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