DOUBLE GAMMA FUNCTION. 
357 
Tran.'^formation Theory. 
§ 67. We shall now consider the theory of the transformation of the parameters 
of the double gamma function. It must not be supposed that we intend to consider 
the general linear transformation. 
There exists no such theory for the present functions—at any I'ate, no theory 
having the simplicity and elegance which is characteristic of the elliptic functions, 
and the reason is obvious—the change of into Wj + wo makes no difference of form 
in such a series as 
30 » 1 
- S 
,ill= — 00 711-2= —00 "f -j- 
but it makes a change of comparatively great complexity in such a series as 
■30 00 1 
V V___ 
„ii = 0 ,/i,j=0 (- + 
The former series is the basis of those occurring in the theory of elliptic functions, 
the latter of those occurring in double gamma functions. We shall then limit our 
consideration to transformations which result from tlie change of and into 
ojj/p and ojy'q respectively, p and q being positive integers. 
By definition we have 
a»^) = 
CO 00 1 
2 S S — ■ 
„i, = 0 W2 = 0 + tt)” 
where H = + rn.^co.o. 
Hence i//o 
Wj (Oo 
p’ q 
00 00 
= - 2 N ^ 
i/ii=o 111 . 1=0 
P 
+ 
j)-i a-i 
'2 S S 
,-=0 s =0 
V V__ 
“' 0 *" / , 'ro)j so)o , 
"'-y p ^ ^ F 
8=0 8=0 \ p (q 
it being understood that the parameters when not expiessed are always and w.,. 
On integrating successively three times witli respect to «, we shall hnd 
log To 
Ct)| 0)2 \ 
p' uj 
yo (.t;) + log 11 
/■ = 0 
where yo ( 2 ) is aii algelwaical polynomial in : of order 2. 
As has been stated, it is possible to obtain y., ( 2 ) by jjurely algebraical jjrocesses, 
use being made of the limit theorems previously established. We may, however, 
obtain its value as follows ;—^ 
