ISRN Discrete MathematicsVolume 2012 (2012), Article ID 956594, 29 pagesdoi:10.5402/2012/956594

Research Article

Explicit Evaluations of Cubic and Quartic Theta-Functions

Nipen Saikia

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

Received 24 February 2012; Accepted 8 April 2012

Academic Editors: H.-J. Kreowski and W. Liu

Copyright © 2012 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 explicit values of cubic and quartic theta-functions and their quotients by parameterizations. In the process, we also find some transformation formulas of these theta-functions.

1. Introduction

For any complex number , define

Ramanujan’s general theta-function is given by where . If we set , , and , where is complex and , then , where [1, page 464] denotes one of the classical theta-functions in its standard notation.

We also define the following three special cases of :

If with , then , where denotes the classical Dedekind eta-function.

In his famous paper [2] and [3, pages 23–39], Ramanujan offered 17 elegant series for and remarked that 14 of these series belong to the “corresponding theories” in which the base in classical theory of elliptic functions is replaced by one or other of the functions: where 3, 4, and 6, where denotes the Gaussian hypergeometric function. In the classical theory, the variable . Ramanujan did not offer any proof of these 14 series for or any of his theorems in the “corresponding” or “alternative” theories. In 1987, J. M. Borwein and P. B. Borwein [4] proved the formulas for . However, in his second notebook [5, Vol. II], Ramanujan recorded, without proof, some of his theorems in alternative theories which were first proved by Berndt et al. [6] in 1995. These theories are now known as the theory of signature , where 3, 4, and 6. In particular, the theories of signature 3 and 4 are called cubic and quartic theories, respectively. An account of this work may also be found in Berndt’s book [7].

In Ramanujan’s cubic theory, the theta-functions , , and are defined by where . These theta-functions were first introduced by J. M. Borwein and P. B. Borwein [8], who also proved that Cubic theta-functions and are related with the Dedekind eta-function by [7, page 109, Lemma 5.1]:

The Borwein brothers [8, ] also established the following three transformation formulas: where . Cooper [9] also found alternate proofs of (1.8)–(1.10).

In quartic theory, Berndt et al. [6] (see also [7, page 146, (9.7)]) established a “transfer” principle of Ramanujan by which formulas in this theory can be derived from those of the classical theory. Taking place of , , and in cubic theory is the functions , , and [10], defined by which also satisfy the equality: Berndt et al. [10] used (1.12) to establish the inversion formula: where is given by (1.4). Therefore, they were able to prove the theorems in the quartic theory directly.

The quartic analogues of (1.7) are given by [10, page 139, Theorem 3.1]

While proving the explicit values of and recorded by Ramanujan in his notebooks, Berndt [7], explicitly determined the value of cubic theta-function [7, page 328, Corollary 3], namely, where is classical [1]. Certain quotients of , , and were also evaluated by Berndt et al. [10] while deriving the series for associated with the theory of signature 4.

In this paper, we find several new explicit values of cubic and quartic theta-functions and their quotients by parameterizations. In the process, we also find some transformation formulas of these theta-functions.

We now define some parameters of Dedekind eta-function and Ramanujan’s theta-functions and . For positive real numbers and , define The parameters and are defined by Yi [11]. She also evaluated several explicit values of and by using eta-function identities and transformation formulas.

In his lost notebook [12, page 212], Ramanujan defined Closely related to is the parameter defined by Ramanathan [13] as

From the definitions of , , , and , we note that and . Ramanujan [12] also provided a list of eleven recorded values of and ten unrecorded values of . All 21 values of and several new were established by Berndt et al. [14]. Yi [11], and Baruah and Saikia [15, 16] also found several new values of parameters and .

In [11], Yi also introduced the following two parameterizations and along with and : where and are positive real numbers. Employing modular transformation formulas and theta-function identities, Yi evaluated several many explicit values of and to find explicit values of and their quotients.

Motivated by Yi’s work, for any positive real numbers and , Baruah and Saikia [17] defined the parameters and by In [17], they proved many properties of the parameterizations and and established their relationship with Yi’s parameters , , , , and Weber-Ramanujan class-invariants and , where and defined by They also found several values of the parameters and .

In Section 2, we record some known values of above parameters, which will be used in this paper.

In Sections 3 and 4, we deal with explicit evaluations of cubic theta-functions and their quotients. In Sections 5 and 6, we find explicit values of the quartic theta-functions and their quotients.

2. Explicit Values of Parameters

Lemma 2.1. If is as defined in (1.16), then For values of , and see [18]. For remaining values we refer to [11] or [17]. We also note that

Lemma 2.2. One has

We refer to [19, page 19, Theorem 5.4] or [11, page 150, Theorem 9.2.4] for proofs of the above assertions.

Lemma 2.3. One has

For proofs (i)–(vi), see [19, page 21, Theorem 5.6] or [11, page 152, Theorem 9.2.6]. For proof of (vii), see [19, page 15, Theorem 4.11] or [11, page 145, Theorems 9.1.10].

Lemma 2.4. One has where

For proofs we refer to [17, page 1781, Theorem 6.7].

3. Theorems on Explicit Evaluation of , , and

In this section, we present some general formulas for the explicit evaluations of cubic theta-functions and their quotients by parameterizations given in Section 1. In the process, we also establish some transformation formulas of quotients of cubic theta-functions.

Theorem 3.1. For any positive real number , one has where and are as defined in (1.16) and (1.19), respectively.

Proof. Using the definitions of and from (1.7), one has Setting and then employing the definitions of and , we finish the proof.

Remark 3.2. Replacing by in Theorem 3.1 and noting that from (2.2), we also have Thus, if we know the value of one quotient of (3.3), then the other quotient follows readily.

From Theorem 3.1 and (1.6), the following theorem is apparent.

Theorem 3.3. One has

Theorem 3.4. For any positive real number , one has

Proof. From the definitions and in (1.7), we observe that Setting in (3.6) and then employing the definition of , we arrive at the desired result.

Remark 3.5. Noting that from (2.2) and using Theorem 3.4, we find that Now, from (3.7), it is obvious that if we know the value of one quotient then the other quotient can easily be evaluated.

In the next theorem, we give a relation between and the parameter as defined in (1.21).

Theorem 3.6. For any positive real number , one has

Proof. From [10, page 111, Lemma 5.5], we note that Now applying the definition of , with , in (3.9), we complete the proof.

The next theorem connects with the parameter defined in (1.16).

Theorem 3.7. For any positive real number , one has

Proof. From [20, page 196, ], we note that where .Setting and then applying (3.3) in (3.11), we obtain which on simplification gives the required result.

Theorem 3.8. One has

Proof. From [7, page 93, (2.8)], one has Setting in (3.14), we readily complete the proof.

Theorem 3.9. For any positive real number , one has

Proof. Setting and in (1.7), we readily arrive at (i) and (ii), respectively.

Theorem 3.10. For all positive real numbers , one has where the parameters and are defined in (1.16) and (1.17), respectively.

Proof. We rewrite in (1.7) as Setting and employing the definition of , we arrive at (i). To prove (ii), we replace by in (3.17) and then use the definition of .

Theorem 3.11. For all positive real number , we have

Proof. It follows readily from (1.7) with and .

Theorem 3.12. For all positive real number , one has

Proof. We set in (1.7) and then employ the definition of the parameter to finish the proof.

4. Explicit Values of , , and

In this section, we find explicit values of cubic theta-functions and their quotients by using the results established in the previous section.

Theorem 4.1. One has

Proof. It follows directly from Theorem 3.1 and the corresponding values of listed in Lemma 2.1.

More values can be calculated by employing Theorem 3.1 and the corresponding values of evaluated in [15, 16].

Theorem 4.2. One has

Proof. It follows easily from (3.3) and the corresponding values of listed in Lemma 2.1.

Theorem 4.3. One has

Proof. It follows from Theorem 3.4 and the corresponding values of in listed in Lemma 2.1.

Theorem 4.4. One has

Proof. We set and 3 in Theorem 3.6 and then employ the values of and from Lemma 2.3(vii) and (iii), respectively, to finish the proof.

For the remaining part of this paper, we set .

Lemma 4.5. One has

For a proof of the lemma, we refer to [7, page 326, Entry 6] and [11, page 125–129].

Theorem 4.6. One has

Proof. To prove (i)–(v), we set , 2, 4, 1/3, and 2/3, respectively, in Theorem 3.9(i) and use the corresponding values of from Lemma 4.5.To prove (vi) and (vii), we set 1 and 2, respectively, in Theorem 3.9(ii) and then use the corresponding values from Lemma 4.5.

Theorem 4.7. One has

Proof. To prove (i)–(v), we set , and 1/3, respectively, in (1.9) and then apply the corresponding values of from Theorem 4.6.To prove (vi), we set in Theorem 3.11 and use the corresponding values of from Lemma 4.5. At last, (vii) follows from Theorems 4.7(v) and 4.4(ii).

Remark 4.8. Setting in (1.10) and then employing the value of from Theorem 4.7(vi), we can also evaluate .

Theorem 4.9. One has

Proof. To prove (i), we set in Theorem 3.7 and use from Lemma 4.5 and the values of from Lemma 2.1.To prove (ii), we set in (1.8) and then employ Theorem 4.9(i).To prove (iii), we set in Theorem 3.8 and then employ the values of and from Theorems 4.9(i) and 4.6(ii), respectively.To prove (iv), we set in (1.8) and use the value of .

5. Theorems on Explicit Evaluations of , , and

In this section, we use the parameters , , and defined in (1.16), (1.20), and (1.23), respectively, to establish some formulas for the explicit evaluations of quartic theta-functions and their quotients.

Theorem 5.1. For any positive real number , one has

Proof. Employing the definition of and given in (1.14), we find that Setting in (5.2) and then using the definition of , we arrive at the first equality. Second equality readily follows from (1.24) and (5.2).

Remark 5.2. From Theorem 5.1 and (2.2), we have the following transformation formula: Thus, if we know the value of one of the quotient of (5.3), then the other one follows immediately.

Theorem 5.3. One has

Proof. Theorem follows easily from (1.14) and the definition of with .

Remark 5.4. Using the fact that in Theorem 5.3, we have the following transformation formula Hence, if we know one quotient of (5.5) then the other quotient follows immediately.

Lemma 5.5. One has where the parameters , , , , and are as defined in (1.16), (1.20), (1.21), (1.23), and (1.24), respectively.

For proofs of (i) and (ii), we refer to [11, page 150] or [19]. For proofs of (iii) and (iv), we refer to [17, Theorem 6.2(ii)] and [17, Theorem 6.3(ii)], respectively.

Theorem 5.6. For any positive real number , one has where is as defined in (1.21).

Proof. From [21, page 39, Entry 24(iii)], we note that Employing (5.8) in (1.14), we obtain Setting in (5.9) and then employing Lemma 5.5(i), we arrive at (i).To prove (ii), we replace by in (i) and employ the result , which is easily derivable from (2.2).

Theorem 5.7. One has where is as defined in (1.21).

Proof. Replacing by in (5.9) and setting , we have Employing Lemma 5.5(ii) in (5.11), we finish the proof of (i).To prove (ii), we replace by in (i) and use the results [19] and .

Remark 5.8. The following transformation formula is apparent from Theorem 5.7(i) and (ii),

Theorem 5.9. For any positive real number , one has where is as defined in (1.23).

Proof. From [21, page 39, Entry 24(iii)], we notice that Thus, from (5.14) and (1.14), we find that Setting in (5.15) and employing Lemma 5.5(iii), we easily complete the proof.

Theorem 5.10. One has

Proof. Applying (5.14) in the definition of given in (1.14) and setting , we find that Now, employing Lemma 5.5(iv) in (5.17), we finish the proof.

6. Explicit Values of Quartic Theta-Functions

In this section, we find explicit values of the quartic theta-functions , , and , and also their quotients by using the results established in the previous section.

Theorem 6.1. One has

Proof. We employ the values of from Lemma 2.1 in Theorem 5.1 to finish the proof.

Theorem 6.2. One has

Proof. It follows easily from Theorem 5.3 and the values of from Lemma 2.1.

Theorem 6.3. One has

Proof. (i)–(vi) follow readily from the first equality of Theorem 5.6(i) and the corresponding values of in Lemma 2.3(i)–(vi), respectively. To prove (vii) and (viii), we employ the corresponding values of listed in Lemma 2.1 to the second equality of Theorem 5.6(i).

Theorem 6.4. One has

Proof. We set 2, 3, and 6, respectively, in Theorem 5.6(ii) and then use the corresponding values of from Lemma 2.1 to complete the proofs.

Theorem 6.5. One has

Proof. We employ the values of given in Lemma 2.2 in Theorem 5.7(i) to finish the proof.

Theorem 6.6. One has

Proof. We use the values of from Lemma 2.2 in Theorem 5.7(ii).

Theorem 6.7. One has where is given in Lemma 2.4(vii).

Proof. The proof of the theorem follows from Theorem 5.9 and the values of from Lemma 2.4.

Theorem 6.8. One has

Proof. We set , and 6 in Theorem 5.10 and then employ the corresponding values of listed in Lemma 2.1.

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