Concrete mathematics : a foundation for computer science

89 Prove that (5.19) has an infinite counterpart,
5 EXERCISES 241
t (mlr)Xk?Jm-k = x (ir) (-X)k(X+y)“pk, integer m,
k>m k>m
if 1x1 < Iy/ and Ix/ < Ix + y/. Differentiate this identity n times with
respect to y and express it in terms of hypergeometrics; what relation do
you get?
90 Problem 1 in Section 5.2 considers tkaO (3 /(l) when r and s are integers
with s 3 r 3 0. What is the value of this sum if r and s aren’t integers?
91 Prove Whipple’s identity,
F
ia, ;a+;, l-a-b-c
l+a-b, l+a-c
= (1 -z)“F
by showing that both sides satisfy the same differential equation.
92 Prove Clausen’s product identities
F
:+a, $+b
1 +a+b
What identities result
formulas are equated?
=F(
93 Show that the indefinite sum
f(i)+a)
$, $+a-b, i--a+b
l+a+b, l-a-b
when the coefficients of 2” on both sides of these
has a (fairly) simple form, given any function f and any constant a.
94 Show that if w = e2ni/3 we have
k+l&x3n (k,~m)2WL+2m = (n,;In) ’ integer n ’ ”