Editing Umbral calculus (section)

==Modern umbral calculus==

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

:<math>L(z^n)= B_n(0)= B_n.</math>

Then, using the definition of the Bernoulli polynomials and the definition and linearity of ''L'', one can write

:<math>\begin{align}
B_n(x) &= \sum_{k=0}^n{n\choose k}B_{n-k}x^k \\
       &= \sum_{k=0}^n{n\choose k}L\left(z^{n-k}\right)x^k \\
       &= L\left(\sum_{k=0}^n{n\choose k}z^{n-k}x^k\right) \\
       &= L\left((z+x)^n\right)
\end{align}</math>

This enables one to replace occurrences of <math>B_n(x)</math> by <math>L((z+x)^n)</math>, that is, move the ''n'' from a subscript to a superscript (the key operation of umbral calculus). For instance, we can now prove that:

:<math>\begin{align}
\sum_{k=0}^n{n\choose k}B_{n-k}(y) x^k 
&= \sum_{k=0}^n{n\choose k}L\left((z+y)^{n-k}\right) x^k \\
&= L\left(\sum_{k=0}^n {n\choose k} (z+y)^{n-k} x^k \right) \\
&= L\left((z+x+y)^n\right) \\
&= B_n(x+y).
\end{align}</math>

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. -->

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.

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

:<math>\left \langle L_1 L_2 | x^n \right \rangle = \sum_{k=0}^n {n \choose k} \left \langle L_1 | x^k \right \rangle \left \langle L_2 | x^{n-k} \right \rangle.</math>

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 journal | last1 = Rota | first1 = G. C. | last2 = Kahaner | first2 = D. | last3 = Odlyzko | first3 = A. | doi = 10.1016/0022-247X(73)90172-8 | title = On the foundations of combinatorial theory. VIII. Finite operator calculus | journal = Journal of Mathematical Analysis and Applications | volume = 42 | issue = 3 | pages = 684 | year = 1973 | doi-access = free }}</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]].

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|Journal of Combinatorial Theory, Series A]], 91:283–304, 2000.</ref>