in combinatory analysis 215 



Associating, as before, the second part of each term with the first 

 part of the succeeding term, we obtain 



H {X) = ,rq (1 - ^Y/) jl- ccY' (1 - ^^q') 1 ^ 



'M i^ ^?)(l_^)(l_5.)(l_,/) + ' 

 = xg{l — x(f) G {x(f) (4 ). 



II now we write K{x) = ^-^--^^^^^- , 



we obtain, from (3) and (4), 



andso A-(^)=l+fl^^^ (5). 



In particular we have 



1^A_ t^- 1 _ {\-q)0{q) .... 



1+ T + 1 + ... K{\) G{1) ^ " 



or 



I q f _ 1 — g — g* + g' + g^^ — 



(7). 



1 + 1 + 1 + . . . 1 - ^2 _ ^3 ^ ^9 + gll 



This equation may also be written in the form 



1 <!_ f__{i^q){i_::q')iX-j')(i^)i^-_thi: 



1+1+1 + ... {\-f){\-q^){l-q^){l-cf){\-t'-)... 



(8)- 



If we write 



;,,._ G J^) 



^ ' {\-xq){l- xq') (1 - xf) ...' 

 then (4) becomes F (x) = F (xq) + xq F {xq"), 

 from which it readily follows that 



