Concrete mathematics : a foundation for computer science

338 GENERATING FUNCTIONS
Table 337 is the database we need. The identities in this table are not
difficult to prove, so we needn’t dwell on them; this table is primarily for
reference when we meet a new problem. But there’s a nice proof of the first
formula, (7.43), that deserves mention: We start with the identity
1
= t (xy)zn
(1 -2)x+’ n
and differentiate it with respect to x. On the left, (1 - z)-~-’ is equal to
elx+l~ln~llll-rll so d/dx contributes a factor of ln(l/( 1 - 2)). On the right,
the numerator df (“‘-,“) is (x +n) . . . (x + 1 ), and d/dx splits this into n terms
whose sum is equivalent to :multiplying (“‘,“) by
-+...+-
1 1
= H x+Tl - H, .
x+n x+1
Replacing x by m gives (7.43). Notice that H,+n - H, is meaningful even
when x is not an integer.
By the way, this method of differentiating a complicated product - leaving
it as a product-is usually better than expressing the derivative as a sum.
For example the right side of
$(i x+n)“...(x+l)‘)
= (x+n)n...(x+l)’
( *+...+A >
would be a lot messier written out as a sum.
The general identities in Table 337 include many important special cases.
For example, (7.43) simplifies to the generating function for H, when m = 0:
&ln& = tH,z”. (7.57)
n
This equation can also be derived in other ways; for example, we can take the
power series for ln(l/( 1 - z)) and divide it by 1 - z to get cumulative sums.
Identities (7.51) and (7.52) involve the respective ratios {,~,}/(“~‘)
and [,“‘J /(“c’), which have the undefined form O/O when n 3 m. However,
there is a way to give them a proper meaning using the Stirling polynomials
of (6.45), because we have
{mmn}/(m~l) .= (-l)n+‘n!mo,(n-m);
[m~n]/(m~l) = n!mo,(m). (7.59)