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Chapter 16 of Ramanujan's Second Notebook Theta Functions by C. Adiga, B. Berndt, S. Bhargava, G. Watson

By C. Adiga, B. Berndt, S. Bhargava, G. Watson

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2) (-q;q2)ro = T r n n=l 00 •q2""1) = TT L ± ^ n=l 1 + q n 2^ , 2n -n q n=l (1 - q ) ( l + q 2 n ) ~ ( q ; q 2 ) J - q 2 ^ 2 ) ^ 1 which is a famous identity of Euler. 1), we complete the proof of (i). CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK PROOF OF (ii). For f(q,q 3 ) = 1 + I l k=0 |q| < 1, I q 2k(2k-l)/2 k=l k=l k odd 33 q"« + »" + J q 2k(2k + l)/2 k=l + I k=0 k even q' q k(k+D/2 ) which proves the first equality. 2). PROOF OF (iii). 1) of f(a,b). By Entry 19, The first equality follows immediately from the definition f(-q,-q2) = (q;q 3 )Jq 2 ;q 3 )Jq 3 ;q 3 L = ( q ^ L ENTRY 23.

4), we see that it suffices to show that Log G(n) = 4 J ^ W ^ k • 1 )n) k=l (2k - 1)(1 - q 4k ~^) Like the calculation of Log F(n) in the previous proof, the proof of the equality above is quite straightforward. Ramanujan now states two "corollaries". We have not been able to discern why the appellation "corollary" has been given to these two results. Moreover, the "corollaries" are incorrect. We shall give two corrected versions of each corollary. Firstly, we prove versions where the "right sides" are corrected; secondly, we establish reformulations when the "left sides" are corrected.

N > i. n f(bq n+1 ,aq n ) ~ n+1 v x/. 1), and iteration, we deduce that f(b,a) _ -, fTbqTH - bq , m^WZ - bq ! aq + 2 f(bq2>aq) f(bq 3 ,ac 2 ) ^ • ^ - 1+ bq 2 3 N bq bq 1 - aq + 1 - aq + 1 - aq +••• The convergence of this continued fraction follows in the same way as in the proof of Entry 13. This completes the proof. COROLLARY. oo k=Q If |q| < 1, then k k(k+l) (q) kk 2 IM^ k k = 1 aq aq 2 " 1 + 1 + 1 aq3 + T + - k=0 PROOF. Set a = 0 in Entry 15 and then replace b by a. The corollary now readily follows.

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