6.5
BERNOULLI
NUMBERS
= 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,
Here’s some more neat stuff that you’ll probably want to skim through the first time. -Friend/y TA
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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 -L+; = -
zeZ+l 2 eL-1
z eLi2 + ecL12 z coth z = =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 ez - e-2 sinhz = -; 2
eL + ecz coshz = ~ 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: z c o t h z = -&+; = xB2,s = UP,,,&, . ( 6 . 8 5 ) II>0
nk0
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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