DOUBLE GAMMA FUNCTION. 
303 
Substituting in the relation 
Ug ^ (z) = (27?) _ Q. 
CO, 
we 
find 
^ z z 
ur 5 l("n "0) + 2720 (coi, CO.) = - log 277 + ~ dr “ 0~, + 1 ^-2 
_COj \-SCOj^CO. -COj -CO., / 
_ 9 
2 ?? 277 l 
V ~ 9 ~ 9-) + ® ^ 9 s • 
.^COnCO., ZCO. ^CO.,/ CO., -ICO.," 
And hence equating coefficients of 2: and we find 
721 (cOi, CO 2 ) 
^ [log coo - 2 m 7 nj + , 
'IcOyW^ *■ ^ 
ico.,“ 
1 \ Cl 
722 ("n <^2) = ~ 9 ^ [log 277CO3 — 2a777i} — — {log co. + 2777771] + — 
-:coi 
Thus^ 
:co., 
CO., 
D (t) = — Wi~7oi (coi, CO.) + {log CO. — log r] — — — 2 mvL, 
CO., u CO, 
C (r) = - COi7.22(cOi, CO2) — + 9^;j [^^-g ^ ~ 2777771 } + 7, 
which are the relations ref[uired. 
Since log co, — log r — 2777771 = log coi — 'Ziu'tti, 
we may evidently write these relations in the form 
D(t) = — CO^jy.i (coi, CO.) + {log COi — 2777 ' 77 t] — , 
Cc>o ^ 
C(t) = — COi72.2{cOi, CO.2) “ (^2 + [log Wi — 2777 Vtj + 7. 
§ 28 . We will now show that, when the parameters coi and oj. are equal, w'e have 
721 ") = 
CO" 
loo- OJ — 1 
77' 
7 
and 
700 (co, co) = - [7 — ^ — log co]. 
CO 
By the definition formula of 72i(coi, co^), we at once have 
CO 
T21 oj)= —Lt 
1 , no) 
-- - - / -A ~ log 
'/!=00 _mi =0 7)72 = 0 ('^^1 + 
VI VI 
s' V' 
Group together all terms for which 777i + m.^ = c, and we have 
VI n 
S' s'. 
(?o.i + rrt,)2 
V 
o 
.= i e~ 
+ 1 
^ (» + 1)'= ^ (>7 + 2)’- ^ ^ (n + nf ’ 
*' Genesis of the Double Gamma Functions,” § G. 
