5 EXERCISES 235 47 The sum tk (rkk+s)
(‘“;~~~“) doesn’t depend on s.
is a polynomial in r and s. Show that it
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 lim e+~ F(-m + E, -2m - 1; -2m + e; 2)?
52 Prove that if N is a nonnegative integer, br]. = a, N. . .
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?
5 4 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.
