284 
MR. E. W. BARNES ON THE THEORY OF THE 
z 
(1 — “ 1 -) (1 — f “ 
"(1 
z 
— C~ "1-) 
■r i*- 
as tlie expaiision from which the odd double Benioullian numbers may be derived. 
Finally if we integrate the fundamental expansion of § 15 with re.spect to a 
between 0 and a, we obtain 
1 — 
(1 — 6'“ "ip (1 — e” 
or, as we may write it, 
1 — e 
(c.) - oSf^) (o) ,, 
T* ~ 1-2^ 1 “ 2'^ 1 (<5) J H- ^ -T ■ ■ ■ 
. . . + 
n . 
U, ~ 2^0 + • • • + 
(_)«-yS„ («) 
n ! 
t' + 
2Tri 
and 
27rt 
&>! 
"2 
(1 — “ip (1 — “2p Wj 
as the expansion from which the double Benioullian functions themselves are at once 
obtained. 
All these expansions are valid within the circle whose radius is the smaller of the 
two quantities 
§ 17. Hitherto we have considered the double Benioullian function as defined by 
one of two difierence equations, each of which involves the sini])le Bernoullian 
function. 
We proceed now to prove tliat, to a linear function of o, nS„(a) is the only rational 
integral algebraic function of a satisfying the difierence equation 
/ [<i + + Wo) — / (n + Wj) — / (a fi- Wo) + / (a) = o". 
In the first place it is at once evident that oS,,(a) does satisfy this equation. 
Again the difierence of any two solutions is a solution of 
J [d fi- Wj + Wo) — / {a fi- wi) — / (« + Wo) + / {a) = 0. 
Putting f{a + wj) —/(«) = </>(«), 
we have 4> {a + Wo) — <p (a) = 0. 
Hence, if /’(«) is the difierence of two algebraic solutions of the original equation, 
(^((x) will be an algebraic sinqily periodic function, and therefore a constant. 
And thus we shall have 
-f Wj) — f{a) = constant, 
so that if/'(«) is to be an algeliraic polynomial, it must be of the form 
Xu —h jjL, 
where X and g are constant with respect to a. 
