DOUBLE GAMMA FUNCTION. 
•305 
It is an interesting piece of work to show that these results accord witli those 
previously obtained for the G function of parameter unity. 
§ 29 . We proceed now to write do’wn the expansion of log ro(.i) and the first few 
of the derived functions in the vicinity of z = 0 . 
We have, by definition, 
1 w.) = — 2 S S' -"g, where 12 = -f- 
-,)l = 0,ii2=oG + 
Since the series on the right-hand side is absolutely convergent, \\ e may ex})and 
in the form 
( 3 ) = - - - ds o, - s2' y+ 5 — -- 
1.2 n-' 
Hence, integrating, 
'I'f ( 2 ) = - - (<"1. «■-') - 2 s S' ” + 3 2 S' - 4 S_S' . 
0 
the constant being determined by making « 
0 . 
Thus integrating again, and determining the constant in the same manner, 
- 7ii ^ - 733 (<^1> " 3 ) - s S' ^ -h S S' - . . . 
and finally. 
r, (z) = - lug 3 - zy,, - - y - SS' y,, + S S' 
■s' V W _ 
V 5 n> 
the expansion holding good within a circle of radius just less than the least \ alue of 
'/Uj co^ d" 
= 0 , 1 , 2 , . . . 
m. = 0, 1, 2, . . . 
X 
X 
excluded. 
We note that by Eisensteix’s Theorem each coefficient in the series is an alisolutely 
convergent series. 
§ 3 U. We jiroceed now to the expressions for the double gamma functions as simply 
infinite products of sini2:)le gamma functions. 
Consider the product 
P (z) = r, (2 1 0,1) ri 
?/l2 “ 1 .. 
ly (s: -j- )>in0).2 I &)j) ^ 1 ("'owj I wj) 
The typical term may be written 
— 1 “1) + " - 'l'l(‘)(“2“2 I Wi) + . . . 
4! 
and the series S I are absolutely convergent when ?’> 3 . The product 
=1 
is therefore in general absolutely convergent. 
VOL. CXCVI.-A. 2 E 
