By Moa Apagodu, Shalosh B. Ekhad, and Patrick Gaskill
If you want to see all the terms of total degree ≥ r-10 in the expansion
of (N+n)^r, the
.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
input
gives the output.
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