Concrete mathematics : a foundation for computer science

258 SPECIAL NUMBERS
Table 258 Stirline convolution formulas.
rs f Ok(T) 0,-k(S) = (r + s)on(r + s)
k=O
(6.46)
S f k&(T) (T&(s) = no,(r+ S) (6.47)
k=O
rS&(l.+k)On&S-4-k) = (?'+S)D,(l-+S+n)
k=O
n
SE kCTk(T+k)G,~k(Si- n-k) = no,(r+S+n)
k=O
n
[ m I
(6.48)
(6a)
= (-l)""-l(mn!,),~"-,(m) (6.50)
I
= imT , )! k,(n) (6.51)
It turns out that these polynomials satisfy two very pretty identities:
zez ’
- = XpJ,(X)2Q
( ez - 1 ) TX>0
(iln&--- = x&o,(x+n)zn;
/
(6.52)
(6.53)
Therefore we can obtain general convolution formulas for Stirling numbers, as
we did for binomial coefficients in Table 202; the results appear in Table 258.
When a sum of Stirling numbers doesn’t fit the identities of Table 250 or 251,
Table 258 may be just the ticket. (An example appears later in this chapter,
following equation (6.100). Elxercise 7.19 discusses the general principles of
convolutions based on identit:ies like (6.52) and (6.53).)
6.3 HARMONIC NUMBERS
It’s time now to take a closer look at harmonic numbers, which we
first met back in Chapter 2:
H, = ,+;+;+...+; = f;, integer n 3 0.
k=l
(6.54)
These numbers appear so often in the analysis of algorithms that computer
scientists need a special notation for them. We use H,, the ‘H’ standing for