Abstract
We prove two identities for Ramanujan’s cubic continued fraction and a continued fraction of Ramanujan, which are analogues of Ramanujan’s identities for the Rogers-Ramanujan continued fraction. We further derive Eisenstein series identities associated with Ramanujan’s cubic continued fraction and Ramanujan’s continued fraction of order six.
Highlights
Throughout this paper, we assume that |q| < 1 and for each positive integer n, we use the standard product notation (a; q)0 := 1, n−1(a; q)n := ∏ (1 − aqj), n ≥ 1, j=0 (1) ∞(a; q)∞ := ∏ (1 − aqj) . j=0Srinivasa Ramanujan made some significant contributions to the theory of continued fraction expansions
We prove two identities for Ramanujan’s cubic continued fraction and a continued fraction of Ramanujan, which are analogues of Ramanujan’s identities for the Rogers-Ramanujan continued fraction
Throughout this paper, we assume that |q| < 1 and for each positive integer n, we use the standard product notation (a; q)0 := 1, n−1 (a; q)n := ∏ (1 − aqj), n ≥ 1, j=0 (1)
Summary
We further derive Eisenstein series identities associated with Ramanujan’s cubic continued fraction and Ramanujan’s continued fraction of order six. Liu [4] and Chan et al [5] have established several new identities associated with the Rogers-Ramanujan continued fraction R(q) including Eisenstein series identities involving R(q). The beautiful Ramanujan’s cubic continued fraction G(q), first introduced by Srinivasa Ramanujan in his second letter to Hardy [2, page xxvii], is defined by q1/3 f (−q, −q5) f (−q3, −q3)
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