2304144
- Blissard, John (1861), "Theory of generic equations", The quarterly journal of pure and applied mathematics, 4: 279–305
- Roman, Steven M.; Rota, Gian-Carlo (1978), "The umbral calculus", Advances in Mathematics, 27 (2): 95–188, doi:10.1016/0001-8708(78)90087-7, ISSN 0001-8708, MR 0485417
- G.-C. Rota, D. Kahaner, and A. Odlyzko, "Finite Operator Calculus," Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.
- Roman, Steven (1984), The umbral calculus, Pure and Applied Mathematics, 111, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-594380-2, MR 0741185. Reprinted by Dover, 2005.
- Roman, S. (2001) , "Umbral calculus", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4