DOUBLE GAMMA FUNCTION. 
287 
We shall show that hy successive integrations we may determine r 2 (t|wj, 0 ) 2 ) as a 
function symmetrical in and Wo such that 
Fg + <»i) fl(^| ^2) ^- 2 mmS'iC I W2) 
^ 2 “^*) 81 (^ 2 ) 
Fg + Mo) I ](^l t«Jl) _2m'77iS'i(« I wa 
where ^](w) = {‘Itt j( d) (“ Theory of tlie Gamma Function,” § 31), and m and m' are 
integers (unity or zero), to l)e determined in accordance with the detailed theory 
which we proceed to give. 
And the function so determined will be unique, provided 
Lf[zFo(z I a»_^, w^)] = 1. 
£ = 0 
§ 20. We readily see that the function 
(O 
u 
CD CO 
_ 9 V S' 
mi = 0 m2=0 
1 
(2 + U)'' ’ 
where Q = -j- uIo&Jo satisfies the two difference equations 
+ "l) = ( 2 ) — I Wo) 
+ Wo) = (z) - wj. 
where here, as always, we suppress the parameters of the functions ^ 2 ''\z) 
{r = I, 2, ... ) and Fo( 2 :) when these parameters are supposed to exist in perfectly 
general form. 
For we have at once from the definition-series 
+ Ci) - = - 
_ V 
n!2=0 G + 
= — I Wo). 
(“ Gamma Function,” § 2) 
Next, we may show that the function 
CO 00 
xljo]^^\z\a}^ W2) — — yoi (w^. Wo) + ^2 “ 1 " ^ 
1 
,=0 )rt ,=0 L(^ + 
satisfies the two difference equations 
+ "j) — ~ 
+ (^-2) = W^), 
whatever he the value of the constant yo^(wi, Wo). 
For the series for <^o^^^( 2 ) is absolutely convergent so long as 
1 1 
{z + ny~ 
1 
ns 
