## 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 , shadow'' identities are obtained when the polynomials are changed to discrete values and the exponent in is changed to the Pochhammer Symbol .

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

 (1)

with looks suspiciously like a finite analog of the Taylor Series expansion
 (2)

where is the Differential Operator. Similarly, the Chu-Vandermonde Identity
 (3)

with a Binomial Coefficient, looks suspiciously like an analog of the Binomial Theorem
 (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.

© 1996-9 Eric W. Weisstein
1999-05-26