Umbral Calculus - The 19th-century Umbral Calculus

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.

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 variable b is an "umbra" (Latin for shadow).

See also Faulhaber's formula.