Editing Umbral calculus (section)

==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]]):

:<math>(y+x)^n=\sum_{k=0}^n{n\choose k}y^{n-k} x^k</math>

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

:<math>B_n(y+x)=\sum_{k=0}^n{n\choose k}B_{n-k}(y) x^k.</math>

Compare also the ordinary derivative

:<math> \frac{d}{dx} x^n = nx^{n-1} </math>

to a very similar-looking relation on the Bernoulli polynomials: 
  
:<math> \frac{d}{dx} B_n(x) = nB_{n-1}(x).</math>

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''&nbsp;&minus;&nbsp;''k'' is an exponent:

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

and then differentiating, one gets the desired result:

:<math>B_n'(x)=n(b+x)^{n-1}=nB_{n-1}(x).</math>

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

See also [[Faulhaber's formula]].