OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 509 
this formula which holds for all real and complex values of n is a generalization of a 
well-known formula, namely the case in which n is a positive integer, in which case 
the series is a finite one, since P n m (p) = 0, when m and n are positive integers such 
that m > n. 
Case II. 
u > log mod. \J 
P + 1 
p — 1 
In this case the expansion of (p + \/p~ — 1 cos (<f> — \jj — lu}’ 1 in powers of w 
consists of powers whose indices differ from n by a real integer; thus 
(p + \/p' — 1 cos (<f> — xfj — iu)}’ 1 = %u m e m ® * lW)l 
where m has the values n, n — 1, n — 2, . . . 
To determine u m multiply both sides of the equation by ; then, since 
f (<{,-<!>±m) i ( i^ _ w p eil m> m ' are different real integers, we have 
J 0 
l r 2rr _ 
U m = — j {p + \/ ( p 2 — 1 COS (<f> — xjj — Lu)} n (2(f) 
= n ( + m) { P "-" M - ~ • e ~"‘ sin n * • Q “ (p)}> • • b y ( 93 )- 
We have thus obtained the expansion 
{p + \/p 2 — 1 cos (<£ — xfj — lu)} h 
= 2 nA+^j { P *“W - smnjr.Q„’*P)}e“<*-<-‘* > . . (99), 
when m has the values n, n — 1, p — 2, . . . and the expansion holds for all real or 
complex values of n ; in the special case in which n is a positive integer, we have 
{p + v^p 2 — 1 cos (<£ — xfj — Lu)} n = 
m = n ik» 
m=0 n (ft + TO) 
P,r (ft) • e 
an (4>—\jj—iu) 
( 100 ). 
When p is a negative integer, change it into — n — 1 ; we thus find, on using the 
formula (97) 
{p + \/p 2 — 1 cos (<£ — i// — m)} n 1 
(~1) B+1 S _1_ 
n (ti) ** n (to — — i) n (to + p) 
Q , ro (p)e 
—t»l (<f) — lp —IV,) 
( 101 ), 
where m has the values p + 1, p -j- 2, n + 3, . . . 
