Umbral Calculus

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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 a^n, "shadow" identities are obtained when the polynomials are changed to discrete values and the exponent in a^n is changed to the falling factorial (a)_n=a(a-1)...(a-n+1).

For example, Newton's forward difference formula written in the form

with f(x+a)=f_(x+a) looks suspiciously like a finite analog of the Taylor series expansion

where D^~ is the differential operator. Similarly, the Chu-Vandermonde identity

with (n; k) a binomial coefficient, looks suspiciously like an analog of the binomial theorem

(Di Bucchianico and Loeb).

See also

Appell Sequence, Binomial Theorem, Chu-Vandermonde Identity, Combinatorics, Faà di Bruno's Formula, Finite Difference, Sheffer Sequence

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References

Bell, E. T. "Postulational Basis for the Umbral Calculus." Amer. J. Math. 62, 717-724, 1940.Di Bucchianico, A. and Loeb, D. "A Selected Survey of Umbral Calculus." Electronic J. Combinatorics Dynamical Survey DS3, 1-34, April 2000. http://www.combinatorics.org/Surveys/#DS3.Loeb, D. E. "Umbral Calculus." http://dept-info.labri.u-bordeaux.fr/~loeb/umbral.html.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.Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.

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Umbral Calculus

Cite this as:

Weisstein, Eric W. "Umbral Calculus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UmbralCalculus.html

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