The Bell Differential Polynomials

Abstract

Let A be an associative algebra with a unit element 1. If A is commutative then there holds the well-known binomial law

$$ (x + y)n = \sum\limits_{k = 0}^n {(_k^n} {)_x}n - {k_y}k $$

((1))

where formally x ⁰ = 1, y ⁰ = 1. Our aim is to generalize the binomial law to the case where A is not necessarily commutative. This is possible if the successive commutators

$$ y' = [x,y],y = [x,y'],y' = [x,y], \cdots $$

are considered to be known. Namely, a factor x can be shifted from the left to the right by means of identities like

$$ xy = y' + yx,xy' = y'' + y'x, \cdots $$

We obtain

$$ {(x + y)^2} = {x^2} + 2yx + (y' + {y^2}) $$

((2))

,

$$ {(x + y)^3} = {x^3} + 3y{x^2} + 3(y' + {y^2})x + (y'' + 2yy' + y'y + {y^3}) $$

((3))

and so on. Some notions have to be introduced in order to describe the general procedure.