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

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

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Additional resources for Chapter 16 of Ramanujan's Second Notebook Theta Functions and Q-Series

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We have given a slightly more explicit version of the corollary above than did Ramanujan. ), mm, nn ^> n0. In Entry 26 and its two corollaries, we quote from the notebooks [60, vol. 2, p. 198]. ENTRY 26. G(q) is a perfect, complete, pure, double series of 1/2 a degree. COROLLARY (i).

See Andrews' book [ 6 , pp. 9-12, 14] f o r an elementary proof and further r e f e r ences. Note that ( i v ) i s only a d e f i n i t i o n of x(q)« PROOF OF ( i ) . 1) of The f i r s t equality follows immediately from the d e f i n i t i o n f(a,b). 1) f(q,q) = (-q;q2)2(q2;q2) . 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).

ENTRY 24. /n) - (iv) f3(-q2) = v(-q)*2(q), f ( q) - X(q) - 3 - M q T I (-1) k (2k + 1 ) q k ( k + 1 ) / 2 , k=0 /^TqT _y(q) _ f(-q2) and x(q)x(-q) = x(-q 2 >- CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK 35 PROOF OF (i). Using Entries 22(iii), (ii), (iv), and (i), respectively, 2 2 (-q;q )CO /(q;q )CO we find that each of the given ratios is equal to PROOF OF (ii). 3, V (-q)^(q) = 2. 2). The second equality in ( i i ) i s another famous theorem of Jacobi [48] and i s a l i m i t i n g case of his t r i p l e product i d e n t i t y .

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