DOUBLE GAMMA FUNCTION. 
273 
{n-\-2) {n-\-l) n „ {n-k-l)n 
- + 
?i\Bi 
3 ! 
O) 
2 ! 
and so on 
U *^"+1 Cl/; \ 1 / 9 
On solving these e(tnations successively we readily obtain 
1 
a 
(vi + 1) (/I + 2) 
+ (y.i 
'^«+l — 
and, since 
Thus 
j C/J^, - 
— 1 '2 
•,n+2 
2(/l +1) 0)^C02 ’ 
3 I 3 I o 
^ Ct )| + 0)^2 “T 00)1^0)2 
COiWo 
+ &).,) , &)i“ -t O).," + 
- - -i-_J_ (-/Jt -1-^- 1 -: 
(/I F1) (?i F 2) 2(yi F 12c()|(i)o 
-^+ . • . 
Further terms can he calculated if necessary. It will be seen, however, that they 
form what we propose to call double Bernoullian numbers, whose })ro})erties may be 
investigated without tlie necessity of their formal evaluation. 
Corollary. AVe note that 
And hence 
'l 0/ I \ 0 I o , »> 
Ci / \ \ Cl’ Cl~\U). F Wo) I COi F Wo F'^w,Wo 
2S,(«|wi,a;o) = T - -- / - --+ ft i -- 
Gw^wo ^CO^Wo 
12a)jWo 
0* 
S\(«| W^, Wo) = 
a{oyY i- coo) ^ 
12w^a)o 
1 W^, C-Jo) = 
a 
Wi F Wo 
w, w 
:aiiC0 
oS/3)(c/, 1 OJI, W 2 ) = 
a)i&)o 
It will be found that these expressions are of constant occurrence 
the present investigation. 
Note also that 
ok 
So( 
a 1 w^. Wo) = 
WjWo 
a (w^ F Wo) 
2w^W2 
in the course of 
5. We will now prove that, if u —/j>0, h > 0, 
oS,/^'b''a. 1 W], w.,) = 
n ! 
We have, when n — Z; > 0 aud h < 0, 
I w^, Wo) + I c^o ^ 2 )- 
08//^) (« + W,) - 28/) («) = 8/> (u| Wo) 
ni f 
= 8//'''^ (0 1 Wo) + y_ '^n-lc{u 1 Wo) 
(“ Theory of the Gamma Function,'’ § 14). 
2 N 
VOL. CXCVI.-A. 
