Date: Apr 1, 2013 12:45 PM
Author: Timothy Murphy
Subject: Re: 1 + 2 + ... + n a polynomial how?

Jussi Piitulainen wrote:

> Is it obvious that 1 + 2 + ... + n is a polynomial of degree 2? How?
> I mean the sum of the first n positive integers. I would like to see
> that it is a polynomial of degree 2 _without using_ the fact that it
> is equal to n(n + 1)/2. Zeilberger (his new Opinion 129) says Gauss
> could have used the polynomiality of the sum to support the equality,
> rather than the other way around.
> Thanks for any insight.

I guess Zeilberger was referring to the difference calculus,
as used in numerical analysis.
If you have a sequence a(n) then Da(n) = a(n) - a(n-1).
Also if A(n) = sum_{r<=n} then DA(n) = a(n).
So in this case the second difference D^A(n) = D(n) = 1 is constant,
and it follows that A(n) is a polynomial of degree 2.
This goes back long before Gauss, to Newton at least.

Timothy Murphy
e-mail: gayleard /at/
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College Dublin