Concrete mathematics : a foundation for computer science

Here’s some more
neat stuff that
you’ll probably
want to skim
through the first
time.
-Friend/y TA
I Start
Skimming
6.5 BERNOULLI NUMBERS 271
= o~,~(m~l)k~,(~~~k)Bj--r+(m+l)A
= o~m~(m~l)o~~~i(m~~~k)~~+~~+~~A
. .
[m-k=Ol+(m+l)A
= nm” + (m+ l)A, where A = S,,,(n) -g,(n).
(This derivation is a good review of the standard manipulations we learned
in Chapter 5.) Thus A = 0 and S,,,(n) = S,(n), QED.
In Chapter 7 we’ll use generating functions to obtain a much simpler
proof of (6.78). The key idea will be to show that the Bernoulli numbers are
the coefficients of the power series
(6.81)
Let’s simply assume for now that equation (6.81) holds, so that we can derive
some of its amazing consequences. If we add ;Z to both sides, thereby
cancelling the term Blz/l! = -;z from the right, we get
zeZ+l z eLi2 + ecL12 z coth z
-L+; = - - = - =-
2 eL-1 2 p/2 - e-z/2 2 2’
(6.82)
Here coth is the “hyperbolic cotangent” function, otherwise known in calculus
books as cash z/sinh z; we have
sinhz = ez - e-2 eL + ecz
-; coshz = ~
2
2
Changing z to --z gives (7) coth( y) = f coth 5; hence every odd-numbered
coefficient of 5 coth i must be zero, and we have
B3 = Bs = B, = B9 = B,, = B,3 = ... = 0. (6.84)
Furthermore (6.82) leads to a closed form for the coefficients of coth:
zcothz = -&+; = xB2,s = UP,,,&, . (6.85)
II>0
But there isn’t much of a market for hyperbolic functions; people are more
interested in the “real” functions of trigonometry. We can express ordinary
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