DOUBLE GA-MI^IA FUNCTION. 
‘2()9 
of Stirling’s tlieorem. By the aid of this theory it is possible to express the 
logarithm of tlie double gamma fimctioii and the double gamma modular constants as 
contour integrals similar to those given in Part III. of the ‘ Theory of the Gamma 
Function.’ 
In Part IV. I consider the mnltiplication and transformation theories of double 
gamma functions as ^yell as certain curious integral formulie, which correspond to 
IIaabe’s theorem for the simple gamma function, and are elementary cases of a 
general theorem connecting successive similar transcendents of higher orders. 
In Part V. the asymptotic expansion of the double gamma function is obtained, 
and it is shown that the function cannot arise as the solution of a dilferentlal equation 
Avhose coefficients are more simple transcendents. 
There exist similar functions of any number of parameters, and these transcendents 
I propose to call multiple gamma functions. I reserve the formal expressl(m of their 
properties for puldication elsewhere. I have worked out the theoiy for doidde 
gamma functions independently inasmnch as, the complex variable being Gvo 
dimensional, there are many points in whicli a liigher analogy breaks down ; and also, 
since many proofs in the higher theory are, in their simplest form, inductive and, 
to be rigorous, require a knowledge of tlie theorem for the two slm])lest cases. Not 
only so, hut in the case of the double gamma functions it is possible to give easily an 
algebraical theory (such as that worked out in Part II.), which is more simple than 
if one derived all the f()rmulfe from the fundamental consideration of certain contonr 
integrals. 
I ajipend a statement of the notation ado})ted in this ])a])er, mentioning the j)lace 
in the j)resent series of investigations where such notation is used for the tirst time. 
I )orivation. 
Name. 
Symbol. 
First occurrence. 
-Algeln'aic solution of 
f{a + to) - /(tt) = 
Simple Bernoullian func¬ 
tion 
S,((tf 1 to) 
“ Gamma Function,” 11. 
1 oj) 
11 
Simple Bernoullian num¬ 
ber 
iB,j(oj) 
“Gamma Function,” § 15. 
! 
Algebraic solution of 
t\a + o^i) - /(tt) 
= S„G 1 ^n)+ + 
n-\-l 
1 )oul)le Bernoullian 
function 
1 toj, (Oo) 
For the case of etpuil para¬ 
meters : 
“ G Function,” § 15. 
In general: 
“ Double Gamma Function,” 
§2. 
2 ^'nip 1 (Oi, Wo) 
11 
Double Bernoullian num¬ 
ber 
oBn-nCwi) W 2 ) 
“DoubleGamma Function,” 
§7. 
ww iyoj/ 
Simple gamma function 
FiCe 1 w) 
“ Gamma Function,” § 2. 
