DOUBLE GAMMA FUNCTION. 283 
Finally for all positive integral values of ')h, can only differ from . 2 ^'^{a ] wj, 
by a constant,— i.e., let us say, 
I Wo) + 
Differentiate the expansion thus obtained with respect to «, and we tind 
(1 — — c““^) 
and hence we have finally (§ 6) 
= +...+(-)■ 
n ! 
+ . . . 
ze 
{1 — e "‘h (1 — e “ 2 ^) 
1 ! 
+ (-)»-> ="^'4"- 2 “ + 
111 
which is the expansion I'equired. 
This expansion may be used to define the double Bernoullian numbers, and tdl their 
properties may be deduced from it. A procedure analogous to tlie one here suggested 
will he the one employed in the general theory of multi})le Bernoullian functions. 
§ 16. Several expansions of constant occurrence may he deduced from the one just 
obtained. 
In the first place, note that we may write the expansion in the form 
ze~ 
(1 — (1 _ 
Wj + CO., a „S'] («) 
+ . --h — S + . . . 
C0|(0.,S -SCOjCO, co^to., 1 ! 
1 
JlCO.i- 
+ (-)"■'vr""+' • ■ 
Put now a = 0, and we have by definition of the double Bernoidlian numbers. 
(1 — cojcoo^ 
We thus have (§11) 
1 I + "3 I T> / \ I 
+ .... .. ' + 3-t^i (*^n + • • • 
4. / V<-1 
^ ' { n - 1)1 ’ 
(1 — “iq (1 
-—-p _ ‘ _ 
— cojco,* -'^1*^3 
*^l “8 COo , 2^1 (*^l! ^2) _ , 2^15 (*^1’ ^3)..,3 
+ 
2 + 
! 
2^2,^ + ! (^1, ^ 2 ) ^2n + 1 
{2n) ! 
+ - ( —) 
})l = 1 
And the last series may be written 
, ^ B„Acod"‘~^ + coA'^-B^-" 
2 . (2w) ! 
+ 
1 — e ““ 1 " 1 — e 
1 1 
CO, COo 
Hence we find 
2 o 2 
