Concrete mathematics : a foundation for computer science

458 ASYMPTOTICS
(The Bernoulli polynomial E&(x) is defined by the equation
B,(x) = (y)Boxm+ (~)B,x~-’ +...+ (~)B,x~ (9.72)
in general, hence Br (x) = x - i in particular.) In other words, we want to
prove that
f(O) + f(l)
2 = /;f(x)dx+l:jx-;)f’lx)dx.
But this is just a special case of the formula
1
1
u(xMx) 11, = u(x) dv(x) + 4x1 du(x) (9.73)
J0
J0
for integration by parts, with u(x) = f(x) and v(x) = x - i. Hence the case
n = 1 is easy.
To pass from m - 1 to m and complete the induction when m > 1, we
need to show that R,-l = (B,/m!)f(mP1’(~)l~ + R,, namely that
This reduces to the equation
(-l)mBmf(mpli (x)1’
0
= m J’B,- (x)Grnp’)(x) dx + JIB,,,(xlGml(x) dx.
0 0
Once again (9.73) applies to these two integrals, with u(x) = f(“- ‘l(x) and Will the authors
v(x) = B,(x), because the d.erivative of the Bernoulli polynomial (9.72) is never get serious?
= mB,-l(x). (9.74)
(The absorption identity (5.7) was useful here.) Therefore the required formula
will hold if and only if
(-l)“‘B,,,f(“~‘) (x)1; = B,(x)f’mpl)(x)l;.