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 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.