The Binomial Theorem for (N+n)r (where Nf(n)=f(n+1))

By Moa Apagodu, Shalosh B. Ekhad, and Patrick Gaskill

.pdf   .ps   .tex  
Written: Dec. 8. 2011.
ADDED Dec. 13, 2011: Christoph KOUTSCHAN kindly informed us that we were scooped by himself, Viktor Levandovskyy, and Oleksandr Motsak. They have much more than we do, and of course, they were first, so of course, we are not publishing in a "real" journal, but the beauty of the internet is that you can still publish rediscoveries (and even rediscoveries of special cases of already proved results), as long as you reference the original discoverers, because each attempt adds something new.
We all know the binomial theorem, i.e. to expand (x+y)r as a sum of monomials xi yr-i, when x and y commute. There is also a formula, where the coefficients of each monomial can be expressed in closed form, for (x+p)r where x is position, p is momentum, and [p,x]=h, according to Heisenberg. But this is (probably) not the case for (N+n)r, where N is the shift operator. So we do what we can.

Maple Package

Important: This article is accompanied by Maple package

Sample Input and Output for Nnr

If you want to see all the terms of total degree ≥ r-10 in the expansion of (N+n)^r, the
input gives the output.

Doron Zeilberger's List of Papers

Doron Zeilberger's Home Page