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

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

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

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Entry 17 can be reformulated in a more compact s e t t i n g . the d e f i n i t i o n of {c) k for every and = (c; ^k integer -az/q, (^qj^ = , k. k by defining , > In Entry 17, now replace respectively. o Lastly, replace written in the form (a). (az) (q/az) (q) (b/a) k kV V OO ' OO ^ OO OO I TET* ^ k " ( z U b /az)Jb ) o o (q/a) c k= _» where |b/a| < |z| < 1. We f i r s t extend q a, B, and z by 2 1/a, b/q , by q. 1) can be 29 CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK COROLLARY. If |nq| < |z| < 1/|nq|, then 2 «> (l/n;q )k(-nq)k(zk + z'k) 1 + I (^z;q 2 M-q/z;q 2 Mq 2 ;q 2 Mn 2 q 2 ;q 2 ) o (nq 2 ;q 2) k k=l PROOF.

A direct proof depending on Cauchy's theorem will be found in Mr. 1) is found in Ramanujan's quarterly reports. 1), consult Hardy's book [39, p. 194] or Berndt's account of the quarterly reports [22]. R. 1) is a q-analogue of the beta-function. R. L. 1). In recent years, there has been much research related to Entry 14. 1). More general work in this direction has been accomplished by Askey and Wilson [15], where a plethora of references may be found. For a discussion of the q-gamma function see Askey's papers [12] and [13].

ADIGA, B. C. BERNDT, S. BHARGAVA, AND G. N. WATSON o (ii) (-q)= 4qip(q ), (iii) *(q)*(-q) = <£ 2 (-q 2 ), *(q)iK-q) = *(q2)2{q) - *>2{-q) = 8q ip2(q4), (vi) *>2(q) + / ( - q ) = 2^ 2 (q 2 ), and (vii) * 4 (q) " * 4 (-q) = 16q * 4 ( q 2 ) . PROOF OF (i). By Entry 22(i), *(q) + *(-q) = 2 + 2 £ (qk + (-q)k ) k=l 2 oo = 2+4 I k=l q 4 k = 2*>(q4). PROOF OF (ii). By Entries 22(i) and (ii), *(q) - *(-q) = 4 I q(2k"1) k=l r A , 8,k(k-l)/2 . ,, 8* k=1 PROOF OF ( i i i ) .

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