The term umbral calculus has two related but distinct meanings. In mathematics, before the 1970s, umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove them. These techniques were introduced in 1861 by John Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively. The use of shadowy techniques was put on a solid mathematical footing starting in the 1970s, and the resulting mathematical theory is also referred to as "umbral calculus".
History In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing, however his attempt in making this kind of argument logically rigorous was unsuccessful. The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s, used techniques of this sort extensively. In the 1970s, Gian-Carlo Rota, with Steven Roman and others, developed the umbral calculus by means of linear functions 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 systematic correspondence techniques of the calculus of finite differences.
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 (which contains a binomial coefficient):
( y + x ) n = ∑ k = 0 n ( n k ) y n − k x k {\displaystyle (y+x)^{n}=\sum _{k=0}^{n}{n \choose k}y^{n-k}x^{k}}
and the remarkably similar-looking relation on the Bernoulli polynomials:
B n ( y + x ) = ∑ k = 0 n ( n k ) B n − k ( y ) x k . {\displaystyle B_{n}(y+x)=\sum _{k=0}^{n}{n \choose k}B_{n-k}(y)x^{k}.}
Compare also the ordinary derivative
d d x x n = n x n − 1 {\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}
to a very similar-looking relation on the Bernoulli polynomials:
d d x B n ( x ) = n B n − 1 ( x ) . {\displaystyle {\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 ) = ∑ k = 0 n ( n k ) b n − k x k = ( b + x ) n , {\displaystyle 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 = n B n − 1 ( x ) . {\displaystyle B_{n}'(x)=n(b+x)^{n-1}=nB_{n-1}(x).}
In the above, the variable b is an "umbra" (Latin for shadow). See also Faulhaber's formula.
Umbral Taylor series In differential calculus, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. That is, a real or complex-valued function f of a single variable that is analytic at a {\displaystyle a} can be written as:
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