# Desummation

From: Lee Corbin (lcorbin@ricochet.net)
Date: Thu Apr 12 2001 - 19:57:14 MDT

> Given the recent discussion of mathematics on this list, I
> thought you might find the following paper interesting:
> http://www.chess3d.com/desum.htm
> It tries to answer the question "Given a polynomial of one
> variable to whatever degree, is it possible to find the
> polynomial that when summed equals the polynomial given?"

Actually, what the author calls the Desummation operator
(and signifies with a backwards summation sign) is already
known in mathematics as the difference operator, and is
symbolized by a capital delta.  So, for example, Delta(x^2)
equals 2x-1.  The author of the web page explains this
very well, and gives good examples.

I'm sure that Newton and Wallis knew how to calculate the
difference of a polynomial the very first time that they
seriously thought about it, and I once saw credit being
given to someone in 1711 for one of the nicest theorems,
namely how you can do the reverse operation (e.g., a
mechanical way to go from 2x+1 to x^2 + C for any polynomial.
But vast amounts about all this were known by 1851 when Boole
wrote "The Calculus of Finite Differences".  A wonderful
modern reference, though probably not containing everything
in Boole's book, is Knuth's "Concrete Mathematics".

Lee Corbin

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