DOUBLE GAMMA FUNCTION. 
279 
and hence oS'„ + oj^) = oS'„(&ji) + oS',; (w^) — oS'„ (o) 
= (o) “h S^( (o I Wo) + S „ (o I Wj). 
Thus, since (“ Gamma Function,” § 15) 
S'„ (o I oj) = 0 when n is odd 
n 
= ( —) “~^ oj”~^ when 'll is even, 
we see that 2 S',; (w^ + w^) = oS'„ ( 0 ), when n is odd ; 
and oS'„(wi + Wo) = oS',, ( 0 ) + (-)-”' B,^(wh'"^ + when n is even. 
2 
But our former relation gives us, when n is even, 
(W| + Wo) = - oS',; ( 0 ). 
Hence, when n is even, 
,S'„(..) = (-)^ .1/ ■ (<“ 1 -'+ «/-'). 
which is e(|uivalent to the relation re(|uired. 
§ 12. We now proceed to show that 
“,S„(a) ,1a = (a,,, 
0 n -t I 
We have 
oS;, (rt "h Wj) — .oS„ (a) — S„ {(I j ojo) -h 
S'„^l(ol Wo) 
/G+1 • 
Hence, integrating with res2)ect to a 
f a + u)i ra ru, ra Q' ) 
oS, («) da - oS„ («) da = oS„ (o) da + S. {<i \ wo) da + « ' 
0 " ■'o'' h) ~ Jn a + 1 
But (“Gamma Function,” § 19) 
[ S„ (a I Wo ) da + 
Jn 
a 
n + 1 
'l l 4- 1 
so that, if /’(«) = (^0 Biis function is an algebraic solution of the difference 
J 0 
ecpiation 
f(a + 0 ,,) -/(«) = f ,S„ («)</« + 
.'0 71 + i 
But this difference efjuation is evidently satisfied hy 
a 
[ oS„(«) da 
Jn 
h K+2 (c I Wo) 
+ 71+1 
WiLJo'^ ^ ' ( 7 ^ + 1 ) (?t + 2) J n-\- 
Ci‘ / \ ”1 I + l 1 £• 
