Continued Fractions of Order Six and New Eisenstein Series Identities

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. 1. Introduction Throughout this paper, we assume that and for each positive integer , we use the standard product notation Srinivasa Ramanujan made some significant contributions to the theory of continued fraction expansions. The most beautiful continued fraction expansions can be found in Chapters 12 and 16 of his second notebook [1]. The celebrated Rogers-Ramanujan continued fraction is defined by [2] where is Ramanujan’s general theta function. Ramanujan eventually found several generalizations and ramifications of which can be found in his notebooks [1] and “lost notebook” [3]. Recently, Liu [4] and Chan et al. [5] have established several new identities associated with the Rogers-Ramanujan continued fraction including Eisenstein series identities involving . The beautiful Ramanujan’s cubic continued fraction , first introduced by Srinivasa Ramanujan in his second letter to Hardy [2, page xxvii], is defined by Adiga et al. [6], Bhargava et al. [7], Chan [8], and Vasuki et al. [9] have proved several elegant theorems for , many of which are analogues of well-known properties satisfied by the Rogers-Ramanujan continued fraction. Recently, Vasuki et al. [10] have studied the following continued fraction of order six: The continued fraction (5) is a special case of a fascinating continued fraction identity recorded by Ramanujan in his second notebook [1], [11, page 24]. Furthermore, they have established modular relations between the continued fractions and for , and 11. In Section 3 of this paper, we establish two new identities associated with the continued fractions and , using the quintuple product identity. In Section 4, we derive Eisenstein series identities associated with and . 2. Definitions and Preliminary Results In this section, we present some basic definitions and preliminary results. One of the most interesting special cases of is [11, Entry 22] Note that the Dedekind eta function , where , . We need the following three lemmas to prove our main results. Lemma 1 (see [11, Entry 30, page 46]). One has Lemma 2 (see [11, page 80]). One has Lemma 3 (see [12, Lemma 2(ii)]). Let , , , and , . Here denotes the largest integer less than or equal to . Then(i) ;(ii) . 3. Main Results The Jacobi triple product