288 
MR. E. W. BARNES ON THE THEORY OF THE 
is regarded as one term, and we may subtract two absolutely convergent series by a 
term-by-term process. Hence we have immediately 
(2 + wj) — ( 2 ) = 
m 2=0 + ’'' 2 *^ 2 )" 
and therefore (“ Theory of the Gamma Function,” § 2 ) 
[Z (z) = -. 
Similarly, (2 -f- ( 2 ) = — (2 | Wj). 
§ 21 . It may now be shown that the function 
*0 CO 
- (21 W2) = 2 721 (oji, Wo) + 723 ("n ^^2) + 4 + - ^ , o 
)ni=0 7)12=0 L* 
satisfies the two difference relations 
^■ 2 ^’ (2 + Wi) — ^ 2 ^'' (z) = — V’ (Z I Wo) + 
,>/■> (2 -f Wo) - lAo'^> (2) = - ^i<*> (2 I Wi) + 
for certain values of the numbers m and m', provided 
721 ("n '^ 3 ) = — Lt 
2mi7L 
Wo 
.m TTL 
W, 
71 n 1 1 
S S' - -logn 
^1 = 0 )n2 = 0^^ ^2 
1 
Ct)| &>.T 
H - \ log (Wi -f Wo) — log Wi — log Wg 1 I, 
the principal values of the logarithms being taken. 
We may write 1//0 *'^( 2 ) in the form 
- Lt 
n = a 
and now we obtain at once 
rf^i^Z + CO,) - 
1 ’* ” ( 1 1 z 
2721 ("n "3) + 723 ("1. "2) + - + - - 1 TT?) “ o + fr2 
77i, = 0l71j = 0 L'* “T il 12 
n 1 
V 
_ s' 
V V' "1 I 
— — 721 (^ 1 ) ^'’) “h / 1 . /-V • 
7i = » L7)ia=0 2 + 7)ij = 0Z + ()l "f- IJWj + Wo 01^=0 
Hence we may take 
+ Wj) — l/io'^^( 2 ) = ~ l//iO> (2 I Wo) d- 
2 - 
mirc 
CO, 
provided 
/ \ I 111/ I \ 7mri 
"i 721 ("n " 3 ) = 7i ( 2 1 W 2 ) — 2- 
CO, 
-\-Lt 
=2 00 i )/», 
n 
1 
n 
_ V 
1 
~ i" llloWj ,,^^-qZ + 7ll,CO, (?^ + 1) Wi 
- V' -0 
171 , = 0 7 ) 1 , 
= on*J 
