DOUBLE GAMMA FUNCTION. 
271 
Part I. 
The Theory of Double Bernoullian Funcfio'us and Numbers. 
§ 2. In the “ Theory of the Gamma Function,” Part II., we have defined tlie simple 
Bernoullian function | &j) as that solution of the difference equation 
f{a + w) — f{a) = a\ 
where n is a positive integer, wliich is such that it is an algebraical polynomial and 
S„(o I m) = 0. And it was j)roved that such a solution does exist. 
In exactly the same manner it may he pi'oved that the difference equation 
f{a + Wi) -f{a) = S„(« l w.) +- 
Sh+i(c I &).,) 
+ 1 
has an algebraic solution, which is a rational integral polynomial of degree n + 2. 
The difierence l)etween any two solutions will he a simply periodic function of 
period w, and will therefore be a constant if the solutions are both algebraic 
polynomials. 
There thus exists a unique algebraical polynomial of degree n + 2, which is a 
solution of the difference eipiation. 
/(« + wj) —f{a) — 1 w,) + - 
n -\- 1 
v/ith the condition f[o) — 0. 
This solution we call the double Bernoullian function of a with parameters Wj and 
Wo, and we denote it l)y oS„(rt | oj|,wo). By symmetry with this notation the simple 
Bernoullian function would be denoted by | w). 
We shall often omit the parameters w^ and Wo, when there is no doubt as to their 
existence, and write the function simply oS,;(«). 
§ 3. We now proceed to show that the double Bernoullian function of a of order n 
is also the uni(|ue algebraical polynomial which is the solution of the difference 
ecpiation 
4.1(01 wd 
f (« + f'u) ~J ('"0 — 1 *^ 1 ) + 
‘It 4" 1 
with the condition f{(>) = U. 
For since 
„s„(« -I- «,) - .s„(«) = S.(«1<»,)+ 
7i. + 1 
(^•) 
we liave at once 
2S,;(a -p OJji + 6J2) — oS„(« -p Wo) — oS„(>‘ -p Wi) + 2S,;(«) 
