Concrete mathematics : a foundation for computer science

276 SPECIAL NUMBERS
Gnethingwecandoisreplace {y} by {~~~}-(j+l){j~,}. The (j+l)
nicely cancels with the awkward denominator, and the left-hand side becomes
x.{~~"}[i;']~~ - &{j;l}[i;'](-l)j+l-k
,
The second sum is zero, when k < m, by (6.31). That leaves us with the first
sum, which cries out for a change in notation; let’s rename all variables so
that the index of summation is k, and so that the other parameters are m
and n. Then identity (6.99) is equivalent to
F {E} [L] “y” == ~(~)B,-,, + [m=n- 11. (6.100)
Good, we have something that looks more pleasant-although Table 251 still
doesn’t suggest any obvious next step.
The convolution formulas in Table 258 now come to the rescue. We can
use (6.51) and (6.50) to rewrite the summand in terms of Stirling polynomials:
k!
(,-,)!hm(k);
%k(-k) ok-m(k).
Things are looking good; the convolution in (6.48) yields
g o,--k(-k) uk-,,,(k) := nc o,-,-k(-n + [n-m-k)) ok(m + k)
k=O k=O
:= (~l,l-“ni unprn (m - n + (n-m)) .
Formula (6.100) is now verified, and we find that Bernoulli numbers are related
to ttre
constant terms in the Stirling polynomials:
6.6
(-l)m~~‘mom(0) = 2 + [m=l]. (6.101)
FIBONACCI NUMBERS
Now we come to a special sequence of numbers that is perhaps the
most pleasant of all, the Fibonacci sequence (F,):
Fli 0 0 1 1 2 1 3 2 4 3 5 5 6 8 13 7 21 8 34 9 55 10 89 11 144 12 233 13 377 14