DOUBLE GAMMA FUNCTION. 
333 
2 S^®(a + <^1 + 4 " 4 " 9.^2) ~ 4 " 
— oSi^^' (a -{- ojo) 70J0 log qo).^. 
[The analogy with the matrix notation would be more complete if q^ and q wei’e 
replaced by and The convention adopted here is, however, (juite natural.] 
And now our asymptotic equality may be vaitten 
jm qn 
log IT n (a + niyO}^ 4“ '^^■ 2 ^:) 
J7ii = 0 m.2 = 0 
C2 (^5 1 ^ 1 ? ^-2} 
5 = 0 
+ pq [n~ log n — ?i2 (i _j_ A) ] -P (p _p q) [_n log n — ii\ 
4- vFo[oS/’n")(P"r^log(p")] 4- ®(a + c.)(pa;)logOjc.)] 
4- Fo [o^i' (« 4- ") log (pw) ] 4- log n [I — oS/ {o) ] 
. ^ ('~ ) ” i p J 2^'« (^ + ttj) + 2D/7i4 1 1 
for, as may be readily proved, 
[o( -p 4“ (p^i 4“ *2^’) — {pt A cjj] p^i — 2^/'^ 4“ ^■'’)~ P 4~ p* 
When the variable a is positive with respect to the w’s, we note that the part of 
the absolute term in this asymptotic e(|uality which is equal to 
mav be written 
— I 
iTT 
0^, ^2 <^ 1^0 ^2) 
-4" 1 {7 + iog(- A } (I 
s = 0 
2 M 77 qS; ((() , 
IL (1 — e~ "1-) (1 _ c - “.iq 
which is the expression which has been proved e(pial to oSg (a) 2 (^n + m') ttl 
_ loo- - - 
P2 (« 1 , cop 
integration. 
by the process of differentiation under the sign of contour 
§ 50 . But if we take the expression for log To («) which has been obtained in § 24 , 
— log To (r() — — y^i (^Jj, cjo) 4” ^^y -22 ^2) 4“ Fg <7 
00 00 
and write it 
+ N N 
r/ij = 0 in.2 = 0 
log {a + n) 
log n 
~ bo (a) = ^ yoi (wj, w.) 4- 0722 ("n "2) 4“ log a 
pn qn 
+ ht 1 1 ' 
n = 00 = 0 T)u = 0 L 
log (a + n) 
log VL 
a a~ 
Jl 2 TU 
a a- 
D 2 rp 
we may obtain this expression independently. 
