Umbral Calculus

The study of certain properties of Finite Differences. The term was coined by Sylvester from the word ``umbra'' (meaning ``shadow'' in Latin), and reflects the fact that for many types of identities involving sequences of polynomials with Powers $a^n$, ``shadow'' identities are obtained when the polynomials are changed to discrete values and the exponent in $a^n$ is changed to the Pochhammer Symbol $(a)_n\equiv a(a-1)\cdots(a-n+1)$.

For example, Newton's Forward Difference Formula written in the form

$$f(x+a)=\sum_{n=0}^\infty {(a)_n \Delta^n f(x)\over n!}$$ (1)

with $f(x+a)\equiv f_{x+a}$ looks suspiciously like a finite analog of the Taylor Series expansion

$$f(x+a)=\sum_{n=0}^\infty {a^n \tilde D^n f(x)\over n!},$$ (2)

where $\tilde D$ is the Differential Operator. Similarly, the Chu-Vandermonde Identity

$$(x+a)_n=\sum_{k=0}^\infty {n\choose k}(a)_k(x)_{n-k}$$ (3)

with ${n\choose k}$ a Binomial Coefficient, looks suspiciously like an analog of the Binomial Theorem

$$(x+a)^n=\sum_{k=0}^\infty{n\choose k} a^k x^{n-k}$$ (4)

(Di Bucchianico and Loeb).

See also Binomial Theorem, Chu-Vandermonde Identity, Finite Difference

References

Roman, S. and Rota, G.-C. ``The Umbral Calculus.'' Adv. Math. 27, 95-188, 1978.

Roman, S. The Umbral Calculus. New York: Academic Press, 1984.