ABSTRACT

The solution F(a,b;c;z) corresponds to the exponent 0, provided that c # 0,-1,-2, ... . The method of $4 shows that the solution corresponding to the other exponent at z = 0 is zl-'F(1 +a-c, I + b-c; 2-c;z), provided that, now, c # 2,3,4, ... . Again, it is sometimes more convenient to adopt as second solution

since this exists for all c. When c is not an integer or zero, the limiting forms of F, G, and their derivatives

Accordingly, the Wronskian of F(a, b; c; z) and G(a, b; c; z) is given by sin (nc) W(F, G) = -Z - ~ ( l -z)~-~-b-l . *

compare (1.10). Analytic continuation immediately extends this identity to all values of c. From this result and Theorem 1.2 it is seen that F and G are linearly independent, except when c is an integer or zero. In these exceptional cases an independent series solution, involving a logarithm, can be constructed by Frobenius' method ($5.3); see Exercise 10.3 below.