OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
483 
Now 
fjl+, J2 + , 1-.2 2 -) 
(v — l )' 1 m V 11 m 1 dv = &' 2,li+1)iu £ (qi — ffl-f 1 r — n — m) 
= —e (, - 2,n )". ( —!) r . 
_ „— 2 mnL (-n-m)...(r-n-m- 1 ) 
(—n —m) (1—n—m)... (r—n—m—1) , , x 
n(n—m) n (—%—m—i) 
. 47rsin(w — 'J?i)7rsin(«-|- W ) 77 ‘ 
n (— 2 m) 
(1 — 2 m) ... (r—2m) 
= 2 2 " 1 . 27 rsin 2 w 27 rsin(n—771)77 
II(ji— m) n(m-i)n(m-l) (—n—m)... (r—n—m—1) 
Tl(n + rn) II ( —J) (1 —2m)... (r—2m) 
Thus the expression becomes 
e 2 " 1,rt . 27rsin2m7rsin(n— 777)77 
n(«-i) n( »-lj (ft *_!)-,« p (T_ m> -*-m, l-2m, 1. 
n(-i) 
As in Art. 21, this expression can be shown to be equal to 
e m . 47t' cos mu . sin (n — m) n . —- -fi— -— — V, ” (a), 
v ' n (n + m) n ( - i) 2 m vr/ ’ 
hence we have the formula 
P--W = “ 
2"‘e 2 ” 
1 
n (n + m) n ( - i) (/U - Iff” 
47 r 2 cos mu sin ( n—m) u II (?i — m) II (m — i) 
r (l+,2 2 +, l-,2 2 -) 
» 72 + 777 +1 
771—+ / 1 
(1 — W) u “ w 1 
— 7»—Wl— 1 
c/« . (55), 
and in particular when m — 0 
^ . 1 1 ru+.*+.i-,*-) f u 
P„ (a) = - -- ■:-— u 2 (1 — w)" (1 - 7 
vr/ 47T 2 sin VlT Z n+i J V ’ \ Z 2 
— /I— 1 
du . (5G). 
A £/wVcZ cZass off definite integrals which represent the function Pff" (g), Q™ (g). 
27. If we put f = f , we find that the differential equation (2) becomes, when f 
is made the independent variable, 
/ z, /\ d 2 W , ( A 2m + 3 ,\ dW , (n - m) (n + m + 1) AA7 
^ ^ - + -4 W = °- 
4 
