`Journal of MathematicsVolume 2013 (2013), Article ID 620756, 5 pageshttp://dx.doi.org/10.1155/2013/620756`
Research Article

A Product of Theta-Functions Analogous to Ramanujan's Remarkable Product of Theta-Functions and Applications

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

Received 24 September 2012; Accepted 12 November 2012

Copyright © 2013 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 define a product for any positive real numbers and involving Ramanujan's theta-functions and which is analogous to Ramanujan's remarkable product of theta-functions recorded by Ramanujan (1957) and study its several properties. We prove general theorems for the explicit evaluations of and find some explicit values. As application of the product , we also offer explicit formulas for explicit values of Ramanujan's continued fraction in terms of and give examples.

1. Introduction

Ramanujan’s theta-functions , , and are defined as where .

On page 338 of his first notebook, Ramanujan [1] defined the remarkable product of theta-functions as where and are positive real numbers. He then, on pages 338 and 339, offered a list of eighteen particular values of the product . All these eighteen values are proved by Berndt et al. [2]. An account of these can be found in Berndt’s book [3]. Naika and Dharmendra [4] also established some general theorems for explicit evaluations of the product and found some new explicit values therefrom. Further results on can be found in [5, 6].

In [7], Mahadeva Naika et al. defined the product They established general theorems for explicit evaluations of and obtained some particular values. Mahadeva Naika et al. [8] established general formulas for explicit values of Ramanujan’s cubic continued fraction in terms of the products and defined above, where and found some particular values of .

Motivated by the above work, in this paper, we define the product of theta-functions as where and are positive real numbers. We establish several properties of the product . We prove general formulas for explicit evaluations of and find its explicit values. As application of the product , we also offer explicit formulas for explicit values of Ramanujan’s cubic continued fraction in terms of and give examples.

In Section 2, we collect some preliminary results. In Section 3, we prove several properties of the product . Section 4, is devoted to find explicit values of . Finally in Section 5, we offer explicit formulas for explicit evaluations of continued fraction in terms of with examples.

To end this introduction, we define Ramanujan’s modular equation. Let , , and denote the complete elliptic integrals of the first kind associated with the moduli , , , and , respectively. Suppose that the equality holds for some positive integer . Then a modular equation of degree is a relation between the moduli and which is implied by (7). Ramanujan recorded his modular equations in terms of and , where and . We say that has degree over . By denoting , where ,  , the multiplier connecting and is defined by .

2. Preliminary Results

Lemma 1 (see [9, p. 43, Entry 27]). If then

Lemma 2 (see [10, p. 1049, (1.13)]). If then

Lemma 3 (see [9, p. 122, Entry ]). One has

Lemma 4 (see [9, p. 123, Entry ]). One has

Lemma 5 (see [9, p. 124, Entry , ]). One has One also notes that if we replace by in the Lemmas 3 and 4, then and will be replaced by and , respectively, where has degree over .

Lemma 6 (see [9, p. 345, Entry , ]). If is as defined in (5), then

Lemma 7 (see [9, p. 347]). If is as defined in (5), then

Lemma 8 (see [9, p. 231, Entry ]). Let and , then where has degree 3 over .

Lemma 9 (see [9, p. 282, Entry ]). Let and , then where has degree 5 over .

3. Some Properties of

In this section, we study some properties of the product .

Theorem 10. For all positive real numbers and , one has (i), (ii), (iii).

Proof. Using the definition of and Lemmas 1 and 2, we easily arrive at (i). Replacing by in and using Lemmas 1 and 2, we find that which completes the proof of (ii). To prove (iii), we interchange and in .

Remark 11. By using the definitions of , , and , it can be seen that has positive real value and that the values of increases as increases when . Thus, by Theorem 10(i), for all if .

Theorem 12. For all positive real numbers k, m, and n, one has

Proof. Using the definition of , we find that Applying Lemmas 1 and 2 in the denominator of right hand side of (20) and simplifying using Theorem 10(ii) and (iii), we complete the proof.

Corollary 13. For all positive real numbers k and n, one has

Proof. Setting in Theorem 12 and simplifying using Theorem 10(ii), we obtain Replacing by , we complete the proof.

Theorem 14. Let k, a, b, c, and d be positive real numbers such that . Then

Proof. From the definition of and using for positive real numbers , and , we deduce that Rearranging the terms in (24) we arrive at the desired result.

Corollary 15. For any positive real numbers and , one has

Proof. The result follows immediately from Theorem 14 with and .

Theorem 16. For all positive real numbers , , , and , one has

Proof. Applying Theorem 10(iii) in Theorem 12, we deduce that, for all positive real numbers , , and Now setting and again employing Theorems 12 and 10(iii) in (27), we complete the proof.

Theorem 17. For all positive real numbers and , one has (i), (ii), (iii).

Proof. By using Theorems 10(ii) and 16, we find that So we complete the proof of (i). Setting in (27), we find that Now (ii) follows from (28) and (29) and Theorem 10(ii).
By using Theorems 10(ii) and 12, we find that Similarly, we find that From (30), (31), and Theorem 10(ii) and (iii), we complete the proof of (iii).

4. Explicit Values of

In this section, we prove general theorems for explicit evaluations of and find some explicit values. First we define Ramanujan’s class invariants. Ramanujan’s two class invariants and which are defined by where and is a positive rational number. Employing Lemma 5 in (32) it follows that Also, if has degree over , then In his notebooks [1] and paper [11], Ramanujan recorded a total of 116 class invariants or monic polynomials satisfied by them. An account of these can also be found in Berndt’s book [3]. For further references, see [2, 1217].

Theorem 18. One has

Proof. Employing Lemmas 3 and 4 in the definition of , we find that Again from (32) and (33) with , we find that Combining (36) and (37), we complete the proof.

From Theorem 18, it is obvious that if we know or then explicit values of can easily be determined. For example, we find some values of in next theorem by employing the values of or from literature.

Theorem 19. One has(i), (ii), (iii), (iv)(v), (vi), (vii), (viii), (ix).

Proof. For (i)–(vi), the corresponding values of can be found in equations , , , , , and , respectively of [12]. For (vii) to (ix), the corresponding values of can be found in equations , , and , respectively of [3].

Corollary 20. One has

Proof. We set in Theorem 18 and use the result , to complete the proof.

From Corollary 20, it is obvious explicit values of can easily be determined if the corresponding values of the class invariants are known. We give some examples in the next theorem.

Theorem 21. One has(i), (ii), (iii), (iv), (v), (vi).

Proof. For (i) and (ii), we use the values of and from [18, p. 114-115, Theorem (ii) (vi)]. For (iii)–(vi), we use the corresponding values of from [3, p. 189-193].

Ramanujan’s Schalfli type modular equations of prime degree can also be used to explicit values of the product . We offer two theorems as examples.

Theorem 22. One has

Proof. We use the definitions of class invariants and in Lemma 8, to complete the proof.

Theorem 23. One has

Proof. We use the definitions of class invariants and in Lemma 9, to complete the proof.

5. Evaluations of Ramanujan’s Cubic Continued Fraction

In this section, we prove two formulas of explicit evaluations Ramanujan’s cubic continued fraction in terms of the product . We also give examples.

Theorem 24. One has

Proof. Replacing by in (14) and Lemma 7, we obtain Dividing (42) by (43), setting and simplifying using the definition of , we obtain Solving (2) for and noting that and the fact in Remark 11, we arrive at the desired result.

From Theorem 24, it is clear that to find explicit values of it is enough to know the explicit values of . For example, noting from Theorem 10(i) we evaluate

Theorem 25. One has

Proof. Replacing by and then replacing by in (13) and (15), we obtain Dividing (47) by (48), setting and simplifying by using the definition of , we arrive at Solving (49) for and noting and the fact in Remark 11, we complete the proof.

From Theorem 25, it is obvious that if we know the values of the corresponding values of can easily be evaluated. For example, by noting from Theorem 10(i) we calculate Similarly, one can find explicit values of by using the value of from Theorem 21(vi).

Acknowledgments

The author is thankful to University Grants Commission, New Delhi, India for partially supporting the research work under the grant no. F. No. 41-1394/2012(SR).

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