Prof. Rogers & Mr RamanKJan, Proof of certain identities 211 



P roof of certain identities in combinatory analysis: (1) by Prof. 

 L. J. Rogers ; (2) by S. Ramanujan, B.A., Trinity College. (Com- 

 municated, with a prefatory note, by Mr G. H. Hardy.) 



[Received 3 October 1918 : read 28 October 1918.] 



[The identities in question are those numbered (10) and (11) in 

 each of the two following notes, viz. 



q q* q^ 



+ l-l + (I-g)(l-5'=)'^(l-5)(l-f/)(l-9^) + -" 



= 1 (1) 



and 



q^ q^ 512 



+ n^ + (i -q){i- f) + j\-~-^(y-f) (1 - f) + • • • 



= ^ ...(2). 



(1 - r/) (1 - (/) (1 - f) (1 - (/) (1 - q^-^) (1 - f/0 ^ ^ 



On the left-hand side the indices of the powers of q in the 

 numerators are n^ and n (n + 1 ), while in each of the products on 

 the right hand side the indices of the powers of q form two arith- 

 metical progressions with difference 5. 



The formulae were first discovered by Prof Rogers, and are 

 contained in a paper published by him in 1894*. In this paper 

 they appear as corollaries of a series of general theorems, and, 

 possibly for this reason, they seem to have escaped notice, in spite 

 of their obvious interest and beauty. They were rediscovered 

 nearly 20 years later by Mr Ramanujan, who communicated them 

 to me in a letter from India in February 1913. Mr Ramanujan 

 had then no proof of the formulae, which he had found by a process 

 of induction. I communicated them in turn to Major MacMahon 

 and to Prof. O. Perron of Tubingen ; but none of us were able to 

 suggest a proof; and they appear, unproved, in Ch. 3, Vol. 2, 1916, 

 of Major MacMahon's Combinatory Analysis'^. 



Since 1916 three further proofs have been published, one by 



* L. J. Rogers, ' Second memoir 011 the expansion of certain infinite products', 

 Proc. London Math. Soc, ser. 1, vol. 25, 1894, pi?. 318—343 (§ 5, pp. 328—329, 

 formulae (1) and (2)). 



t Pp. 33, 35. 



