DOUBLE GAMMA FUNCTION. 
301 
We also have 
rr'(^) = 
CO CO 
(^n , + --y.2 _ 2. n n' 
JJlj = 0 3»2 = ® L 
1 + ^)e » »> 
SO that on comparison of the two products we find 
r3-^(z) = + 
w, 
where a, / 3 , and K are suitable functions of and wo. 
Now G(2 :|t) satisfies the diflPerence equation 
and hence G 
G(clT) = r-i - G(2+ ijr), 
CO 
T ] satisfies the relation 
f{z + ojj) = Fj {z I w.)/(z). 
Hence a solution of the difference equation 
f{z + OO,) = ^{coJ'Ztt) . 
IS 
T{z) = oo. 
2ajjt02 
2 coi 
And it is evident that the coefficient of Zirnn in the last exponential may be 
written 380(2 |ajj, 03). 
The general solution of the difference equation is 
r3-i(2) X p(2|0Ji), 
where ^9(2] Wj) is a function of 2 simply periodic of period Wj. 
G) 
Hence 
TG) 
= p {A^\)- 
G-'G) 
But —- has been seen to be an expression of the form where K, a, and 
T(z) 
/3 are independent of 2. We thus have 
(s + 0)1)2 + ^ (3 + Mj) + pz 
Zr 
SO that a = 0 and /8 = where r is some integer. 
CO 
Hence 
and since 
we have at once 
2?*7rt2 “2 / 
r3“^(2) — KeW (co3e~™’"‘)2®‘'^"^(27r)-“iG ( 
[2r2(2)] = 1, Li 
z=0 
z=0 
G(-|t 
" il / 
V«i 
= 6 ).- 
(1)> 
■ 2 -> 
K = C03. 
