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New Properties for The Ramanujan’S Continued Fraction of Order 12

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

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

Abstract: In this paper, we derive new identities involving a continued fraction of Ramanujan of order twelve that are similar to those of the Ramanujan-Göllnitz-Gordon continued fraction.

Keywords: continued fraction, power series expansion

Received June 07, 2014; Revised June 27, 2014; Accepted July 03, 2014

Citation: Chandrashekar Adiga, M. S. Surekha, A. Vanitha, New Properties for The Ramanujan’S Continued Fraction of Order 12. Turkish Journal of Analysis and Number Theory, 2 (3). 90-95. January 2014. doi:10.12691/tjant-2-3-7

1. Introduction

The celebrated Ramanujan-Göllnitz-Gordon continued fraction is defined by

[picture] | (1)

On page 229 of his second notebook [10], Ramanujan recorded a product representation of namely

[picture] | (2)

Throughout this paper, we assume that and use the standard notation

For convenience, we often write for in the sequel.

Ramanujan’s general theta function is defined by

[picture] | (3)

The Jacobi triple product identity [1] in Ramanujan’s notation is

[picture] | (4)

Jacobi’s triple product identity is a special case of summation formula [1] due to Ramanujan. S. Bhargava et al. [6] made use of Ramanujan’s summation formula to prove a convolution identity for certain coefficientsgenerated by the quotient of two infinite products. As special cases of this identity, they deduced several results including, for example, the convolution identities given earlier by Kung - Wei Yang [15]. In [6], Bhargava et al. raised an interesting question concerning the derivability of convolution identity as a conseuence of summation formula instead of the and sums. This question was completely answered by H. M. Srivastava[12].

The four important special cases of [1] are

[picture] | (5)

[picture] | (6)

[picture] | (7)

and

[picture] | (8)

Also, after Ramanujan, define

Ramanujan recorded many identities involving and

The famous Göllnitz-Gordon functions and are defined by

[picture] | (9)

and

[picture] | (10)

where the two equalities on the righthand sides of (9) and (10) are the celebrated Göllinitz-Gordon identities [7,8].

We note that

Without any knowledge of Ramanujan’s work, H. Göllnitz [7] and B. Gordon [8] rediscovered and proved (2) independently. Later G. E. Andrew [3] proved (2) as a corollary of a more general result. Moreover, Ramanujan established two further identities for [5, 10], namely,

[picture] | (11)

and

[picture] | (12)

In [6], Boonrod Yuttanan found that

[picture] | (13)

and

[picture] | (14)

The following continued fraction (15) was established by M. S. M. Naika et al. [9] as a special case of a fascinating continued fraction identity recorded by Ramanujan in his Second Notebook [10]:

[picture] | (15)

Now, we have

[picture] | (16)

where

We define

[picture] | (17)

Motivated by the identities (11)-(14) involving the Ramanujan-Göllnitz-Gordon continued fraction, in this paper, we establish several new identities and properties of

2. Preliminaries

The following Lemmas are useful to prove our main results.

Lemma 2.1. [9]. We have

[picture] | (18)

[picture] | (19)

Lemma 2.2. [2]. We have

[picture] | (20)

Lemma 2.3. [1]. We have

[picture] | (21)

[picture] | (22)

[picture] | (23)

[picture] | (24)

Lemma 2.4. [4]. We have

[picture] | (25)

Lemma 2.5. [9]. We have

[picture] | (26)

Lemma 2.6. [11]. We have

[picture] | (27)

Lemma 2.7. [1]. We have

[picture] | (28)

and

[picture] | (29)

where

Lemma 2.8. [12]. We have

[picture] | (30)

Lemma 2.9. [1]. Let and for each integer Then

[picture] | (31)

for every positive integer

Lemma 2.10. [1]. We have

and if is an integer, then

[picture] | (32)

3. Main Results

In this section, we establish some new identities involving the continued fraction which resemble (11)-(14) and Theorem 3.1 of [6].

Theorem 3.1. We have

[picture] | (33)

[picture] | (34)

[picture] | (35)

[picture] | (36)

Proof. By Lemma we have

[picture] | (37)

On simplification of (37), we obtain

[picture] | (38)

Employing (20) in (38), we can deduce that

[picture] | (39)

Using (22) in (39), we establish that

By Lemma we have

[picture] | (40)

On simplification of (40), we obtain

[picture] | (41)

Employing (20) and (25) in (41), we get

[picture] | (42)

Using (22) in (42), we deduce that

[picture] | (43)

It is easy to check that

[picture] | (44)

[picture] | (45)

[picture] | (46)

Employing (44), (45) and (46), in (43), we obtain (34).

Using the definition of we have

[picture] | (47)

Putting and in (21), we obtain

[picture] | (48)

Employing (48) and Lemma 2.1. in (47), we get

[picture] | (49)

Employing (20) and (23) in (49), we obtain (35).

The proof of (36) is similar to the proof of (35).

Remark 3.2. Squaring (35) and (36) and after some simplifications, we obtain

The results in the following corollary are simply the squares (or product) of the identities (33) and (34).

Corollary 3.3. We have

[picture] | (50)

[picture] | (51)

[picture] | (52)

Theorem 3.4. We have

[picture] | (53)

where and

Proof. From Lemma we have

[picture] | (54)

Using (26) and Lemma 2.6 in (54), we obtain (53).

Theorem 3.5. For

[picture] | (55)

[picture] | (56)

or equivalently

[picture] | (57)

[picture] | (58)

Proof. By the definitions of and we have

[picture] | (59)

Employing (18) and (19) in (59) and after some simplifications, we deduce

[picture] | (60)

Employing (25) and (28) in (60), we obtain (55).

The proof of (56) is similar to the proof of (55).

Dividing both sides of (55) and (56), by and then employing (18), (19) and (20) in resulting identities, we arrive at

[picture] | (61)

[picture] | (62)

Next, by adding (61) to (62), we obtain

[picture] | (63)

Subtracting (62) from (61), we get

[picture] | (64)

Employing (29), (5), (6), (7), (8) and definition of in above two identities, we get (57) and (58). The arguments above can be easily reversed to show that (57) and (58) imply (55) and (56).

Theorem 3.6. We have

[picture] | (65)

[picture] | (66)

[picture] | (67)

Proof. By Lemma we have

[picture] | (68)

On simplification of (68), we obtain

[picture] | (69)

Employing (20), (26) and (30) in (69), we deduce that

[picture] | (70)

Using (22) in (70), we get (65).

By Lemma 2.1, we have

[picture] | (71)

On simplification of (71), we deduce

[picture] | (72)

Employing (26) and (30) in (72) and after some simplifications, we get (66).

The proof of (67) is similar to the proof of (66).

Now, we shall establish identities involving which are similar to the identities in Theorem 3.2 proved by B. Yuttanan [6].

Theorem 3.7. We have

[picture] | (73)

and

[picture] | (74)

where and

Proof. By the definition of we have

[picture] | (75)

Now, putting and in (31), where we obtain

[picture] | (76)

Put and in (32), to get

[picture] | (77)

Again, setting and in (32), we deduce

[picture] | (78)

[picture] | (79)

Note that so It follows that

[picture] | (80)

Now, using (80), Lemma and Lemma in (75), we obtain

[picture] | (81)

By Jacobi’s triple product identity,

Since and we have

[picture] | (82)

Similarly, by Jacobi’s triple product identity, we have

[picture] | (83)

Note that

[picture] | (84)

Substituting (82), (83) and (84) in (81), we get (73).

To prove (74), employ and and in (31).

Acknowledgements

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 UGC-BSR fellowship. The third author is thankful to DST, New Delhi for awarding INSPIRE Fellowship [No. DST/INSPIRE Fellowship/2012/122], under which this work has been done.

References

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