27(3 
MR. E. AV. BARNES ON THE THEORY OF THE 
§ 7. We now introduce doul^le Bernoulli an numbers analogous to the simple 
Bernoullian numl^ers introduced in the theory of the gamma function. 
In that theory the simple rdli Bernoullian number was defined by the relation 
T) / \ I 
1^4") = —— > 
and now the ath double Bernoullian number is given by 
T> /■ \ 2 ^ I (Mo) 
oi>/i (CJi, (Oo) — . 
-Vi-/ 
We note that by the tlmorem of § G we may put 
3^«+i (^n *^ 2 ) ~ _|_ /.yi 3^4/.' (0 1 Wj, oj^), 
and therefore 
oS,/'''* ( 0 ) = ("n " 3 )- 
§ 8. At this point ^ve may conveniently note the reduction which takes place in 
tlie double Bernoullian functions and numbers when the parameters are ecpial to one 
anothei". 
If we put = oj, = (o we have as the single difference equation of the rdli double 
Bernoullian function the relation 
/(a -b oj) — /(a) — S„(rt4) + iT1«+i(<w)5 
and tlie function is now defined as the algebraical solution of tins equation with the 
condition o>S„(o | w, w) = 0. 
Put now 
y (n) ~ — bt;, (n| oj) -j- (a — oj) S„(u | oj) -|- (c 
bh+i(ol&)) 
and 
we Jiave 
^ (a -{- co) — f ('0 “ a) 8 „(o 1 <y) “b 
71 + 1 ’ 
I oj) 
?l + 1 
Hence, the otlier definition conditions being satisfied, we see that 
ob,, {(( w, oj) — - (n oj) — - ^ oj) +-—j—, 
■ ' ' ' 0 ) \ I / ^ 1 i \ I / CO 71 + 1 
that is, the double Bernoullian function when the parameters are equal reduces to 
simple Bernoullian functions and numbers. 
It will be seen later that it is for this reason that it was possible to obtain all the 
expansions in the theory of the G function in terms of simple Bernoullian functions. 
Note that the above relation may be writte]i ,; _ 
.S„(« I », a,) = S„ («1 <.) - i S,,,, (a I «,) + - ,B„, (<«). 
