DOUBLE GAMMA FUNCTION. 
281 
/(«+ -/{«)=*- 
3-1 
I =0 
3-1 ■ 
l=o\ 
3^4 rt + j — oS„^a + ^ y 
S a + 
ho 2 
•^3 ) “1“ iB,,+i((y2.) 
= S„ a 
CO 
"f" {“ Gamma Function,” § 18). 
Thus f{a) is an algebraic solution of the difference equation satisfied by 
The two solutions can then only differ by a constant, and thus 
o S ;j ( Ct 
P ' 9. J 
oS^f a 
(Oi «. 
p-l 7-1 
J,-) = ‘S 2 „s„ A + 
i? 2 / 1=0 ( = o“ \ p '1 
w, 
, Ct)o I —h It#, 
where R# is independent of a. 
To determine this constant, let us integrate between 0 and We find 
r"H 
J f! 
a 
, da = t I nSA + 
P qj i=o-'o \ q 
CO, 
„ coG^ + ^R 
p 
and therefore by § 6 
CO.i 
, 1"''43( , 
CO, T-. / <^1 <^2 \ ^ \q 
_Tl D 
p " q 
?i, + 1 
5 
— qo)^ ("n " 3 ) “b - 
,_l2S# + i( ~ 'j _ + 
P 
I =0 
'll .1 
\ ?? / q_ —IR^, by § 12. 
p 
And therefore 
so that 
T) / \ I / ^ \lh»+3l^2) I ,d^“ + 2(®2) I ‘^IT) 
S"! "i) + [pr, - 1) + 3 + iW" 
(“Gamma Function,” § 18). 
p 2^3 + 1 
p q 
— qwi gB^+i (co^, coj) -f ^4 R«, 
R« — pq sB/i+i (^n ^3) 3 ®»+i 
CO] CO 3 
p' q 
On substituting this value we have the theorem enunciated. 
O 
As a coro//rtry we have on making a — 0. 
V V Si j_ ^ 2 
A A oO„ — q- 
1 = 01=0 \ p q 
CO 
n *^3) — 31^3+1 ( ^4 (cii, CI2), 
§ 15. We now proceed to prove the expansion of fundamental importance in the 
theory of double BernouUian functions :—• 
ze 
(1 — e “I*) (1 — e “2^) 
VOL. CXC7I.- 
_ jSi«>(a) + A*r 3 +. . • + + . . 
n ! 
O 
