OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 475 
It follows that, since V l5 V 3 must both be linear functions of P/ ! (p.), Qp (p), 
Q n m (fi)=2 m e mn n(w +? } ° ( v 3 ~ (pi 2 — l ) 4 *2 (w+m+1) F (l+m, n+m+1, n+f, M (35). 
11 ( 51 + 2 ') \ %“ ) 
p . w=2 .^l- n( ° + ( ;ff (*+*»■ «+™+i. »+*.5 
-f 2“ 
II (n — on) II (— -i-) 
—— (f — l) hl z /l m F -f- m, m — n, \ — n, - j . . (36) 
These formulae, (35), (36), are the expressions for Q/' 1 (p), P(p) in series of 
powers of — ; the series are convergent over the whole plane. 
In the particular case m = 0, we have 
Q« (/*) = 
n (5?.) n (- +) 
n (n + i) 
% - (»+1) F 
1 
2 ’ 
H~ b n + 
3 
2) 
(37). 
P ;i (p) = tan mr 
+ 
n (?i) 
n (5i + i)n(-i) 
2 (w+1) Py|, n, + 1,71 + f, 
n (»i) n (- f> 
JL 
2’ 
n, | 
(38). 
The particular cases of (37), (38), in which n is a real integer, are given by Heine.* 
It will be observed, that the case of a real integral value of n is the only one in which 
P rt (p) is represented by a single hypergeometric series. Exceptional cases of the 
four formulae will be considered below. 
A Second Class of Definite Integral Expressions for Vf (p), Q, m (p). 
22. By using the definite integral forms which satisfy the hypergeometric equation, 
we see that the expressions 
satisfy the differential equation (]), the integrals being taken along closed paths, 
such that after a complete description of such path the integrand attains its initial 
value. 
* See ‘ Kiigelfunctionen,’ vol. 1, p. 129. 
3 p 2 
