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.

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

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:(x+y)^n=sum_{k=0}^n{nchoose k}x^{n-k} y^k

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

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:B_n(x+y)=sum_{k=0}^n{nchoose k}B_{n-k}(x) y^k

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Compare also the ordinary derivative

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: rac{d}{dx} x^n = nx^{n-1}

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

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: rac{d}{dx} B_n(x) = nB_{n-1}(x)

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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:

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:B_n(x)=sum_{k=0}^n {nchoose k}b^{n-k}x^k=(b+x)^n,

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and then differentiating, one gets the desired result:

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

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

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