On Ramanujan's cubic continued fraction

Ramanujan’s theories of elliptic functions to alternative bases

“…Concluding remarks It seems inconceivable that Ramanujan could have developed the theory of signature 3 without being aware of the cubic theta function identity (2.5), and in Lemma 2.1 and Theorem 2.2 we showed how (2.5) follows from results of Ramanujan. Heng Huat Chan [19] has found a much shorter proof of (2. Farkas and Kopeliovich [23] have generalized this to a /7-th order identity.…”

“…We employ many results from Ramanujan's second notebook in our proofs, in particular, from Chapters 17,19,20,and 21. Proofs of all of the theorems from Chapters 16-21 of Ramanujan's second notebook can be found in [5].…”

Flowers Which We Cannot Yet See Growing in Ramanujan’s Garden of Hypergeometric Series, Elliptic Functions, and q’S

“…In a fragment published with his lost notebook [51, p. 366], Ramanujan writes "and many results analogous to the previous continued fraction," indicating that there is a theory of the cubic continued fraction similar to that for the Rogers-Ramanujan continued fraction. Piqued by this remark, Chan [24] developed an extensive theory of the cubic continued fraction.…”

On Some Modular Equations and Their Applications Ii

“…Note that the numerical values of G(e −π ), G(e −2π ), and G(−e −π ) were evaluated in [4]. Note also that the numerical values of G(e −π ), G(e −2π ), G(e −3π ), G(−e −2π ), and G(−e −3π ) were evaluated in [9].…”

“…Note also that the numerical values of G(e −π ), G(e −2π ), G(e −3π ), G(−e −2π ), and G(−e −3π ) were evaluated in [9]. Hence the numerical values of G(e −π ) and G(e −2π ) were given in both [4] and [9], but they were evaluated by different proofs. We close this section by evaluating the numerical values of G(e −6π ), G(e −9π ), G(e −12π ), G(e −18π ), G(−e −6π ), and G(−e −9π ).…”