Umbral Calculus
The study of certain properties of Sylvester 
from the word ``umbra'' (meaning ``shadow'' in Latin), and reflects the fact that for many types of identities involvingsequences of polynomials with Powers 
, ``shadow'' identities are obtained when the polynomials arechanged 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) |
looks suspiciously like a finite analog of the Taylor Series expansion | (2) |
is the Differential Operator. Similarly, the Chu-Vandermonde Identity | (3) |
a Binomial Coefficient, looks suspiciously like an analog of the Binomial Theorem | (4) |
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.
- Affine Scheme
- Affine Space
- Affine Transformation
- Affinity
- Affix
- Aggregate
- AGM
- Agnesi's Witch
- Agnésienne
- Agonic Lines
- Ahlfors-Bers Theorem
- A-Integrable
- Airy Differential Equation
- Airy-Fock Functions
- Airy Functions
- Airy Projection
- Aitken Interpolation
- Aitken's Delta Squared Process
- Ajima-Malfatti Points
- Albanese Variety
- Albers Conic Projection
- Albers Equal-Area Conic Projection
- Alcuin's Sequence
- Aleksandrov-Cech Cohomology
- Aleksandrov's Uniqueness Theorem