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.
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  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  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.  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 .
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 , 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  also found alternate proofs of (1.8)–(1.10).
In quartic theory, Berndt et al.  (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 , defined by
which also satisfy the equality:
Berndt et al.  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 , explicitly determined the value of cubic theta-function [7, page 328, Corollary 3], namely,
where is classical . Certain quotients of , , and were also evaluated by Berndt et al.  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 . 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  as
From the definitions of , , , and , we note that and . Ramanujan  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. . Yi , and Baruah and Saikia [15, 16] also found several new values of parameters and .
In , 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  defined the parameters and by
In , 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 . For remaining values we refer to  or . 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
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
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.