Concrete mathematics : a foundation for computer science

So l/x isa
polynomial?
(Sorry about that.)
6.2 EULERIAN NUMBERS 257
Second-order Eulerian numbers are important chiefly because of their
connection with Stirling numbers [119]: We have, by induction on n,
{x"n} = &(($~+n~lyk). integern30; (6.43)
[x”n] = g(~))(“~k) 7
For example,
{Zl} = (1))
[x:1] = (1);
integer n 3 0. (6.44)
{z2} = (7’) +2(g) [x:2] = (:)+2(y);
(,r,} = (“:‘) +8(y) +6(d)
[xx3] = (1) +8(x;‘) +6(x12).
(We already encountered the case n = 1 in (6.7).) These identities hold
whenever x is an integer and n is a nonnegative integer. Since the right-hand
sides are polynomials in x, we can use (6.43) and (6.44) to define Stirling
numbers { .“,} and [,Tn] for arbitrary real (or complex) values of x.
If n > 0, these polynomials { .“,} and [,“J are zero when x = 0, x = 1,
. . . , and x = n; therefore they are divisible by (x-O), (x-l), . . . , and (x-n).
It’s interesting to look at what’s left after these known factors are divided out.
We define the Stirling polynomials o,(x) by the rule
&l(x) = .", /(X(X-l)...(X-TX)). (6.45)
[ 1
(The degree of o,(x) is n - 1.) The first few cases are
q)(x) = l/x;
CT,(x) = l/2;
02(x) = (3x-1)/24;
q(x) = (x2 -x)/48;
Q(X) = (15x3 -30x2+5x+2)/5760.
They can be computed via the second-order Eulerian numbers; for example,
CQ(X) = ((~-4)(x-5)+8(x+1)(x-4) +6(x+2)(x+1))/&