Journal of Numbers
Volume 2014 (2014), Article ID 162759, 6 pages
http://dx.doi.org/10.1155/2014/162759

Research Article

New General Theorems and Explicit Values of the Level 13 Analogue of Rogers-Ramanujan Continued Fraction

Nipen Saikia

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India

Received 4 September 2014; Accepted 6 November 2014; Published 27 November 2014

Academic Editor: Cheon S. Ryoo

Copyright © 2014 Nipen Saikia. 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 general theorems for the explicit evaluations of the level 13 analogue of Rogers-Ramanujan continued fraction and find some new explicit values. This work is a sequel to some recent works of S. Cooper and D. Ye.

1. Introduction

For and , the Dedekind eta function and Ramanujan’s function are defined by

where . The famous Rogers-Ramanujan continued fraction is defined by

This continued fraction was introduced by Rogers [1] in 1894 and rediscovered by Ramanujan in approximately 1912. In his notebooks [2], lost notebook [3], and his first two letters to Hardy [4], Ramanujan recorded several explicit values of . These values are first proved by Watson [5, 6] and Ramanathan [7]. For further references on explicit evaluations of see [8–14].

The Rogers-Ramanujan continued fraction is closely related to the function by the following two beautiful relations:

which are stated by Ramanujan [15, page 267, (11.6)] and first proved by Watson [5].

In his second notebook [2, 15] Ramanujan stated an interesting analogue of . If

then

Proofs of (5) can be found in [16, 17]. The function is studied by Cooper and Ye [18, 19]. In [18] Cooper and Ye evaluated or indicated the explicit values of and for = 1, 2, 3, 5, 7, 9, 13, 15, 31, 55, 69, 129, 231, and 255 by using the methods of reciprocity formulas, modular equations, Ramanujan-Weber class invariant, and Kronecker’s limit formula and in terms of the functions and [18, page 94, (1.7) and (1.8)] which are defined by

From (6) and (7), it is clear that if we know or for any particular value of , then can be determined by solving the corresponding quadratic equations.

Cooper and Ye [18, page 104, Theorem 4.1] further established that

where

and . They also indicated that if we know for any particular , then and can be determined by appealing to (8) and therefrom and can be determined at the same . For this they evaluated for = 5, 9, 13, 69, and 129.

In this paper, we prove some general theorems for the explicit evaluations and by parameterizations of Dedekind eta function and find some old and new explicit values. In Section 2, we record some preliminary results which will be used in the subsequent sections. In Section 3, we prove general theorems for the explicit evaluations of and evaluate some old and new explicit values. In Section 4, we find some new explicit values of by parametrization. Finally, in Section 5, we consider the function .

2. Preliminaries

Lemma 1 (see [15, page 43, Entry 27(iii), (iv), (vi)]). If and are such that the modulus of each exponential argument below is less than 1 and , then

Lemma 2 (see [20, page 211, Entry 57]). If and , then

Lemma 3 (see [20, page 237, Entry 72]). If and , then

3. General Theorems for Explicit Evaluations of ()

In this section we prove general theorems for the explicit evaluations of and find some explicit values.

Theorem 4. One has the following.

(i)For , let

Then

(ii)For , let

Then

Proof. We set in (5) and use the definition of to arrive at (i). To prove (ii) we replace by in (5), set , and use the definition of .

Theorem 5. If and are as defined in Theorem 4, then

Proof. We use the definitions of and and use (10) and (11), respectively, to complete the proof.

Corollary 6. If and are as defined in Theorem 4, then

(i), (ii).

Proof. We replace by in Theorem 4(i) and (ii) and employ Theorem 5 to arrive at (i) and (ii), respectively.

From Theorem 4(i) and Corollary 6(i), it is clear that if we know the explicit values of the parameter , then explicit values of and can be evaluated, respectively. Similarly, if we know the explicit values of the parameter , then explicit values of and can be determined from Theorem 4(ii) and Corollary 6(ii), respectively.

Next we find some explicit values of the parameters and .

Corollary 7. One has

Proof. We set in Theorem 5 to complete the proof.

Remark 8. Setting in Theorem 4(i), employing the values , and solving the resulting equation, we evaluate

Similarly, setting in Theorem 4(ii), employing the values , and solving the resulting equation, we evaluate

The values and are also evaluated by Cooper and Ye [18] by using reciprocity formulas.

Theorem 9. One has

Proof. We use the definition of in Lemma 2 to complete the proof.

Corollary 10. One has

Proof. Setting in Theorem 9 and noting from Corollary 7, we obtain

Set

Employing (26) in (25) and simplifying, we obtain

Employing (27) in (26), solving the resulting equation, and noting , we arrive at (i).
To prove (ii), we set in Theorem 9 and noting from Theorem 5, we obtain

Solving (28) and choosing the appropriate root, we complete the proof.

Remark 11. Employing the value of in Theorem 4(i) and Corollary 6(i) and solving the resulting equations, we evaluate new explicit values

respectively. Similarly, by employing the value of in Theorem 4(i) and Corollary 6(i) and solving the resulting equations we can evaluate explicit values of and , respectively. The value is also indicated by Cooper and Ye [18].

Theorem 12. One has

Proof. We employ the definition of in Lemma 3 to complete the proof.

Theorem 13. One has

Proof. Setting in Theorem 12 and using the result , we obtain

Solving (33) and choosing the appropriate root, we arrive at (i).
Again, setting in Theorem 12 and simplifying using Theorem 5, we obtain

Solving (34) and choosing the appropriate root, we complete the proof of (ii).

Remark 14. Employing the value of in Theorem 4(i) and Corollary 6(i) and solving the resulting equations we evaluate and , respectively. Similarly, by employing the value of in Theorem 4(i) and Corollary 6(i) and solving the resulting equations, we determine the explicit values of and , respectively.

Theorem 15. One has

Proof. We replace by in Lemma 3 and employ the definition of .

Theorem 16. One has

Proof. We set in Theorem 15, employ from Corollary 7, and solve the resulting equation to arrive at (i). To prove (ii) we set , employ the result from Theorem 5, and solve the resulting equation. We complete the proof.

Remark 17. Employing the value of in Theorem 4(ii) and Corollary 6(ii) and solving the resulting equations, we can evaluate the explicit values of and , respectively. Similarly, by employing the value of in Theorem 4(ii) and Corollary 6(ii) and solving the resulting equations, we can calculate the values of and , respectively.

4. Explicit Evaluations of

In this section we evaluate explicit values of the function defined in (9) by parametrization method. Cooper and Ye [18] used Ramanujan-Weber class invariants to evaluate .

Theorem 18. For , let

Then

Proof. We set in (9) and use the definition of to complete the proof.

Theorem 19. If is as defined in Theorem 18, then

(i), (ii).

Proof. (i) follows easily from the definition of and (12) or see [21, Theorem 16(ii)]. To prove (ii) we replace by in Theorem 18 and use part (i).

From Theorems 18 and 19(ii) it is clear that if we know the explicit values of then explicit values of and can be determined. Many explicit values of the parameter are evaluated in [21]. In the next theorem we find some explicit values of by employing the explicit values of from [21, Corollaries 32 and 34] in Theorem 18.

Theorem 20. One has the following:

(i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (ix),

where .

Remark 21. (i) The values and are also evaluated in [18] using Ramanujan’s class invariants . The remaining values of in Theorem 20 are new.
(ii) The values and can be used to evaluate new explicit values and , respectively, by appealing to (6) and (8). Similarly, the explicit values and can be used to evaluate new explicit values and , respectively, by appealing to (6) and (8).

Theorem 22. One has the following:

(i), for ,(ii).

Proof. (i) follows from the definition of and (12). (ii) follows from Theorems 18 and 19(ii) or from part (i) with and .

Remark 23. Theorem 22(ii) implies that if we know explicit value of any one of and , then others can be determined. Cooper and Ye [18, pages 107-108, Theorems 4.4, 4.5, and 4.6] evaluated explicit values of for = 13, 69, and 129 by using Weber-Ramanujan class invariants. Employing these values of in Theorem 22(ii) we can evaluate for 169, 897, and 1677, respectively. For example, setting in Theorem 22(ii) and employing the value

from [18, page 108, Theorem 4.6], we evaluate

The explicit values for 169, 897, and 1677 can be used to calculate new explicit values , , and , respectively, by appealing to (6) and (8). Similarly, we can use these values of to evaluate new explicit values ,  , and , respectively, by appealing to (6) and (8).

5. Explicit Evaluations of

Theorem 24. If is as defined in Theorem 4, then

(i), (ii).

Proof. Replacing by in (6) and employing the definition of we arrive at (i). To prove (ii) we replace by in part (i) and use the result from Theorem 5.

Theorem 25. One has the following:

(i), for ,(ii).

Proof. (i) follows from the definition of from (6) with replaced by and (11). (ii) follows from Theorem 24(i) and (ii) or from part (i) with and .

Remark 26. Theorem 25(ii) implies that if we know explicit value of any one of and , then others can be determined. Using Kronecker’s limit formula Cooper and Ye [18, page 109, Theorem 5.2] evaluated for = 15, 31, 55, 231, and 255. Employing these values of in Theorem 25(ii) we can evaluate explicit value of for 3315, respectively. For example, setting in Theorem 25(ii) and employing the value

we evaluate

The values for = 195, 403, 715, 3003, and 3315 can be used to find new explicit values , , , , and , respectively.

Theorem 27. One has the following:

(i), (ii).

Proof. We employ the definition of in (6) to arrive at (i). To prove (ii) we replace by in part (i) and use the result from Theorem 5.

Theorem 28. One has the following:

(i), for ,(ii).

Proof. (i) follows from the definition of from (6) and (10). (ii) follows from Theorem 27(i) and (ii) or from part (i) with and .

Corollary 29. One has the following:

(i), for ,(ii).

Proof. (i) follows from part (i) of Theorems 25 and 28. To prove (ii) we use part (ii) of Theorems 25 and 28.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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