Journal of Numbers

Volume 2014 (2014), Article ID 643241, 6 pages

http://dx.doi.org/10.1155/2014/643241

Research Article

Continued Fractions of Order Six and New Eisenstein Series Identities

Chandrashekar Adiga, A. Vanitha, and M. S. Surekha

Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, India

Received 25 February 2014; Revised 25 April 2014; Accepted 7 May 2014; Published 27 May 2014

Academic Editor: Ahmed Laghribi

Copyright © 2014 Chandrashekar Adiga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 identity states that

In Ramanujan’s notation, the Jacobi triple product identity takes the form

The Jacobi triple product identity was first proved by Gauss [13]. Using (3), we have

Putting and in (10), we obtain

Putting and , respectively, in (12), we obtain

Multiplying (13) and (14) together and using the identities

in the resulting equation and then after some simplifications, we obtain the following identity:

Theorem 4. Let , , and . Then

Proof. We may rewrite (13) and (14) as follows:

where

Then,

Multiplying the above two equations together and then using (16), in resulting identity, we find that

Subtracting (22) from (21), we obtain

Using (11) in (24), we deduce that

where

Now, by direct computations, we find that

Therefore,

In the right-hand side of the above equation, changing to in the first two summations and also changing to in the last two summations, we obtain

Now, using the definition of in the right-hand side of the above equation, we find that

In the quintuple product identity (8), replacing by and then setting , we find that

Combining (25), (30), and (31), we find that

Dividing both sides of (32) by and then using (23), we obtain

Replacing by and by in (7), we find that

Now, using the above equation in the right-hand side of (33) and then changing to throughout, we obtain (17). Equation (18) follows from the following identity:

This completes the proof of Theorem 4.

4. Eisenstein Series Identities Associated with and

In this section, we present four Eisenstein series identities associated with and .

Theorem 5. Let . Then

Proof. Changing to in the second summation, of the left-hand side of Theorem 5, we have

Using a corollary of Ramanujan’s summation formula [11, Entry 17, page 32]

and Lemma 3 in (37), we find that

Using (4) in (39) and after some simplifications, we obtain (36).

Theorem 6. Let . Then

Proof. Using the identity

the left-hand side of Theorem 6 can be written as

Using (38) in (42), we obtain

Using (5) in (43) and after some simplifications, we obtain (40).

Differentiating both sides of (12) and then setting yield

where denotes the partial derivative of with respect to .

Now we prove a lemma, which is useful to prove Eisenstein series identities associated with and .

Lemma 7. Consider the following:

Proof. For simplicity, we use to denote the logarithmic derivative of with respect to . To prove this lemma we need the following identity, which can be found in [14, Theorem 5], [15, Corollary 2]:

As the proof of this lemma is similar to that of Lemma 1 in [4], we omit the details.

Using (45), we derive the following Eisenstein series identity.

Theorem 8. Let . Then

Proof. Dividing both sides of (45) by and then letting , we obtain

Using Lemma 3 in (48), we complete the proof of Theorem 8.

We use to denote the Legendre symbol modulo . Setting in (45) and noting that , we find that

Recall the identity [16, Eq. (3. 1)]

Using the above identity in (49), we obtain the following Eisenstein series identity.

Theorem 9. Let . Then

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their several helpful comments and suggestions. The first author is thankful to the University Grants Commission, Government of India, for the financial support under the Grant F.510/2/SAP-DRS/2011. The second author is thankful to DST, New Delhi, for awarding INSPIRE Fellowship (no. DST/INSPIRE Fellowship/2012/122). The third author is thankful to UGC, New Delhi, for UGC-BSR fellowship, under which this work has been done.

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