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

-----------------------------------------------------------------
 gopher://quux.org/ conversion by John Goerzen <jgoerzen@complete.org>
 of http://communication.ucsd.edu/A-News/


This Usenet Oldnews Archive
article may be copied and distributed freely, provided:

1. There is no money collected for the text(s) of the articles.

2. The following notice remains appended to each copy:

The Usenet Oldnews Archive: Compilation Copyright (C) 1981, 1996 
 Bruce Jones, Henry Spencer, David Wiseman.