214 Prof. Roger's S Mr Ramanujan, Proof of certain identities 

 Thus 



{l-q){\-q'){\-t) ^ 



2. {By S. Ramanujan.) 



Let 

 G{x) = l 



+ t^ ^^^^^ ^' ""i \i-q){l-q^){l~f)...{l-qn 



If we write 1 -a;g2'' = 1 - (^^ + 5''(1 -^g"), 



every term in (1) is split up into tAvo parts. Associating the second 

 j)art of each term with the first part of the succeeding term, we 

 obtain 



1 —xq 



G (a;) = {1- xhf) - x'^q^ ( 1 - x-'q') 



l-q 



+ ^q (1 ^q ) (i_^)(i_^.>) {-)■ 



G(x) 



Now consider II(x) = ., ^ - G (xq) (3). 



\—xq 



Substituting for the first term from (2) and for the second term 

 from (1), we obtain 



x^q' 

 a;*g" (1 — xq-) 



H {x) = xq - ^3^- {(1 -q) + xc^ (1 - xf)] 



+ 





