Concrete mathematics : a foundation for computer science

5 EXERCISES 235
47 The sum tk (rkk+s) (‘“;~~~“) is a polynomial in r and s. Show that it
doesn’t depend on s.
48 The identity xkGn (“Lk)2pk = 2n can be combined with tk30 (“lk)zk =
l/(1 - 2) n+’ to yield tk>n (“~“)2~” = 2”. What is the hypergeometric
form of the latter identity?
49 Use the hypergeometric method to evaluate
50 Prove Pfaff’s reflection law (5.101) by comparing the coefficients of 2” on
both sides of the equation.
51 The derivation of (5.104) shows that
lime+0 F(-m, -2m - 1 + e; -2m + e; 2) = l/ (-z2) .
In this exercise we will see that slightly different limiting processes lead
to distinctly different answers for the degenerate hypergeometric series
F( -m, -2m - 1; -2m; 2).
a Show that lime+~ F(-m + e, -2m - 1; -2m + 2e; 2) = 0, by using
Pfaff’s reflection law to prove the identity F(a, -2m - 1; 2a; 2) = 0
for all integers m 3 0.
b What is lime+~ F(-m + E, -2m - 1; -2m + e; 2)?
52 Prove that if N is a nonnegative integer,
br].
N
= a, . . .
l-bl-N,.. . , l-b,-N,-N
1-al-N,...,l-am--N
53 If we put b = -5 and z = 1 in Gauss’s identity (5.110), the left side
reduces to -1 while the right side is fl. Why doesn’t this prove that
-1 =+l?
54 Explain how the right-hand side of (5.112) was obtained.
55 If the hypergeometric terms t(k) = F(al , . . . , a,,,; bl, . . , , b,; z)k and
T(k) = F(A,,... ,AM;B~,...,BN;Z)~ satisfy t(k) = c(T(k+ 1) -T(k))
for all k 3 0, show that z = Z and m - n = M - N.
56 Find a general formula for t (i3) 6k using Gosper’s method. Show that
(-l)k-’ [y] [y] is also a solution.