Concrete mathematics : a foundation for computer science

190 BINOMIAL COEFFICIENTS
The Newton series for f(x) is therefore
f(x) = Adf(0) ; +Ad-‘f(0)
0
For example, suppose f(x) = x3. It’s easy to calculate
f(0) = 0, f(1) = 1, f(2) = 8, f(3) = 27;
Af(0) = 1, Af(1) = 7, Af(2) = 19;
A’f(0) = 6, A’f(1) = 12;
A3f(0) = 6.
+-.+.f,O,(;) +f(O)(;)
So the Newton series is x3 = 6(:) +6(l) + 1 (;) + O(i).
Our formula A” f(0) = c, can also be stated in the following way, using
(5.40) with x = 0:
g;)(-uk(Co(~)+cl(;)+c2(~)+...) = (-1)X,
integer n 3 0.
Here (c~,cI,c~,...) is an arbitrary sequence of coefficients; the infinite sum
co(~)+c,(:)+c2(:)+... is actually finite for all k 3 0, so convergence is not
an issue. In particular, we can prove the important identity
wk
L (-l)k(ao+alk+...+a,kn) = (-l)%!a,,
integer n > 0, (5.42)
because the polynomial a0 -t al k + . . . + a,kn can always be written as a
Newton series CO(~) + cl (F) -t . . . + c,(E) with c, = n! a,.
Many sums that appear to be hopeless at first glance can actually be
summed almost trivially by using the idea of nth differences. For example,
let’s consider the identity
c (3 (‘n”“) (-l)k = sn , integer n > 0. (5.43)
This looks very impressive, because it’s quite different from anything we’ve
seen so far. But it really is easy to understand, once we notice the telltale
factor (c)(-l)k in the summand, because the function