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

Moa APAGODU, Shalosh B. EKHAD, and Patrick GASKILL

Added Dec. 13, 2011: The main theorm of this note is contained in Lemma 5 of “On Twogenerated Non-commutative Algebras Subject to the Affine Relation” by Christoph Koutschan, Viktor Levandovskyy, Oleksandr Motsak, http://arxiv.org/abs/1108.1108, who prove many other results, and a stronger version of our main result (using Stirling numbers).

This note has only personal and historical interest, and is only published in the Personal Journal of Ekhad and Zeilberger and arxiv.org .

We all know the binomial theorem

(x + y)^{r} = ∑_{i=0}^{r} \frac{r!}{i!(r − i)!}x^{i}y^{r−i} ,  (1)

where x and y are commuting variables, i.e. yx = xy. The binomial theorem is easily proved by induction on r.

Not as famous is the quantum analog, that goes back to Marco Schützenberger,

(x + y)^{r} = ∑_{i=0}^{r} \frac{[r]!}{[i]![r − i]!}x^{i}y^{r−i} ,  (2)

where x and y are q-commuting variables, i.e. yx = qxy, and [j]! := (1)(1 + q)(1 + q + q^{2}) ⋯ (1 + q + … + q^{j−1}). The quantum binomial theorem is also easily proved by induction on r.

Even more obscure is the binomial theorem for (D + x)^{r}, where D is the differentiation operator \frac{d}{dx} (so Dx = xD + 1):

(x + D)^{r} = ∑_{k=0}^{⌊r/2⌋} \frac{r!}{2^{k}k!} ∑_{j+l=r−2k} \frac{1}{j!l!}x^{j}D^{l} ,  (3)

that is also easily proved by induction.

But nothing analogous is known for (n + N)^{r}, where N is the shift operator Nf(n) := f(n + 1), and n is multiplication by n. Now the commutation relation is Nn = nN + N and things seem to get messier.

Let’s first try to expand (N + n)^{r} in powers of N with coefficients that are polynomials in n:

(N + n)^{r} = ∑_{d=0}^{r} P_{r,d}(n)N^{d} .  (4a)

We claim that the coefficients P_{r,d}(n) are given by the following “explicit” formula

P_{r,d}(n) = ∑_{{p_{0}+...+p_{d}=r−d \atop 0≤p_{0},...p_{d}≤r−d}} ∏_{j=0}^{d}(n + j)^{p_{j}} .  (4b)

In other words, P_{r,d}(n) is the weight-enumerator of compositions of r − d into d + 1 non-negative integers, with

W eight([p_{0}, …, p_{d}]) := ∏_{j=0}^{d}(n + j)^{p_{j}} .

Since (N + n)^{r} = (N + n)(N + n)^{r−1} we have the recurrence:

P_{r,d}(n) = P_{r−1,d−1}(n + 1) + nP_{r−1,d}(n) .

Identity (4) is proved by induction by noting that any composition of r − d into d + 1 non-negative integers either has p_{0} = 0 and beheading it yields a composition of r−d into (d−1)+1 non-negative integers, and the weights get adjusted by replacing n by n + 1, or p_{0} ≥ 1 and subtracting 1 from p_{0} yields a composition of r − d − 1 into d + 1 non-negative integers, and adding the 1 back to the p_{0} term results in multiplying the weight by n.

While formula (4) is very elegant and combinatorial, it would be nice to have an explicit formula, as a linear combination of monomials n^{i}N^{j} analogous to (1), (2) and (3). This does not seem to be possible, but by using the Maple package Nnr, written by Doron Zeilberger, and downloadable

from http://www.math.rutgers.edu/~zeilberg/tokhniot/Nnr, one can get an explicit formula

for the first k highest-degree terms for any desired k, for r ≥ 2k.

Let’s describe the answer for k = 3. First we define,

((n + N))^{r} = ∑_{i=0}^{r} \frac{r!}{i!(r − i)!}n^{i}N^{r−i} ,

in other words the polynomial in the (non-commuting) variables n and N obtained by expanding (n + N)^{m} while pretending that n and N commute. We have (all the algebric computations below should be done in commutative algebra, and at the end each monomial should be expressed with n before N. i.e. in the form n^{i}N^{j})

(N + n)^{r} = ((N + n))^{r} + \binom{r}{2}N((N + n))^{r−2} + \binom{r}{3}N((N + n))^{r−4} (\frac{1}{4} (3 r − 5) N + n)

+\binom{r}{4}N((N + n))^{r−6} (\frac{1}{2} (r − 2) (r − 3) N^{2} + 2 (r − 3) nN + n^{2}) +

(terms − of − total − degree ≤ r − 4) .

To get all the terms of total degree ≥ r−10, see http://www.math.rutgers.edu/~zeilberg/tokhniot/oNnr10.

You are welcome to get all the terms of degree ≥ r − k, for any desired positive integer k, by typing

Mispat(r,k,n,N); in the Maple package Nnr.

Moa Apagodu, Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284, mapagodu [at] vcu [dot] edu .

Shalosh B. Ekhad, Department of Mathematics, Rutgers University, Piscataway, NJ 08854 .

Patrick Gaskill, Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284, gaskillpw [at] vcu [dot] edu .