Concrete mathematics : a foundation for computer science

270 SPECIAL NUMBERS
Bernoulli numbers are defined by an implicit recurrence relation,
B’ = [m==O], for all m 3 0.
For example, (i)Bo + (:)B’ = 0. The first few values turn out to be
(All conjectures about a simple closed form for B, are wiped out by the
appearance of the strange fraction -691/2730.)
We can prove Bernoulli’s formula (6.78) by induction on m, using the
perturbation method (one of the ways we found Sz(n) = El, in Chapter 2):
n-.1
S ,,,+I (n) + nm+’ = 1 (k + l)m+’
k=O
= g z (m:l)k’ = g (m:l)Sj(n). ( 6 . 8 0 )
Let S,(n) be the right-hand side of (6.78); we wish to show that S,,,(n) =
S,(n), assuming that Sj (n) = Sj (n) for 0 < j < m. We begin as we did for
m = 2 in Chapter 2, subtracting S,,,+’ (n) from both sides of (6.80). Then we
expand each Sj (n) using (6.78), and regroup so that the coefficients of powers
of n on the right-hand side are brought together and simplified:
nm+’ = f (m+l)Sj(,i = g (mT1)5j(Tl) + (“z’) A
j=O
= ~(m~')~~~(jk')Bknj+l~'+~m+l)b
= o~~~~(m~l)(i~l)~n’i’~~k+(m+l)A
. .,
= o~~~,,(m~l)(~~~)~nk+l +(m+l)A
, ,,