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) |

(2) |

(3) |

(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.

© 1996-9

1999-05-26