DOITBLE GAMMA FUNCTION. 
341 
wa.s, however, published after my paper had ])een sent to press, aiifl I was therefore 
ignorant of his elegant results. Expressed in the notation which I have adopted, his 
method is as follows. 
It is evident from the expression of the function ^{s, a, w) as an asymptotic limit, 
lhat its imjjortance lies in the fact that it is a solution of the difference ecpiation 
co)-f{a)= - i. 
when .s has any value, real or complex. [From this result we see at once that we 
should expect that, when s is a positive integer, the simple ^ function should be sub¬ 
stantially a derivative of the gamma function, and when s is a negative integer, a 
Bernoullian function.] Now when'll(.s) > 1, the simplest .solution of our difference 
equation is evidently 
I {s, a, oj) 
1 
(c -t- ncoY ‘ 
This solution becomes nugatory when 1, but Melltx has succeeded in finding 
a modified solution by the following ingenious modificatiou of Mtttag-Lefflp]r’s 
process. 
We construct the function 
when 1 > i({(s) < — I', by writing — s in place of m in the wf' simple Bernoidlian 
function 
S„,(a 1 (o) 
(^T/l -(- 1) Cl) 
• 5 
and taking the sum of the first Z’ fi- 1 terms, k being of course a positive integer. 
Thus 
y 1 S 
y-S A- = l 
(a I w) — - - r -f- S 
(1 — -s) « J = 1 \ / 
And now ^(.5, «, co) is defined by the relation 
— s 
a 
^Bj+,. (w). 
t (■'?,«, (o) 
ao 
— — (<"< 1 / “h ^ 
— (o + (w + 1) ftj I o)} + (o + no) I (y) 
= oL(« + 
We readily see that this function formally satisfies our fundamental difference 
ef[uation, and we may at once prove that the series does define a function existing 
over the whole plane. 
For when 5 = 0, — 1, . . 
it is evident that {a \ oj) = [a | oj) -f- constant, 
and therefore (a [ w) — S_sj;{a) — cC^ — 0. 
