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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
x^{i} y^{r-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.

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