DOUBLE GAMMA FUNCTION. 
335 
pn qn 
log n n ““ (i i)] + P"] 
mi = 0 1112=0 
+ (p + q) {_n log n — 7i] + 7 ?Fo [cS/''(w) 'poj logpw] 
+ [1 — 2 (o)] log 11 + log (ai^, Wo) + Fo [oS\ (w) log pw] 
^ ^ p j" oS„; (O)) + pFtfi + l 
„~i vin’'^ (pw)™ 
is so evident as to determine the nomenclature. 
Note in the second place that the fundamental asymptotic expansion (A) of § 38 
may, with the symbolic notation subsequently introduced, be Mudtten 
pn qn 
X S' 
1 
Tii =0 m ^=0 {«- + 
1 
— ^3 ('5) I *^ 2 ) “h 
1 
(.s — 1) (s — 2 ) .n- 
3 ^Fo 
oSit^pro) 
ipcoy-") 
_p. , 1 F [ aS'i + w) 
(5 — 1 ) . «■' “ |_ I " 1 ipoiY 
§ 51 . If in this equality, true for all values of s, a, and wo, we make s = 1 , we 
obtain, since 
FUiS/'WCH; = « 
F.fsSi® (« + <«)! = - 2 Si®(«) ■ 
pn qn 
1 
- = lA 
mi = 0 jn,, =0 C + 7?qa)2 + 
4o(S, o 1 Wj, Wo) + ^ 
, 2 F I 2*^^ (a + co) 
11 ~ [ pco 
+ uF 3[2S2^^H") P" logpwj- — 2S2*“'(n)log n + F2h2S2^“^(« + w) log2:>wl 
+ 
* ( —p” p j” 2 S,„(« + &)) + 2 ^^>« + l 
??l = 1 
71 ‘ 
m +1 
(p&))“+i 
which is equivalent to the result of difterentiating the asymptotic expansion of § 49 . 
If now we make a = 0 , we obtain 
pn qn 
V V' 
= Lt 
= 0 772^=0 “1“ a = 0 
V/ I X , 1 ' 
l{s, «j W2, Wo) + g ^ - 
J = 1 
+ ?iF3{oS2'^^(w)pwlog pw} — oSj^“>(o) log W + Fo{2Sj^“'(w) log 2^w] 
( yy p [ 38m (w) + 
m = 0 
iHl + l 
We now see that 
U 
pn qn 
S' V' 
1 
l_))ii = 0 rii 2 = 0 + '?^qWo 
(p^)m+l 
+ oS2‘"’(o) logw — F|oSi^“^(w) log 2^0j] 
— nFoi^i^^ioj)poj log pw] 
= U 
a = 0 
= 0 
CoCh « i "n "3) + 5 _ ~ 
which is a quantity independent of p and q. 
