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Residue Theorem and Theta Function Identities - Liu (2001) Self-citation (Ramanujan) (Correct)
....the following important number theory identity due to Jacobi [1, 6] 1 16 # # n=1 ( 1) n n 3 q n 1 q n = # # # n= # ( 1) n q n 2 # 8 . 6.11) If we divide both sides of (6. 10) by a and then let a# 0 we obtain the following number theory identity due to Ramanujan [22] and just proved by Ewell [11] # # n=1 n 3 q n 1 q 2n = q # # # n=0 q n(n 1) 2 # 8 . 6.12) In the same way as above from f (z) # 1 (3z a q 3 ) # 2 1 (z q)# 1 (z a q) 6.13) THETA FUNCTIONS IDENTITIES 143 we obtain the following important ....
S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Ramanujan's Formulas For Explicit Evaluation Of The.. - Kang (4 citations) Self-citation (Ramanujan) (Correct)
....q 2 1 q 3 1 Delta Delta Delta and (1.2) S(q) GammaR( Gammaq) denote the Rogers Ramanujan continued fractions. This famous continued fraction was introduced by L. J. Rogers [19] in 1894 and rediscovered by S. Ramanujan in approximately 1912. In his first two letters to G. H. Hardy [16], Ramanujan communicated several results concerning R(q) In particular, he asserted that (1.3) R(e Gamma2 ) s 5 p 5 2 Gamma p 5 1 2 ; 1.4) S(e Gamma ) s 5 Gamma p 5 2 Gamma p 5 Gamma 1 2 ; and (1.5) R(e Gamma2 p 5 ) p 5 1 5 3=4 i p 5 Gamma1 2 j ....
....17 Remark. Corollary 5.1(i) is recorded on page 285 in Ramanujan s first notebook. One of the proofs of this evaluation is given by Berndt and Chan [5] All the other evaluations of theta functions in this section appear to be new. Proof. Let q = e Gamma . From Ramanujan s paper [16], 17] Weber s table [22] p. 722, or [4] Chapter 34, 5.2) G 1 = 1 and G 25 = 1 p 5) 2: Thus by (2.14) 5.3) t 1 = e i=6 G 1 G 25 = e i=6 p 5 Gamma 1 2 : Hence, from Theorem 2.1(iv) 5.4) e Gamma ) e Gamma5 ) 2 = Gamma4 2 p 5 q 81 Gamma 36 p 5 = 5 ....
S. Ramanujan, Collected papers, Chelsea, New York, 1962.
Fragments by Ramanujan on Lambert Series - Berndt Self-citation (Ramanujan) (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Ramanujan's Theories Of Elliptic Functions To Alternative.. - Berndt, Bhargava, Garvan (3 citations) Self-citation (Ramanujan) (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Notes On Ramanujan's Singular Moduli - Berndt, Chan Self-citation (Ramanujan) (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Radicals And Units In Ramanujan's Work - Berndt, Chan, Zhang Self-citation (Ramanujan) (Correct)
....AND UNITS IN RAMANUJAN S WORK Bruce C. Berndt, Heng Huat Chan, and Liang Cheng Zhang In memory of S. Chowla 1. Introduction In problems he submitted to the Journal of the Indian Mathematical Society [16], in his notebooks [15] and in his lost notebook [17] Ramanujan established many intriguing equalities between radicals. In particular, in his extensive calculations of more than 100 class invariants, he frequently needed to establish difficult radical equalities; see two papers [3] 5] by the ....
S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Some Values For The Rogers-Ramanujan Continued Fraction - Berndt, Chan (1995) (2 citations) Self-citation (Ramanujan) (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
The Rogers-Ramanujan Continued Fraction - Berndt, Chan, Huang, Kang, Sohn.. Self-citation (Ramanujan) (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Some Theorems On The Rogers-Ramanujan Continued Fraction .. - Berndt, Huang, Sohn, Son Self-citation (Ramanujan) (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Remarks on some series considered by Ramanujan - Brambilla, Montaldi (1995) Self-citation (Ramanujan) (Correct)
....integer. Section 4 contains a simple proof of an identity involving two infinite series, by means of the Poisson summation formula. Finally, in Section 5, we revisit four remarkable Lambert series and derive other results of a similar nature. II. SERIES RELATED TO i(S) i(S; X) AND L(S) Ramanujan [10] defined OE(s; x) 1 X n=0 ( p x n p x n 1) Gammas (2.1) and obtained for this series a finite expression in terms of generalized zeta functions, when s is an odd integer greater than 1. The corresponding result for the series (s; x) 1 X n=0 ( p x n p x n ....
....p x (2.25) 3; x) 2i( 1 2 ; x) Gamma 1 p x 4 p x (2.26) 5; x) 24i( Gamma 1 2 ; x) 1 p x Gamma 12 p x 16x p x (2.27) and, recalling the functional equation for i(s) OE(3; 0) 3 2 i( 3 2 ) 2.28) OE(5; 0) 15 2 2 i( 5 2 ) 2. 29) in agreement with Ramanujan [10]. He stated the formulas for OE(s; x) and (s; x) in a somewhat disguised form, because he followed a different (and very ingenious) procedure, which however does not apply to the more general cases (2.4) 2.5) If, in (2.4) and (2.5) the terms of the series have alternating signs, it suffices ....
S. Ramanujan, Collected Papers, Chelsea, New York, 1962, 68--71.
An Identity of Ramanujan, and Applications - Hirschhorn (Correct)
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S. Ramanujan Aiyangar, Collected Papers, G. H. Hardy (ed.), New York, Chelsea, 1962.
The Remaining 40% Of Ramanujan's Lost Notebook - Berndt (Correct)
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S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
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