1550251515-Classical_Complex_Analysis__Gonzalez_

646 Chapter^8

From Theorem 8.51 (3) and the four relations above, the remaining

10 contiguous functions relations can be obtained. Show that:

(e) [2a - c + (b -a)z]F = a(l - z)Fa+ - (c - a)Fa-

(f) (a+ b - c)F = a(l - z)Fa+ - (c - b)Fb-

(g) (c - a - b)F = (c - a)Fa-- b(l - z)Fb+

(h) (b - a)(l - z)F = (c - a)Fa-- (c - b)Fb-

(i) [1 - a+ (c - b __: l)z]F = (c - a)Fa-- (c - 1)(1 - z)Fc-

(j) [2b - c +(a - b)z]F = b(l - z)Fb+ - (c - b)H-

(k) [b +(a - c)z]F = b(l - z)Fb+ - c-^1 (c - a)(c - b)zFc+

(1) (b - c + l)F = bFb+ - (c - l)Fc-

(m) [1 - b + (c - a - l)z]F = (c - b)Fb--(c - 1)(1 - z)Fc-

(n) [c-1 +(a+b+ l-2c)z]F = (c-1)(1-z)Fc--c-^1 (c-a)(c-b)zFc+

5. Show that zF' = b(Fb+ - F), and deduce the Legendre relation

dE

2z-=E-]{

dz

where K(z) and E(z) are as defined in Problem 2.

6. Show that

z(l - z)F' = (c-b)Fb-+ (b-c + az)F

and deduce that

d]{

2z(l - z) a;- = E - (1 - z)]{

7. Show that the Laplace transforms of (a, c; t) is given by

(b) Prove that

e-zq,(a, 2a; 2z) = f F(a, -n, 2a; 2) (-~r

n=O n.

and conclude that F( a, -n, 2a; 2z) = 0 if n is a positive odd integer.