16 I. MULTIPLE GAMMA FUNCTION AND ITS GENERALIZATIONS
For 1 . I ; r, put
By definition, we obviously have
Cr(8,UJ,x) -Cr{s,(J,X+U)i) = Cr-l(s, &(Z), x)
if r ^ 2. Hence we obtain the difference equation
Mitt 1 ^(x + ^u;) rP(x,oj) T^jxMl))
(1.13) log ^ - log n i = - l o g rr. .
If r = 1, (1.13) holds with (
logx)2
on the right-hand side. By (1.6), we have an
expression by contour integrals:
(i.i4) i o
g
% ^ = ' ± H(-D^(i) /
e~*:{i°tz)r-
In the case i = 1, (1.14) can be written as
(1.15) log— , . = 7 = / _ / ,„
This is the contour integral representation obtained by Barnes. Though Barnes
studied r
r
(x,a;) and pr(w) separately in great detail, we will have little occasions
to treat them separately.
We define the Bernoulli polynomials by
eXt
=
y - Bm(x) j
el
- 1 ^
We have
e
L
1 ^—' m!
ra=0
B0(z) = 1, B1(x) = x - - , B2{x) = x2 - x + - ,
B3(a;) = x3 - -x2 + -x , B4(x) = x4 - 2x3 + x2 - —,
Suppose that x X)fc=i ^fc^fc 0 with x/~ G R. By (1.7), we see that £r(0,u;,x) is
equal to the constant term when we expand =pp—
fc=1
into the Laurent
llfc=i(l-exp(-a;fc2f))
series of z. This is equal to
^ Bh(l - XX) Bh(l - X2)
Blr(X-xr)..h-l.JL2-l
,.lr~l
^ hi l2\ " lr\
Since Bm(l x) =
(—l)mJ5m(x),
we obtain
THEOREM 1.1.
r
Cr(0,
(CJI,O;2,...
,cjr),y^cjfcxfc)
- ^ 1 ^ 2
fc=l
Zl_! /2_i z^-i^ZiC^i) Sz2(rr2) Bir{xr)
~
( i r
S ~i "^ ~r j j y /r!
Zi,... ,/
r
^0,/i H Hr=r
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