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| In [[mathematics]] before the 1970s, the term '''''umbral calculus''''' referred to the surprising similarity between seemingly unrelated [[polynomial equation]]s and certain shadowy techniques used to 'prove' them. These techniques were introduced by {{harvs|txt|authorlink=John Blissard|first=John|last=Blissard|year=1861}} and are sometimes called '''Blissard's symbolic method'''. They are often attributed to [[Édouard Lucas]] (or [[James Joseph Sylvester]]), who used the technique extensively.<ref>E. T. Bell, "The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life", ''The American Mathematical Monthly'' '''45''':7 (1938), pp. 414–421.</ref>
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| In the 1930s and 1940s, [[Eric Temple Bell]] attempted to set the umbral calculus on a rigorous footing.
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| In the 1970s, [[Steven Roman]], [[Gian-Carlo Rota]], and others developed the umbral calculus by means of [[linear functional]]s on spaces of polynomials. Currently, ''umbral calculus'' refers to the study of [[Sheffer sequence]]s, including polynomial sequences of [[binomial type]] and [[Appell sequence]]s, but may encompass in its penumbra systematic correspondence techniques of the [[calculus of finite differences]].
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| ==The 19th-century umbral calculus==
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| 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.
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| An example involves the [[Bernoulli polynomials]]. Consider, for example, the ordinary [[binomial expansion]] (which contains a [[binomial coefficient]]):
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| :<math>(y+x)^n=\sum_{k=0}^n{n\choose k}y^{n-k} x^k</math>
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| and the remarkably similar-looking relation on the [[Bernoulli polynomials]]: | |
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| :<math>B_n(y+x)=\sum_{k=0}^n{n\choose k}B_{n-k}(y) x^k.</math>
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| Compare also the ordinary derivative
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| :<math> \frac{d}{dx} x^n = nx^{n-1} </math>
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| to a very similar-looking relation on the Bernoulli polynomials:
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|
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| :<math> \frac{d}{dx} B_n(x) = nB_{n-1}(x).</math>
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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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| :<math>B_n(x)=\sum_{k=0}^n {n\choose k}b^{n-k}x^k=(b+x)^n,</math>
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| and then differentiating, one gets the desired result:
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| :<math>B_n'(x)=n(b+x)^{n-1}=nB_{n-1}(x).\, </math>
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| In the above, the variable ''b'' is an "umbra" ([[Latin]] for ''shadow'').
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| See also [[Faulhaber's formula]].
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| ==Umbral Taylor series==
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| Similar relationships were also observed in the theory of [[finite differences]]. The umbral version of the [[Taylor series]] is given by a similar expression involving the ''k'' 'th [[forward difference]]s <math>\Delta^k [f]</math> of a [[polynomial]] function ''f'',
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| :<math>f(x)=\sum_{k=0}^\infty\frac{\Delta^k [f](0)}{k!}(x)_k</math> | |
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| where
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| :<math>(x)_k=x(x-1)(x-2)\cdots(x-k+1)</math>
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| is the [[Pochhammer symbol]] used here for the falling sequential product. A similar relationship holds for the backward differences and rising factorial.
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| This series is also known as the [[Finite difference#Newton's_series|''Newton series'']] or '''Newton's forward difference expansion'''.
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| The analogy to Taylor's expansion is utilized in the [[calculus of finite differences]].
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| ==Bell and Riordan==
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| In the 1930s and 1940s, [[Eric Temple Bell]] tried unsuccessfully to make this kind of argument logically rigorous. The [[combinatorics|combinatorialist]] [[John Riordan (mathematician)|John Riordan]] in his book ''Combinatorial Identities'' published in the 1960s, used techniques of this sort extensively.
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| ==The modern umbral calculus==
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| 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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| :<math>L\left(y^n\right)= B_n(0)= B_n.\,</math>
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| Then one can write
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| :<math>
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| \begin{align}
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| B_n(x)
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| &=\sum_{k=0}^n{n\choose k}B_{n-k}x^k
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| && \text{applying the definition of Bernoulli polynomials}
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| \\
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| &= \sum_{k=0}^n{n\choose k}L\left(y^{n-k}\right)x^k
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| && \text{applying the above definition} | |
| \\
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| &= L\left(\sum_{k=0}^n{n\choose k}y^{n-k}x^k\right)
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| && \text{since L is linear} | |
| \\
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| &= L\left((y+x)^n\right).&&
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| \end{align}
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| </math>
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| This enables one to replace occurrences of <math>B_n(x)</math> by <math>L((y+x)^n)</math>, that is, move the ''n'' from a subscript to a superscript (the key operation of umbral calculus).
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| For instance, we can now prove that
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| :<math>B_n(y+x)=\sum_{k=0}^n{n\choose k}B_{n-k}(y) x^k</math>
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| by expanding the right-hand side as
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| :<math>\sum_{k=0}^n{n\choose k}B_{n-k}(y) x^k = \sum_{k=0}^n{n\choose k}L\left((2y)^{n-k}\right) x^k = L\left(\sum_{k=0}^n{n\choose k}(2y)^{n-k} x^k\right) = L\left((2y+x)^n\right) = B_n(x+y).</math>
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| Rota later stated that much confusion resulted from the failure to distinguish between three [[equivalence relation]]s that occur frequently in this topic, all of which were denoted by "=". <!-- Details need to be added here. -->
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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 [[partition of a set|partitions]] of finite sets.
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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 over a field|algebra]] of linear functionals on the [[vector space]] of polynomials in a variable ''x'', with a product ''L''<sub>1</sub>''L''<sub>2</sub> of linear functionals defined by
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| :<math>\langle L_1 L_2 \mid x^n \rangle = \sum_{k=0}^n {n \choose k}\langle L_1 \mid x^k\rangle \langle L_2 \mid x^{n-k} \rangle.</math>
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| When [[polynomial sequence]]s replace sequences of numbers as images of ''y''<sup>''n''</sup> 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.<ref>{{cite doi|10.1016/0022-247X(73)90172-8|noedit}}</ref> A small sample of that theory can be found in the article on [[binomial type|polynomial sequences of binomial type]]. Another is the article titled [[Sheffer sequence]].
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| Rota later applied umbral calculus extensively in his paper with Shen to study the various combinatorial properties of the [[cumulant]]s.<ref>G.-C. Rota and J. Shen, [http://www.sciencedirect.com/science/article/pii/S0097316599930170 "On the Combinatorics of Cumulants"], Journal of Combinatorial Theory, Series A, 91:283–304, 2000.</ref>
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| ==See also==
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| *[[Pidduck polynomials]]
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| *[[Symbolic method]] in invariant theory
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| ==Notes==
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| <references />
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| ==References==
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| *{{Citation | authorlink=E. T. Bell | last1=Bell | first1=E. T. | title=The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life | jstor=2304144 | publisher=[[Mathematical Association of America]] | year=1938 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=45 | issue=7 | pages=414–421}}
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| *{{Citation | last1=Blissard | first1=John | title=Theory of generic equations | url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0004 | year=1861 | journal=The quarterly journal of pure and applied mathematics | volume=4 | pages=279–305}}
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| *{{Citation | last1=Roman | first1=Steven M. | last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | title=The umbral calculus | doi=10.1016/0001-8708(78)90087-7 | mr=0485417 | year=1978 | journal=Advances in Mathematics | issn=0001-8708 | volume=27 | issue=2 | pages=95–188}}
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| * G.-C. Rota, D. Kahaner, and [[Andrew Odlyzko|A. Odlyzko]], ''"Finite Operator Calculus,"'' Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.
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| *{{Citation | last1=Roman | first1=Steven | title=The umbral calculus | url=http://books.google.com/books?id=JpHjkhFLfpgC | publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | series=Pure and Applied Mathematics | isbn=978-0-12-594380-2 | mr=741185 Reprinted by Dover, 2005 | year=1984 | volume=111}}
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| *{{eom|id=U/u095050|first=S. |last=Roman}}
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| ==External links==
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| * {{MathWorld|urlname=UmbralCalculus|title=Umbral Calculus}}
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| * {{cite journal |author=A. Di Bucchianico, D. Loeb |title=A Selected Survey of Umbral Calculus |journal=[[Electronic Journal of Combinatorics]] |series=Dynamic Surveys |volume=DS3 |year=2000 |url=http://www1.combinatorics.org/Surveys/ds3.pdf}}
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| * Roman, S. (1982), [http://www.romanpress.com/MathArticles/TheoryI.pdf The Theory of the Umbral Calculus, I]
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| {{DEFAULTSORT:Umbral Calculus}}
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| [[Category:Combinatorics]]
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| [[Category:Polynomials]]
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| [[Category:Finite differences]]
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