Concrete mathematics : a foundation for computer science

A ANSWERS TO EXERCISES 525
need r > s+ 1 if the infinite series is going to converge. (If r and s are complex,
the condition is %r > ’31s + 1, because lkZl = km’.) The sum is
F
-r, 1
, _ r(r-s-l)T(-s) s+l
( -S I)- T(r-s)T(-s-l) = s+l-r
by (5.92); this is the same formula we found when r and s were integers.
5.91 (It’s best to use a program like MACSYMA for this.) Incidentally,
when c = (a+ 1)/2, this reduces to an identity that’s equivalent to (5.110), in
view of the Pfaff’s reflection law. For if w = -z/( 1 -2) we have 4w( 1 ~ w) =
-42/(1 - z)‘, and
F
ia, + a+;-b
1 +a-b
4w(l-w)
= (l-z)uF(,;;~bir).
5.92 The identities can be proved, as Clausen proved them more than 150
years ago, by showing that both sides satisfy the same differential equation.
One way to write the resulting equations between coefficients of z” is in terms
of binomial coefficients:
F(
(I;) (3 L’k) Ck)
r+skl/2)(r+s11/2)
n k
Another way is in terms of hypergeometrics:
F
F
a,b, i-a-b-~--n _ (Za)“(a+b)“(2b)“;
i+a+b,l-a-n,l-b-n - (2a+2b)“a”b”
$+a, i tb,a+b-n,-n
l+a+b,i+a-n,i+b-n 1)
1
= (1/2)“(1/2+a-b)“(l/2-a+b)”
(1 +a+b)n(l/4-a)K(1/4-b)” ’
5.93 0~~ ’ n:_, (f(j) + cx)/f(j). (The special case when f is a polynomial of
degree 2 is equivalent to identity (5.133).)