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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
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(1) |
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(2) |
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(3) |
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(4) |
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.