Amhuxa.322
net.general
utcsrgv!utzoo!decvax!ucbvax!mhtsa!eagle!mhuxj!mhuxa!rwhaas
Fri Jan 29 08:04:06 1982
Re: more boring factorials, be forewarned
As it turns out,taking the n-th difference of an n-th degree
polynomial gives n! times the coefficient of the x^n term. So does taking
the n-th derivative. The reason for this is (in the difference case,the
derivative case being fairly obvious) that monomials may be written as
a linear combination of the factorial polynomials,the required coefficients
being known as Stirling numbers of the second kind, and it is fairly easy
to show that the n-th difference of the n-th factorial polynomial is n!.
For example, denoting factorial polynomials by
x^(n)=x(x-1)...(x-n+1) {x^(0)=1}
we have
x^(1)=x
x^(2)=x(x-1)=x^2-x
x^(3)=x(x-1)(x-2)=x^3-3x^2+2x
and so on, and conversely,
x^1=x^(1)
x^2=x^(2)+2x^(1)
x^3=x^(3)+3x^(2)+x^(1)
and in general,
x^n=sum[v(m,n)x^(m)]
where the summation runs from 0 to m,and v(m,n) the Sterling number
of the second kind. If you have some time to waste, prove that v(m,n)
is also the number of ways of partitioning a set of n elements into
m nonempty subsets.
Roy Haas
Bell Labs, Indian Hill
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