Umbral calculus

Umbral calculus in the traditional sense of the term

In mathematics, before the 1970s, the term umbral calculus was understood to mean what is sometimes called Blissard's symbolic method, sometimes attributed to James Joseph Sylvester or to Edouard Lucas. 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

<math>p_n(x)=\sum_{k=0}^n{n\choose k}b_{n-k}x^k</math>

where the Bernoulli numbers b_n are denoted by the lower-case b in order to distinguish them from the Bell numbers B_n. We can derive the identity

<math>p_n'(x)=np_{n-1}(x)</math>

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:

<math>p_n(x)=\sum_{k=0}^n {n\choose k}b^{n-k}x^k=(b+x)^n,</math>

and then differentiate, getting

<math>p_n'(x)=n(b+x)^{n-1}=np_{n-1}(x).</math>

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

<math>L(y^n)=b_n</math>.

Then one can write

<math>p_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),</math>

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 "=". [Details need to be added here.] Rota used umbral methods to establish the recursion formula satisfied by the Bell numbers.

Umbral calculus in a more modern sense of the term

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.