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 modern umbral calculus

Another combinatorialist, Gian-Carlo Rota, pointed out that the mystery vanishes if one considers the linear functional L on polynomials in y defined by

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:L(y^n)=B_n(0)=B_n.

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Then one can write

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

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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 "=".

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In a paper published in 1964, Rota used umbral methods to establish the recursion formula satisfied by the Bell numbers, which enumerate partitions of finite sets.

Related Topics:
Recursion - Bell numbers - Partitions

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In the paper of Roman and Rota cited below, the umbral calculus is characterized as the study of the umbral algebra, defined as the algebra of linear functionals on the vector space of polynomials in a variable x, with a product L1L2 of linear functionals defined by

Related Topics:
Algebra - Vector space

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:langle L_1 L_2 mid x^n angle = sum_{k=0}^n {n choose k}langle L_1 mid x^k angle langle L_2 mid x^{n-k} angle.

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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.

Related Topics:
Polynomial sequence - Polynomial sequences of binomial type - Sheffer sequence

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