212 Prof. Rogers d- Mr Ramanujan, Proof of certain identities 



Prof. Rogers* and two by Prof. I. Schur of Strassburgf, who appears 

 to have rediscovered the formulae once more. 



The proofs which follow are very much simpler than any pub- 

 lished hitherto. The first is extracted from a letter written by 

 Prof Rogers to Major MacMahon in October 1917 ; the second 

 fi-om a letter written by Mr Ramanujan to me in A.pril of this year. 

 They are in principle the same, though the details differ :|:. It 

 seemed to me most desirable that the simplest and most elegant 

 proofs of such very beautiful formulae should be made public with- 

 out delay, and I have therefore obtained the consent of the authors 

 to their insertion here. 



It should be observed that the transformation of the infinite 

 products on the right-hand sides of (1) and (2) into quotients of 

 Theta-series, and the expression of the quotient of the series on the 

 left-hand sides as a continued fraction, exhibited explicitly in Prof 

 Rogers' original paper and in Mr Raman ujan's present note, offer no 

 serious difficulty. All the difficulty lies in the expression of these 

 series as products, or as quotients of Theta-series. — G. H. H.] 



1. {By L. J. Rogers.) 



Suppose that \q\<l, and let F,,^ denote the convergent series 

 (1 - ^»^) - ^«(^«+i-'« (1 - x'"^q"''') C\ 



where 



., _ (1 - ^) (1 - xq) (1 - iC(f) ... (1 - xq'-^) 

 ^'- (i_5)(i_5-.)(i_23^...(l_5.) -■ 



the general term being 



Then 



V,n - Fm-i = ^*"~' (!-«;)- x"q''+'-'^ {(1 -q) + x'>''~Hf"-' (1 - Ar^)} d 



_,_ ^in^m+z-im |(^1 _ ^^2) ^ ^m-i^im-2 (^l _ ^^2^J (J^^ (• 2^_ 



Suppose now that the symbol 77 is defined by the equation 



vf{«^) =f(xq)^ 

 Then ( 1 - f) C, = (1 - x) rj C,-! , (1 - xq^) 0,. = (1 - x) rj C,. 



* L. J. Kogers, ' Ou two theorems of Combinatory Analysis and some allied 

 identities', Proc. London Math. Soc, ser. 2, vol. 16, 1917, pp." 315—336 (pp. 815— 

 317). 



t I. Schnr, ' Ein Beitrag zur additiven Zahlentheorie uud zur Theorie der 

 Kettenbriiche', Berliner Sitzungsberichte, 1917, No. 23, pp. 301—321. 



J I have altered the notation of Mr Eamanujan's letter so as to agree with that 

 of Prof. Rogers. 



