OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
463 
The phases of p + 1 , p — 1 in (p + l)*” 4 , (p — l )'" 1 being restricted to lie between 
7 rand — tt, on changing p into —p, we must put — p — 1 = e T7rt (p + 1 ), —p-f 1 = — 1 ), 
where the upper or lower sign is taken according as the imaginary part of p is 
positive or negative ; we have therefore from (5) 
p, m (-o = 
n (— m) \ p + 1 
1W T7 / ITT 1 + P 
F — ft, n + 1, 1 - ra, — 
on substituting for the series its value given by (i5), we find the relation 
p„- ( - D = e*”‘ P “ 0.) - — 1 >_+”>"• . e -»■ Q « W . . . (20). 
77* 
Again from ( 10 ), we have since (— p ) H+m+1 = p n+m+1 . e :p(,l+m+1 )‘ 7r 5 where the sign is 
chosen as before, 
Q.“ (-/*)= ~e* m Q.’W . (21). 
In the particular case of a real integral value of n, we have 
P/‘ (- /*) = (- 1)" P„* (/*) - ~ (- 1)" sin wi7T . e— . Q/ 2 (p) 
Qa(- p) = (- 1)" +1 Q,“ (^). 
Expression for P„"‘ (p) in powers of —, when mod p > 1. 
13. In the formula (10), the expression for Q/ 2 (p) in a series of powers of — has been 
obtained for the domain of p = 00 ; we shall now employ the relation (18) to express 
P / 2 (p) in a similar manner. We find by changing n into — n — 1 in ( 10 ), 
Q-«-i * M 
_ n (m— 11— l) n (—A ). „ ., _ 
— 2\e mm -— 7 “—— (p-—L)^"p" _ '"F 
= — Z“e‘ 
O Upturn 
n (— n --k) 
n(-i)n ( W -i) 
/m —n + 1 m—n , 1 
V 2 5 2 ’ 2 ’ P 2 
n (a—??i) sin (a—?ft) 
cos tut . 0 „ N1 T - i / fl 7 — % + l m —n . 
(p 3 -i)V' ;! F —^i - n, 
7T 
p" 
Hence we find 
p." M 
_ sin (n + m) 7r 
+ 2 " 
2 ,i+i cos ?i 7 r ’ n (ft+i) n (—j) 
n (n—\) 
U(n+m) oV-ifAF 
P \ 
'n + m + 2 ft + to +1 
, n ~’r f, 
p" 
n (ft-m) n (-i) 
, (p 3 —l)^p' 2 “ w F 
to —ft + l m—n 
2 ’ 2 
■n, 
A 
( 22 ). 
