Roman (1984, p. 2) describes umbral calculus as the study of the class of Sheffer sequences. Umbral calculus provides a formalism for the systematic derivation and classification of almost all classical combinatorial identities for polynomial sequences, along with associated generating functions, expansions, duplication formulas, recurrence relations, inversions, Rodrigues representation, etc., (e.g., the Euler-Maclaurin integration formulas, Boole's summation formula, the Chu-Vandermonde identity, Newton's divided difference interpolation formula, Gregory's formula, Lagrange inversion).
The term "umbral calculus" 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 falling
factorial 
.
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).
