488 
ME. E. AY. BAEXES OX IXTEGEAL FUXCTIOXS. 
", II 1 +7)" 
<7-f 1 
= W <r1 exp 
r r (- o-) + “f~ {log j + 7 T - f £ 
+ 
w Y_ 1 — Tf)' 
J=l- 
<T+1 
sz* 
in which — and cr are hoth integers, and J + is the “ genre ” of the function. 
It is interesting to notice that the constants which enter into the asymptotic 
expansion of this very general function are all values of the Riemann £ function. 
§ 82. When r = 1, the function is to an exponential factor an important function 
which I have proposed to call the cr-ple G function. These G functions are derived 
from the multiple Gamma functions by the coalescence of the parameters. The theory 
of the simple G function has been developed elsewhere* in the second of a series of 
papers on Gamma functions. 
In that development I took 
Gr (z -b 1) = (2 tt) 2 e 
Z . 2d* 1 
n 
'<1=1 
{ 1 + -) e J+ 2 » 
v n 
and obtained! the asymptotic expansion 
log G (s + 1) — ra — log A + 2 log 27r T ~ rVj l°g 2 ~ 4 + ^ 2772^+^2 z 2 *’ 
where A is the Glaisher-Kinkelin constant. 
Putting cr = t = 1, the asymptotic expansion which we have obtained for the 
same function in the present paragraph is 
I log 2 . - *Y+i) _ y+ {( _ x) i ogz + 1 log ,+y _ 
Now J 
and 
+ £(- 1 ) + 
£(-i)= -A. £(o)=-i. 
£ (— .<f — l) = 0, when s is odd, 
G (- r 1 si-*- 1) 
.<=-i 
s~ 
1 B 
(21 + 2 )’ when * = 
* 1 Quarterly Journal of Mathematics/ vol. 31, pp. 264 et seq. 
t Ibid., § 15. + Ibid., § 23. 
