280 
E. W. BARNES ON THE THEORY OF THE 
lienee as these two solutions both vanish when a = 0, we have 
f" oS„ (a) da = - a ,B,„, (»., a,,). 
J 0 /I “h 1 
As a corollary we liave on differentiation 
2 ^ « + l (*^0 ~ i-) 2^" ('"^) d“ 2^ J' + l ('^)- 
§ 13 . The limitIplication theory of double Bernoullian functions may be con¬ 
veniently expressed l>y tlie formula 
S / I \ I""! i I 1 \ 
Ania coi, fo»,d = a , —- . 
~ ~ ~ \ Hi m / 
From the fundamental difference relation we have 
^ m a 4- - 
m 
«i 
Cl, 
CJo ^ — .iS;, -yllia I 61^, cio] ? 
in" 
1 _ 
/oioXn 
\ 
1_ ^ 
in j 
—1 
rH 
+ 
0.^ 
Hence — oS„ ima I oi,, oj.d satisfies the difference equation 
—/{«) = 
n + 1 
and is the only algeliraical solution such tliat J\o) — 0. 
Ct)o 
llei 
ice 
J. 
IIV 
- oS„ {ma 101^, 01.) = .S, Hi --j, 
which is the relation required. 
As a corollary we see that the ?dh double Bernoullian number is homogeneous 
and of degree [n —1) in the w’s. 
For in § 9 we liave seen that in the expansion of 
i^a j Clj, Cl.i^j 
the part of the coefficient of which involves the w’s is .,B^.(ajj, &i^). 
§ 14. The transfoimation of the jiarameters of the double Bernoullian function is 
given by the relation 
aij &)., 
V’ *1 / 
A=o r=o p 
01 
n " 2 ) 
+ P(/ 2 th + i (o>i, 01 .) — ^B„ + J ( 
as we proceed to prove. 
Let d(n) 
oH, (o q- -p ‘ oi|, ojo), 
/.■=0i = 0 V <l\ 
tlien we obtain at once 
