Concrete mathematics : a foundation for computer science

204 BINOMIAL COEFFICIENTS
This holds because the coefficient of zk in (-z)“+“B2(-~)“~‘/~~ is
= (-,)n+l[Zk n-11
= (-1 )n+l(-, )km n 1 [Zkmnpl] B2(Z)n+’
= (-1y 2(k-n-l)+n+l
k--n- 1
dixz
= (-l)kr;I;I-;) = (-,)k('"-;-')
n - k
=( k )
= ,z”, %-I (Z)n+’
JiTz
when k > n. The terms nicely cancel each other out. We can now use (5.68)
and (5.69) to obtain the closed form
integer n > 0. (5.74)
(The special case z = -1 came up in Problem 3 of Section 5.2. Since the
numbers $(l f G) are sixth roots of unity, the sums tks,, (“ik)(-l)k
have the periodic behavior we observed in that problem.) Similarly we can
combine (5.70) with (5.71) to cancel the large coefficients and get
(l+yG)‘+(l-ywz)y
integer n > 0. (5.75)
5.5 HYPERGEOMETRIC FUNCTIONS
The methods we’ve been applying to binomial coefficients are very
effective, when they work, but we must admit that they often appear to be
ad hoc-more like tricks than techniques. When we’re working on a problem,
we often have many directions to pursue, and we might find ourselves going They’re even more
around in circles. Binomial coefficients are like chameleons, changing their versatile than
chameleons; we
appearance easily. Therefore it’s natural to ask if there isn’t some unifying
can dissect them
principle that will systematically handle a great variety of binomial coefficient and put them
summations all at once. Fortunately, the answer is yes. The unifying principle back together in
is based on the theory of certain infinite sums called hypergeometric series.
different ways.