Concrete mathematics : a foundation for computer science

214 BINOMIAL COEFFICIENTS
Now sin( x + y ) = sin x cos y + cos x sin y ; so this ratio of sines is
cos 2n7t sin 2~
cos n7t sin c7r
= (-qn(2 + O(e)) ,
by the methods of Chapter 9. Therefore, by (5.86), we have
(-n-4! = 2(-l),r(2n) = ,(-,),P-l)! n Vn)!
!‘_mo (-2n - 2e)!
r(n) (n-l)! = (-‘) 7’
as desired.
Let’s complete our survey by restating the other identities we’ve seen so
far in this chapter, clothing them in hypergeometric garb. The triple-binomial
sum in (5.29) can be written
F
1 --a-2n, 1 -b-211, -2n ,
a, b 1)
(2n)! (a+b+2n-2)”
= (-l)nn!- ak’,‘i ’ integer n 3 0.
When this one is generalized to complex numbers, it is called Dixon’s formula:
F
a, b, c = (c/2)! (c-a)*(c-b)*
1 fc-a, 1 fc-b , c! (c-a-b)* ’
fla+Rb < 1 +Rc/2.
b6)
One of the most general formulas we’ve encountered is the triple-binomial
sum (5.28), which yields Saalschiitz’s identity:
F
a, b, --n = (c-a)K(c-b)”
c, afb-c-n+1 c”(c-a-b)K
(a - c)n (b - c)E
= (-c)s(a+b-c)n’
integer n 3 0.
This formula gives the value at z = 1 of the general hypergeometric series
with three upper parameters and two lower parameters, provided that one
of the upper parameters is a nonpositive integer and that bl + bz = al +
a2 + a3 + 1. (If the sum of the lower parameters exceeds the sum of the
upper parameters by 2 instead of by 1, the formula of exercise 25 can be used
to express F(al , a2, as; bl , b2; 1) in terms of two hypergeometrics that satisfy
Saalschiitz’s identity.)
Our hard-won identity in Problem 8 of Section 5.2 reduces to
1 x+1, n+l, -n
---F 1 = (-‘)nX”X-n=l.
1+x ( 1, x+2 1)