OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
527 
we also obtain the expression 
S 3 = - 2- (- i)»sinW. A{ n ^ o Vi)n + (-i ) sin (Po + I + “)"■ 
e -(p„+m+l)<r + 1 ^ 0 + |+ + 2, e _2<r ) j, 
since cos (p + |) tt = — (— 1 ) Po (p — p 0 ), in the limit. 
We have now, on the whole, picking out the terms in S 2 , S 3 , obtained by differen¬ 
tiating the exponential function 
(1) The finite series S l5 
(2) 2 m+1 . sin (m + |-) tt . 
n (2 j p + w + I) 
ii ( Po + i)n(-i) 
o-.sinh'" i o- . e~ (Po+m+i)a ' 
F (m + p a + m + p Q -f- 2, e - ^), 
(p 0 +m+l)<r 
(3) - 2" 1 008 (m + n ' sinW ■ c ' 
F (m + A m + p 0 4-1, + 2, e 2<r ), 
II( 
n (Po m 2) 
[ir(0) n'fm — J) n' ( p 0 + to + 
» 0 + TO + I) 
-r -r a; . / | T \ “ V"/ ^ V"- 2 / “ W 
-i)n(p 0 + i) ( +2 ) |n(0) n(m-|) 
+ n (p° ft + T) } smh 47 • e _ (Po+m+t)<7 F (m + A, m+_p 0 + |, p 0 + 2, c 2 "), 
in the case of the ordinary ring functions (m integral), the first term vanishes on 
account of the factor cos (m + i) 7T - 
(4) 
2 m sin (to + J) 7 r 
” nT- IT 
sinh" 
<7 
0-(/’o+«l + tTO 
5 up (p 0 + to + 5 + ^-) n (to + ^ — 2) (i , i I 4- — 
8 =i n (p 0 + s + i) n (m — |) n (s) j 1 2 ‘' s 
_... 1 + -L- + ... + —I_ 
TO -{• 2 TO 5 - TO + S — + 2 —(— s —f- 1 
_ - __ _ __ - _ \e~ 2s,r . 
p 0 + m + f ‘ ' _p 0 + TO + S + 
Confining ourselves to the case in which m is integral, we can simplify the 
expression in (2); we have 
IF (Pq + 1 ) _ _J_ J_ 4 - -1 _i_ n/ (°) 
n (i^o + 1 ) + i i 5 i n (0) 
if(to - _*> _j_ i , 1 , n-(-i) 
n (to - J-) TO - 4 F TO - I "F • • • F i "F n (- 1) 
IF (to +p 0 + I) _1_ , 1 4_n'h|) 
n (to + p 0 + i) _ TO + p 0 +1 “*■ • • • 1 ' n(- j) ’ 
