Umbral calculus

In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to 'prove' them. These techniques were introduced in the 19th century and are sometimes called Blissard's symbolic method, and sometimes attributed to James Joseph Sylvester, who used the technique extensively, or to Edouard Lucas.

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. Currently, umbral calculus is understood primarily to mean the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences.

The 19th-century umbral calculus

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. Consider, for example, the ordinary binomial expansion

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}$

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

$B_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}B_{n-k}(x)y^{k}$

Compare also the ordinary derivative

${\frac {d}{dx}}x^{n}=nx^{n-1}$

to a very similar-looking relation on the Bernoulli polynomials:

${\frac {d}{dx}}B_{n}(x)=nB_{n-1}(x)$

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:

$B_{n}(x)=\sum _{k=0}^{n}{n \choose k}b^{n-k}x^{k}=(b+x)^{n},$

and then differentiating, one gets the desired result:

$B_{n}'(x)=n(b+x)^{n-1}=nB_{n-1}(x).$

In the above, the variable b is an "umbra" (Latin for shadow).

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

Another combinatorialist, Gian-Carlo Rota, pointed out that the mystery vanishes if one considers the linear functional L on polynomials in y defined by

$L(y^{n})=B_{n}(0)=B_{n}$.

Then one can write

$B_{n}(x)=\sum _{k=0}^{n}{n \choose k}B_{n-k}x^{k}=\sum _{k=0}^{n}{n \choose k}L(y^{n-k})x^{k}=L\left(\sum _{k=0}^{n}{n \choose k}y^{n-k}x^{k}\right)=L((y+x)^{n}),$

etc. 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

$\langle L_{1}L_{2}\mid x^{n}\rangle =\sum _{k=0}^{n}{n \choose k}\langle L_{1}\mid x^{k}\rangle \langle L_{2}\mid x^{n-k}\rangle .$

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.

References

• Steven Roman and Gian-Carlo Rota, "The Umbral Calculus", Advances in Mathematics, volume 27, pages 95 - 188, (1978).
• 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.
• Steven Roman, The Umbral Calculus, Dover Publications.