DOUBLE GAMMA FUNCTION. 
sr. 
When 2 > ( 5 ) > — Jc, I: being some positive integer, we form a modified solution 
by the introduction of the function («| w.i), formed as follows. A¥e take the 
mi\i double Bernoullian function, («j w,, w.,), write —s in place of m, and then 
take the sum of the first [h + 2) terms. 
We thus have 
{ci I Ct»o) — 
W, + 
(1 — s )(2 — s)o)y(o^o/ “ 2(1 — s)o)iW^ct’ 
and then ^0 (s, a \ oj^, is given by 
I 2^1 \ ~lk±i 
-1 /yf i r / rC-''- ’ 
y = 0 
00 CO . 
oS_jg-(a)— 2 S l)(yj^d-(niod-l)(ao} 
//i| = 0 Yi'l2 — 0 
— “k “k 1) ^1 “k 
— [a + + ( ~k f) aj., ^ “k ~k h 
— {a + m^oj^ + moWn)"''] , 
an expression which for shortness we shall sometimes write 
00 x> 
(a) — N - X 1 'b k) • 
nij = 0 //Jo = 0 
It is at once evident that the function so defined formally satisfies the fundamental 
difference equation, and we may readily prove that the series is in general con¬ 
vergent. 
For when s = 0 , — f, 2 , . . ., — /j, — (/j + 1 ), obviously 
z^-sjc {<^1' 1 ! <^ 0 ) = -z^s 1 <^1 > “k “k P- 
where X and fx are constants. 
And therefore 
x(5:|i',^') vanishes when = 0, — 1, ~ 2, . . . , — k, — {k + 1). 
When 2 is large this expression admits of expansion in the form 
, Mb) 
+ ■ 
Avhere Pq (.!>), (6'), . . . are integral polynomials in s of degree indicated by their 
suffixes, jji’ovided that the logarithms which intervene in defining the many-valued 
functions A\'ith s as index are such that, when e is small compared with 2 , 
log 2 + log (^1 -k = log (s + e). 
If the axes of oj^ and include the axis of — 1, this will not be the case for 
terms of the double series, for which the numbers and in the term 
2 = ct -J- m^o)^ + ruM.^ are large, unless the logarithms have their 2 )rinci})al value 
with respect to some line between the axes of — Wj and — XVe take this line 
to be — (cij| “k a;.,). 
