Table of Contents
Journal of Complex Analysis
Volume 2013, Article ID 538592, 7 pages
http://dx.doi.org/10.1155/2013/538592
Research Article

Some New Explicit Values of Quotients of Theta-Function and Applications to Ramanujan's Continued Fractions

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

Received 14 November 2012; Accepted 23 January 2013

Academic Editor: Nikolai Tarkhanov

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 find some new explicit values of the parameter for positive real numbers and involving Ramanujan's theta-function and give some applications of these new values for the explicit evaluations of Ramanujan's continued fractions. In the process, we also establish two new identities for by using modular equations.

1. Introduction

For , Im, define Ramanujan's theta-function as where [1, page 464] is one of the classical theta-functions.

In his notebook [2, volume I, page 248], Ramanujan recorded several explicit values of theta-functions and its quotients which are proved by Berndt and Chan [3]. They also found some new explicit values. An account of these can also be found in Berndt's book [4]. Yi [5] also evaluated many new values of by finding explicit values of the parameters and for positive real numbers and which are defined by Yi [5] established several properties of these parameters and found their explicit values by appealing to transformation formulas and theta-function identities for . Recently, Saikia [6] found many new explicit values of quotients of by finding explicit values of the parameter which is a particular case of the parameter where . Saikia [6] also established some new theta-function identities for .

In the sequel of the previous work, in this paper we find some new explicit values of the parameters and which are particular cases of the parameter by using some properties of established in [5] and two new theta-function identities for . In addition, we give some applications of these new values of for the explicit evaluations of Ramanujan's continued fractions and defined by The continued fraction is studied by Adiga and Anitha [7]. For explicit evaluations of , see [6]. The continued fraction is called Ramanujan-Gllnitz-Gordon continued fraction [4, page 50]. For further references on , see [8, 9].

In Section 2, we record some preliminary results. Section 3 is devoted to prove two new identities for theta-function . In Section 4, we find some new explicit values of the parameter . In Section 5, we evaluate some new values of the parameter . Finally in Section 6, we give applications of these values of and for the explicit evaluations of Ramanujan's continued fractions and .

We end the introduction by defining Ramanujan's modular equation. The complete elliptic integral of the first kind is defined by where , denotes the ordinary or Gaussian hypergeometric function, and The number is called the modulus of , and is called the complementary modulus. 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).

If we set we see that (7) is equivalent to the relation . Thus, a modular equation can be viewed as an identity involving theta-functions at the arguments and . Ramanujan recorded his modular equations in terms of and , where and . We say that has degree over . The multiplier connecting and is defined by where .

2. Preliminary Results

Lemma 1 (see [5, page 385, Theorem 2.2]). For positive real numbers and , (i), (ii).

Lemma 2 (see [5, page 387, Corollary 2.6]). For positive real numbers and ,

Lemma 3 (see [5, page 392, Theorem 4.6]). One has for any positive real number n.

Lemma 4 (see [10, page 122, Entry 10 (i) and (v)]). One has(i), (ii).

Lemma 5 (see [10, page 280-281, Entry 13(vii)]). If has degree 5 over , then

Lemma 6 (see [10, page 314, Entry 19(i)]). If has degree 7 over , then

3. Two New Identities for

In this section, we prove two new identities for theta-function by using Ramanujan's modular equations and transformation formulas.

Theorem 7. If , , then

Proof. Transcribing and using Lemma 4(i) and (iv) and then simplifying, we get where has degree 5 over .
Equivalently, Now by Lemma 5, we have Squaring (16) and simplifying, we arrive at where Squaring (17) and simplifying, we obtain where .
Squaring (19) and simplifying, we obtain Again squaring (20), we obtain Now employing (14) and (15) and factorizing using Mathematica, we deduce that where By examining the behavior of the first factor and the last factor of the left-hand side of (22) near , it can be seen that there is a neighborhood about the origin, where these factors are not zero. Then the second factor is zero in this neighborhood. By the identity theorem, this factor is identically zero. Hence, we complete the proof.

Theorem 8. If , , then

Proof. Transcribing and using Lemma 4(i) and (iv) and then simplifying, we obtain where has degree 7 over .
Equivalently, Now by Lemma 6, we have Squaring (27) and simplifying, we obtain where .
Squaring (28) and simplifying, we obtain Squaring (29) and simplifying, we obtain Again, squaring (30), we arrive at Employing (25) and (26) in (31) and simplifying with the help of Mathematica, we complete the proof.

4. New Values of

In this section, we find some new values of and by using theta-function identities proved in Section 3 and the properties of listed in Lemmas 1 and 2. We begin with following remarks.

Remark 9. The values of are real and for all if . We also note that the values of decrease as increases when . In view of this, in the following theorem we have where and are evaluated in [6, Theorem 4.3]. Yi [5] also evaluated the value of .

Theorem 10. One has(i), (ii), (iii), (iv), (v), (vi), (vii), (viii),
 where and  −  +  − .

Proof. For (i) and (ii), setting and employing the definition of in Theorem 7, we get Setting in (32), applying to (13), and then simplifying using Lemma 1(ii), we obtain Equivalently, where Solving (34) for and noting that has positive real value greater than 1, we obtain Invoking (36) in (35), solving for , and using the fact in Remark 9, we complete the proof of (i). Noting from Lemma 1(ii), we arrive at (ii).
For proofs of (iii) and (iv), we set in (32), applying to (13) and then simplifying using Lemma 1(i), we arrive at
Equivalently, where Solving (38) for and noting that has positive real value greater than 1, we obtain Invoking (40) in (39), solving for , and using the fact in Remark 9, we complete the proof of (iii). Noting from Lemma 1(iv), we prove (ii).
To prove (v) and (vi), applying the definition of in Theorem 8, we get Setting in (41), applying in (24), and then simplifying using Lemma 1(ii), we arrive at Dividing (42) by and simplifying, we get Equivalently, where Solving (44) by using Mathematica and noting that has positive real value greater that 1 satisfying the fact in Remark 9, we obtain Employing (46) in (45), solving for , and using the fact in Remark 9, we complete the proof of (v). Noting , we arrive at (vi).
For proofs of (vii) and (viii), setting in (41), applying in (24), and simplifying using Lemma 1(ii), we arrive at Equivalently, where Solving (48) for and noting that has positive real value greater than 1, we obtain where and  −  +  − .
Invoking (50) in (49), solving for , and using the fact in Remark 9, we arrive at (vi). Noting from Lemma 1(ii), we complete the proof of (vii).

5. New Values of

In this section, we find some new values of the parameter by using the values of evaluated in Section 4 and in [6].

Theorem 11. One has(i), (ii), (iii), (iv).

Proof. Setting and in Lemma 2, we deduce that From [6, page 174, Theorem 4.3(vii)], we have Combining (51) and (52), we obtain Next, setting in Lemma 3 and simplifying using Lemma 1(ii), we obtain Invoking (53) in (54) and simplifying, we deduce that Multiplying (53) and (55) and simplifying, we complete the proof of (i). Dividing (53) by (55) and simplifying, we arrive at (ii). (iii) and (iv) follow from (i) and (ii), respectively, and Lemma 1(ii).

The proofs of Theorems 1215 are identical to the proof of Theorem 11. So we omit details and give only references of the required results to prove them.

Theorem 12. One has(i), (ii), (iii), (iv),
 where .

Proof of Theorem 12 follows from Theorem 10(i), Lemma 2 with and , Lemma 1(ii), and Lemma 3 with .

Theorem 13. One has(i), (ii), (iii), (iv),
   where .

To prove Theorem 13, we use Theorem 10(iii), Lemma 2 with and , Lemma 1(ii), and Lemma 3 with .

Theorem 14. One has(i), (ii), (iii), (iv).

We employ Theorem 10(v), Lemma 2 with and , Lemma 1(ii), and Lemma 3 with to prove Theorem 14.

Theorem 15. One has(i), (ii), (iii), (iv),
 where a and b are given in Theorem 10(viii).

Proof follows from Theorem 10(vii), Lemma 2 with and , Lemma 1(ii), and Lemma 3 with .

6. Applications of and

In this section, we use the new values of the parameters and to find explicit values of continued fractions and defined in (3) and (4), respectively.

The parameter is useful in finding explicit values of the continued fraction . If we know values of the parameter for any positive real number , then explicit values of can be calculated by appealing to the following theorem.

Theorem 16 (see [6, page 177, Theorem 5.1]). One has where is any positive real number.

For example, employing the value of from Theorem 10(i) in Theorem 16, we obtain Similarly, we can find new values of , , , , ,  , and by employing the values of from Theorem 10(ii)–(viii), respectively, in Theorem 16. Since it is a routine calculation, we omit details.

Next, the parameter is connected to continued fraction by the following theorem.

Theorem 17 (see [8, page 281, Theorem 4.1]). For any positive real number n, one has

From Theorem 17, we note that if the values of are known, then the values of can easily be evaluated. For example, using the value of from Theorem 11(i) in Theorem 17, we evaluate

Similarly, we can evaluate new values of for = 3/2, 1/6, 2/3, 10, 5/2, 1/10, 2/5, 50, 25/2, 1/50, 2/25, 14, 7/2, 1/14, 2/7, 98, 49/2, and 2/49 by using Theorem 17 and the values of evaluated in Theorems 1115.

Acknowledgment

The author is thankful to the 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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