DOUBLE GAMMA FUNCTION. 
275 
and integrate with respect to a between 0 and we obtain 
— oS/“^'(o) = — oJ^n-k{o)da + nS/^(o), 
( n — /d) i j f) 
so 
that 
<0^ .C («) <«) + (,. + 2 -/.). —‘A I -.)■ 
Write now n for [n — k), as is evidently allowable, since both n and k are positive 
integers, and we have 
)„ =s. («) 'la = - ( 0 ) + 
We thus see that 
n 
— (O] 07,, ojJ 
{11 + /i-’) ! 
is independent of k, since this is the only time in the relation just obtained which 
depends on k. 
Putting then k = 1, we have when k > 0 and n > k. 
c'ik'i I I \ T 7 )! , , I . 
I 07 ^) - "+1 1 *^ 1 ’ ^ 3 )’ 
which is one of the relations re<piired. 
And also 
t o / \ 7 O' / \ I ^ « + 2 (o I (O.t) 
(«) da = — -: ..b „,, [o) + ,-—-—— . 
another of the given relations. The second integral formula of course may Ije 
written down by symmetry. 
We notice that in the notation formerly introduced (“ Gamma Function,” § 15) 
we have 
h ,,^.0 (0 I COj) _ ll'«+2(iy2) 
{n + 1 ). (71 + 2 ) 71 + 2 ’ 
and therefore that each of tliese expressions 
= 0 when n is odd, 
(->'B|+. 07/+' 
(a + 1 ). ( 77 , + 2 ) 
Thus we see that when n is odd 
when n is even. 
3 S, (a) da = - aS'«+i ( 0 ) 
2 N 2 
