Umbral calculus

The umbral calculus or umbral calculus (English umbral calculus , also Latin umbra , "the shadow") is a sub-area of combinatorics that arose from the observation of formal similarity in the derivation of polynomial identities, in which indices were treated as exponents . Since no explanation was found for a long time, the term shadow calculus (Umbral calculus) was used.

The techniques go back to the 19th century, in particular John Blissard (1861), after whom Blissard's symbolic method was spoken of, but they were also used by Édouard Lucas (who called it the symbolic method) and James Joseph Sylvester , among others . The name umbral comes from New Year's Eve . In the 1930s, Eric Temple Bell tried to give the methods a strict foundation (with little success in enforcing them); this was only achieved by Gian-Carlo Rota and Steven Roman in the 1970s. But they were already used in combinatorics by John Riordan, for example .

Examples

An example are the Bernoulli polynomials , for which the following applies:

${\displaystyle B_{n}(y+x)=\sum _{k=0}^{n}{n \choose k}B_{n-k}(y)x^{k}.}$

This has a remarkable similarity to the binomial expansion:

${\displaystyle (y+x)^{n}=\sum _{k=0}^{n}{n \choose k}y^{n-k}x^{k}}$

if you swap indices and exponents.

The following applies to the derivation:

${\displaystyle {\frac {d}{dx}}B_{n}(x)=nB_{n-1}(x).}$

a similarity to the derivation of powers:

${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}$

if you swap indices and exponents here too.

This gave rise to umbral evidence that, although it cannot be strictly justified, nevertheless works . For example, if one uses the index formally as an exponent for the Bernoulli polynomials in the formula given above:

${\displaystyle B_{n}(x)=\sum _{k=0}^{n}{n \choose k}b^{n-k}x^{k}=(b+x)^{n},}$

and differentiated, you get the correct result:

${\displaystyle B_{n}'(x)=n(b+x)^{n-1}=nB_{n-1}(x).}$

The variable b is a shadow (umbra).

Another example is Newton's formula of calculus of differences :

${\displaystyle f(x+a)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](x)}{k!}}(a)_{k}}$

With

${\displaystyle (a)_{k}=a(a-1)(a-2)\cdots (a-k+1)}$

the falling factorial ( Pochhammer symbol ).

Replacing the differences due to the discharge and by obtaining the Taylor series : ${\displaystyle (a)_{k}}$ ${\displaystyle a^{k}}$

${\displaystyle f(x+a)=\sum _{k=0}^{\infty }{\frac {D^{k}[f](x)}{k!}}a^{k}}$

There is also a formal similarity between the binomial formula :

${\displaystyle (x+a)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}a^{k}}$

and the formula of Vandermonde and Chu for decreasing factorials:

${\displaystyle {(x+a)}_{n}=\sum _{k=0}^{n}{\binom {n}{k}}{(x)}_{n-k}{(a)}_{k}}$

Here, too, indices are formally replaced by exponents.

Theory of Rota and Roman

According to Rota, the similarities can be explained if one considers linear functionals L on polynomials.

Applied to Bernoulli polynomials, one first defines:

${\displaystyle L(z^{n})=B_{n}(0)=B_{n}.}$

and has for the Bernoulli polynomials

${\displaystyle {\begin{aligned}B_{n}(x)&=\sum _{k=0}^{n}{n \choose k}B_{n-k}x^{k}\\&=\sum _{k=0}^{n}{n \choose k}L\left(z^{n-k}\right)x^{k}\\&=L\left(\sum _{k=0}^{n}{n \choose k}z^{n-k}x^{k}\right)\\&=L\left((z+x)^{n}\right)\end{aligned}}}$

which can be replaced by . Indices are now in the form of exponents, the essential step in the umbra calculus. This can be used to show, for example: ${\displaystyle B_{n}(x)}$ ${\displaystyle L((z+x)^{n})}$

${\displaystyle {\begin{aligned}\sum _{k=0}^{n}{n \choose k}B_{n-k}(y)x^{k}&=\sum _{k=0}^{n}{n \choose k}L\left((z+y)^{n-k}\right)x^{k}\\&=L\left(\sum _{k=0}^{n}{n \choose k}(z+y)^{n-k}x^{k}\right)\\&=L\left((z+x+y)^{n}\right)\\&=B_{n}(x+y).\end{aligned}}}$

Rota used the Umbralkule in 1964 to derive recursion formulas for Bell's numbers . With J. Shen he applied the umbrella calculus to the study of the combinatorial properties of cumulants .

According to Roman and Rota, the umbral calculus examines the umbral algebra of linear functionals on the space of polynomials in one variable . The product of two functionals is defined as follows: ${\displaystyle x}$

${\displaystyle \left\langle L_{1}L_{2}|x^{n}\right\rangle =\sum _{k=0}^{n}{n \choose k}\left\langle L_{1}|x^{k}\right\rangle \left\langle L_{2}|x^{n-k}\right\rangle .}$

literature

• Steven Roman: The Umbral Calculus , Academic Press 1984, Dover 2005, online
• Gian-Carlo Rota, Steven Roman: The umbral calculus , Advances in Mathematics, Volume 27, 1978, pp. 95-188
• Gian-Carlo Rota: Finite Operator Calculus, Academic Press 1976

Web links

• Eric Weisstein, Umbral calculus , mathworld
• Roman, Umbral Calculus , Encyclopedia of Mathematics

Individual evidence

1. ↑ Eric Temple Bell, The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life, The American Mathematical Monthly, Volume 45, No. 7, 1938, pp. 414-421
2. ↑ The method used to be ascribed to Lucas
3. ^ Bell, Algebraic Arithmetic. American Mathematical Society, 1927, ISBN 0-8218-4601-9
4. ^ Riordan, Combinatorial identities, Wiley 1968
5. ↑ Rota, The number of partitions of a set, American Mathematical Monthly, Volume 71, 1964, pp. 498-504
6. ↑ Rota, Shen, On the Combinatorics of Cumulants, Journal of Combinatorial Theory A, Volume 91, 2000, pp. 283-304