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ISRN Discrete Mathematics
Volume 2012 (2012), Article ID 956594, 29 pages
http://dx.doi.org/10.5402/2012/956594
Research Article

Explicit Evaluations of Cubic and Quartic Theta-Functions

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 𝑎, define (𝑎;𝑞)𝑛=(𝑎;𝑞)(𝑎𝑞𝑛;𝑞),and(𝑎;𝑞)=𝑘=11𝑎𝑞𝑘1.(1.1)

Ramanujan’s general theta-function 𝑓(𝑎,𝑏) is given by 𝑓(𝑎,𝑏)=𝑘=𝑎𝑘(𝑘+1)/2𝑏𝑘(𝑘1)/2,(1.2) where |𝑎𝑏|<1. If we set 𝑎=𝑞𝑒2𝑖𝑧, 𝑏=𝑞𝑒2𝑖𝑧, and 𝑞=𝑒𝜋𝑖𝜏, where 𝑧 is complex and Im(𝜏)>0, then 𝑓(𝑎,𝑏)=𝜗3(𝑧,𝜏), where 𝜗3(𝑧,𝜏) [1, page 464] denotes one of the classical theta-functions in its standard notation.

We also define the following three special cases of 𝑓(𝑎,𝑏):𝜙(𝑞)=𝑓(𝑞,𝑞)=𝑛=𝑞𝑛2=𝑞;𝑞2𝑞2;𝑞2𝑞;𝑞2𝑞2;𝑞2,𝜓(𝑞)=𝑓𝑞,𝑞3=𝑘=0𝑞𝑘(𝑘+1)/2=𝑞2;𝑞2𝑞;𝑞2,𝑓(𝑞)=𝑓𝑞,𝑞2=𝑛=(1)𝑛𝑞𝑛(3𝑛1)/2=(𝑞;𝑞).(1.3)

If 𝑞=𝑒2𝜋𝑖𝑧 with Im(𝑧)>0, then 𝑓(𝑞)=𝑞1/24𝜂(𝑧), where 𝜂(𝑧) denotes the classical Dedekind eta-function.

In his famous paper [2] and [3, pages 23–39], Ramanujan offered 17 elegant series for 1/𝜋 and remarked that 14 of these series belong to the “corresponding theories” in which the base 𝑞 in classical theory of elliptic functions is replaced by one or other of the functions:𝑞𝑟=𝑞𝑟𝜋(𝑥)=exp𝜋csc𝑟2𝐹1(1/𝑟,(𝑟1)/𝑟,1,1𝑥)2𝐹1(,1/𝑟,(𝑟1)/𝑟,1,𝑥)(1.4) where 𝑟= 3, 4, and 6, where 2𝐹1 denotes the Gaussian hypergeometric function. In the classical theory, the variable 𝑞=𝑞2. Ramanujan did not offer any proof of these 14 series for 1/𝜋 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 1/𝜋. 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 𝑟, where 𝑟= 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 𝑎(𝑞), 𝑏(𝑞), and 𝑐(𝑞) are defined by𝑎(𝑞)=𝑚,𝑛=𝑞𝑚2+𝑚𝑛+𝑛2,𝑏(𝑞)=𝑚,𝑛=𝑤𝑚𝑛𝑞𝑚2+𝑚𝑛+𝑛2,𝑐(𝑞)=𝑚,𝑛=𝑞(𝑚+1/3)2+(𝑚+1/3)(𝑛+1/3)+(𝑛+1/3)2,(1.5) where 𝑤=exp(2𝜋𝑖/3). These theta-functions were first introduced by J. M. Borwein and P. B. Borwein [8], who also proved that𝑎3(𝑞)=𝑏3(𝑞)+𝑐3(𝑞).(1.6) Cubic theta-functions 𝑏(𝑞) and 𝑐(𝑞) are related with the Dedekind eta-function by [7, page 109, Lemma 5.1]:𝑓𝑏(𝑞)=3(𝑞)𝑓𝑞3,𝑐(𝑞)=3𝑞1/3𝑓3𝑞3𝑓.(𝑞)(1.7)

The Borwein brothers [8, (2.2)] also established the following three transformation formulas:𝑎𝑒2𝜋𝑡=1𝑡3𝑎𝑒2𝜋/3𝑡,𝑏𝑒(1.8)2𝜋𝑡=1𝑡3𝑐𝑒2𝜋/3𝑡𝑐𝑒,(1.9)2𝜋𝑡=1𝑡3𝑏𝑒2𝜋/3𝑡,(1.10) where Re(𝑡)>0. 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 𝑎(𝑞), 𝑏(𝑞), and 𝑐(𝑞) in cubic theory is the functions 𝐴(𝑞), 𝐵(𝑞), and 𝐶(𝑞) [10], defined by 𝐴(𝑞)=𝜙4(𝑞)+16𝑞𝜓4𝑞2,𝐵(𝑞)=𝜙4(𝑞)16𝑞𝜓4𝑞2,𝐶(𝑞)=8𝑞𝜙2(𝑞)𝜓2𝑞2,(1.11) which also satisfy the equality: 𝐴2(𝑞)=𝐵2(𝑞)+𝐶2(𝑞).(1.12) Berndt et al. [10] used (1.12) to establish the inversion formula: 𝑧4=2𝐹114,34=;1;𝑥𝐴(𝑞),(1.13) where 𝑞=𝑞4 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]𝑓𝐵(𝑞)=2(𝑞)𝑓𝑞24,𝐶(𝑞)=8𝑞𝑓2𝑞2𝑓(𝑞)4.(1.14)

While proving the explicit values of 𝜙(𝑞) and 𝜓(𝑞) recorded by Ramanujan in his notebooks, Berndt [7], explicitly determined the value of cubic theta-function 𝑎(𝑒2𝜋) [7, page 328, Corollary 3], namely,𝑎𝑒2𝜋𝜙2(𝑒𝜋)=1(12)1/8,31(1.15) where 𝜙(𝑒𝜋)=𝜋1/4/Γ(3/4) is classical [1]. Certain quotients of 𝐴(𝑞), 𝐵(𝑞), and 𝐶(𝑞) were also evaluated by Berndt et al. [10] while deriving the series for 1/𝜋 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 𝑓(𝑞) and Ramanujan’s theta-functions 𝜙(𝑞) and 𝜓(𝑞). For positive real numbers 𝑛 and 𝑘, define𝑟𝑘,𝑛=𝑓(𝑞)𝑘1/4𝑞(𝑘1)/24𝑓𝑞𝑘,𝑞=𝑒2𝜋𝑛/𝑘,𝑟(1.16)𝑘,𝑛=𝑓(𝑞)𝑘1/4𝑞(𝑘1)/24𝑓𝑞𝑘,𝑞=𝑒𝜋𝑛/𝑘.(1.17) The parameters 𝑟𝑘,𝑛 and 𝑟𝑘,𝑛 are defined by Yi [11]. She also evaluated several explicit values of 𝑟𝑘,𝑛 and 𝑟𝑘,𝑛 by using eta-function identities and transformation formulas.

In his lost notebook [12, page 212], Ramanujan defined𝜆𝑛=133𝑓6(𝑞)𝑞𝑓6𝑞3,𝑞=𝑒𝜋𝑛/3.(1.18) Closely related to 𝜆𝑛 is the parameter 𝜇𝑛 defined by Ramanathan [13] as𝜇𝑛=133𝑓6(𝑞)𝑞𝑓6𝑞3,𝑞=𝑒2𝜋𝑛/3.(1.19)

From the definitions of 𝑟𝑘,𝑛, 𝜇𝑛, 𝑟𝑘,𝑛, and 𝜇𝑛, we note that 𝑟63,𝑛=𝜆𝑛 and 𝑟63,𝑛=𝜇𝑛. Ramanujan [12] also provided a list of eleven recorded values of 𝜆𝑛 and ten unrecorded values of 𝜆𝑛. All 21 values of 𝜆𝑛 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 𝜆𝑛 and 𝜇𝑛.

In [11], Yi also introduced the following two parameterizations 𝑘,𝑛 and 𝑘,𝑛 along with 𝑟𝑘,𝑛 and 𝑟𝑘,𝑛: 𝑘,𝑛=𝜙(𝑞)𝑘1/4𝜙𝑞𝑘,𝑞=𝑒𝜋𝑛/𝑘,(1.20)𝑘,𝑛=𝜙(𝑞)𝑘1/4𝜙𝑞𝑘,𝑞=𝑒2𝜋𝑛/𝑘,(1.21) where 𝑘 and 𝑛 are positive real numbers. Employing modular transformation formulas and theta-function identities, Yi evaluated several many explicit values of 𝑘,𝑛 and 𝑘,𝑛 to find explicit values of 𝜙(𝑞) and their quotients.

Motivated by Yi’s work, for any positive real numbers 𝑘 and 𝑛, Baruah and Saikia [17] defined the parameters 𝑔𝑘,𝑛 and 𝑔𝑘,𝑛 by𝑔𝑘,𝑛=𝜓(𝑞)𝑘1/4𝑞(𝑘1)/8𝜓𝑞𝑘,𝑞=𝑒𝜋𝑛/𝑘,𝑔(1.22)𝑘,𝑛=𝜓(𝑞)𝑘1/4𝑞(𝑘1)/8𝜓𝑞𝑘,𝑞=𝑒𝜋𝑛/𝑘.(1.23) In [17], they proved many properties of the parameterizations 𝑔𝑘,𝑛 and 𝑔𝑘,𝑛 and established their relationship with Yi’s parameters 𝑟𝑘,𝑛, 𝑟𝑘,𝑛, 𝑘,𝑛, 𝑘,𝑛, and Weber-Ramanujan class-invariants 𝐺𝑛 and 𝑔𝑛, where 𝐺𝑛 and 𝑔𝑛 defined by𝐺𝑛=21/4𝑞1/24𝑞;𝑞2,𝑔𝑛=21/4𝑞1/24𝑞;𝑞2;𝑞=𝑒𝜋𝑛.(1.24) They also found several values of the parameters 𝑔𝑘,𝑛 and 𝑔𝑘,𝑛.

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 𝑟𝑘,𝑛 is as defined in (1.16), then 𝑟1,1=1,𝑟2,1=1,𝑟2,2=21/8,𝑟2,3=1+21/6,𝑟2,4=21/81+21/8,𝑟2,5=1+52,𝑟2,6=21/243+11/4,𝑟2,7=2+1+22121/2,𝑟2,8=23/161+21/4,𝑟2,9=2+31/3,𝑟2,10=121+55+1+21/4,𝑟2,12=1+25/2421+2+61/8,𝑟2,16=21/81+21/44+2+1021/8,𝑟2,18=1+31/31+3+233/41/3211/24,𝑟2,20=1+55/82+32+51/82,𝑟2,27=1+25/182+21+21/3+1+22/31/3,𝑟2,32=23/161+21/416+1521/4+122+923/41/8,𝑟2,36=21+3522831/8322/3,𝑟2,49=1+7+21422+14+7+2142,𝑟2,50=25/851/4,𝑟12,72=2+31/32+4+23+33/43+11/3213/48215/12,𝑟2,3/2=1+31/427/24,𝑟2,5/2=5+1+21/421/4,𝑟2,7/2=3+71/423/8,𝑟2,9/2=1+3+233/41/3213/24,𝑟2,25/2=51/4+125/8,𝑟2,27/2=1+31/1213+22/331/323/821/311/3,𝑟2,63/2=723+21+3+33+16212771/3213/24312/3371/12,𝑟2,9/4=1+352+2831/821/82+31/3,𝑟2,9/8=25/48215/12311/3321/312+3+33/4261/3,𝑟3,3=31/123+231/12=31/81+31/621/12,𝑟3,4=3+12,𝑟3,5=5+125/6=11+5521/6,𝑟3,7=3+72231/4,𝑟3,8=2+11/32+31/4,𝑟3,9=31/61+21/3+22/31/3=31/621/311/3,𝑟3,18=31/61+25/182+21+21/3+1+22/31/3,𝑟3,25=121+310+5+2310+3102,𝑟3,49=3+32237+32372+49+1332237+83237223,𝑟4,4=25/161+21/4,𝑟4,5=1+5+2+1+521/2,𝑟4,8=21/41+23/84+2+1021/8,𝑟4,9=121+243+3,𝑟4,7=8+371/4,𝑟4,9=12+31/42+32,𝑟4,25=123+45+5+453=45+14,𝑟514,49=144+7+21+87+7+21+872,𝑟5,5=25+1051/6=5+52,𝑟6,6=31/83+11+3+233/41/3213/24.(2.1) For values of 𝑟4,7,𝑟4,9, and 𝑟4,49 see [18]. For remaining values we refer to [11] or [17].
We also note that 𝑟𝑘,1=1,𝑟𝑘,1/𝑛=1𝑟𝑘,𝑛,𝑟𝑘,𝑛=𝑟𝑛,𝑘.(2.2)

Lemma 2.2. One has (i)1,1=1,(ii)2,2=222,(iii)3,3=2331/4=31/83121/4,(iv)4,4=23/44,2+1(v)5,5=52(5,vi)6,6=23/431/821311/64+32+35/4+2333/4+2233/41/3.(2.3)

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 (i)1,1=1,(ii)2,2=21/16211/4,(iii)3,3=21/331/8311/61+3+24331/3,(iv)4,4=21/416+1542+122+94231/8,(v)5,5=124515+5,(vi)6,6=21/431/8211/123+11/613+233/41/3232+35/4+33/41/3,(vii)3,1=21/431.(2.4)

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 (i)𝑔1,1=1,(ii)𝑔2,2=23/8,(iii)𝑔3,3=31/31+3+233/41/31+31/62,(iv)𝑔4,4=23/81+21/2,(v)𝑔5,5=5+51/251/4+12,(vi)𝑔6,6=31/81+35/61+3+233/42/3229/24,(vii)𝑔9,9=𝑎+(2(𝑏2𝑐))1/3+(2(𝑏+2𝑐))1/32,(2.5) where 𝑎=2+231/4+23+233/4,𝑏=82+452+483+25233/4,𝑐=388+47231/4+503+27233/4.(2.6)

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

3. Theorems on Explicit Evaluation of 𝑎(𝑞), 𝑏(𝑞), and 𝑐(𝑞)

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 𝑛, one has 𝑏𝑒2𝜋𝑛/3𝑐𝑒2𝜋𝑛/3=𝑟43,𝑛=𝜇𝑛2/3,(3.1) where 𝑟𝑘,𝑛 and 𝜇𝑛 are as defined in (1.16) and (1.19), respectively.

Proof. Using the definitions of 𝑏(𝑞) and 𝑐(𝑞) from (1.7), one has 3𝑏(𝑞)𝑐=(𝑞)𝑓(𝑞)𝑞1/12𝑓𝑞34.(3.2) Setting 𝑞=𝑒2𝜋𝑛/3 and then employing the definitions of 𝑟𝑘,𝑛 and 𝜇𝑛, we finish the proof.

Remark 3.2. Replacing 𝑛 by 1/𝑛 in Theorem 3.1 and noting that 𝑟3,1/𝑛=1/𝑟3,𝑛 from (2.2), we also have 𝑏𝑒2𝜋𝑛/3𝑐𝑒2𝜋𝑛/3=𝑐𝑒2𝜋/3𝑛𝑏𝑒2𝜋/3𝑛.(3.3) 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 𝑎𝑒2𝜋𝑛/3𝑐𝑒2𝜋𝑛/3=𝑟123,𝑛+11/3.(3.4)

Theorem 3.4. For any positive real number 𝑛, one has 𝑏𝑒2𝜋𝑛𝑐𝑒2𝜋𝑛/3=𝑟9,𝑛3.(3.5)

Proof. From the definitions 𝑏(𝑞) and 𝑐(𝑞) in (1.7), we observe that 𝑏𝑞3=𝑐(𝑞)𝑓(𝑞)3𝑞1/3𝑓𝑞9.(3.6) Setting 𝑞=𝑒2𝜋𝑛/3 in (3.6) and then employing the definition of 𝑟𝑘,𝑛, we arrive at the desired result.

Remark 3.5. Noting that 𝑟9,1/𝑛=1/𝑟9,𝑛 from (2.2) and using Theorem 3.4, we find that 𝑒3𝑏2𝜋𝑛𝑐𝑒2𝜋𝑛/3=𝑐𝑒2𝜋/3𝑛𝑏𝑒2𝜋/𝑛.(3.7) 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 𝑐(𝑞) and the parameter 𝑘,𝑛 as defined in (1.21).

Theorem 3.6. For any positive real number 𝑛, one has 𝑐𝑒8𝜋𝑛/3𝑐𝑒2𝜋𝑛/3=14133,𝑛2.(3.8)

Proof. From [10, page 111, Lemma 5.5], we note that 𝑐𝑞144𝑐=(𝑞)𝜙(𝑞)𝜙𝑞32.(3.9) Now applying the definition of 𝑘,𝑛, with 𝑘=3, in (3.9), we complete the proof.

The next theorem connects 𝑎(𝑞) with the parameter 𝑟𝑘,𝑛 defined in (1.16).

Theorem 3.7. For any positive real number 𝑛, one has 𝑎12𝑒2𝜋𝑛/3=𝑟27123,𝑛+14𝑒2𝜋𝑛/3𝑓24𝑒2𝜋𝑛/3𝑟363,𝑛.(3.10)

Proof. From [20, page 196, (2.9)], we note that 27𝑞𝑓24(𝑞)=𝑎12(𝑞)(1𝛼(𝑞))3𝛼(𝑞),(3.11) where 𝛼(𝑞)=𝑐3(𝑞)/𝑎3(𝑞).
Setting 𝑞=𝑒2𝜋𝑛/3 and then applying (3.3) in (3.11), we obtain 27𝑒2𝜋𝑛/3𝑓24𝑒2𝜋𝑛/3=𝑎12𝑒2𝜋𝑛/311𝑟123,𝑛+131𝑟123,𝑛,+1(3.12) which on simplification gives the required result.

Theorem 3.8. One has 𝑎𝑒3𝑛𝜋=13{𝑎(𝑒𝑛𝜋)+2𝑏(𝑒𝑛𝜋)}.(3.13)

Proof. From [7, page 93, (2.8)], one has 1𝑏(𝑞)=2𝑞3𝑎3.𝑎(𝑞)(3.14) Setting 𝑞=𝑒𝑛𝜋 in (3.14), we readily complete the proof.

Theorem 3.9. For any positive real number 𝑛, one has (i)𝑏(𝑒𝑛𝜋𝑓)=3(𝑒𝑛𝜋)𝑓𝑒3𝑛𝜋,(ii)𝑏(𝑒𝑛𝜋𝑓)=3(𝑒𝑛𝜋)𝑓𝑒3𝑛𝜋.(3.15)

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

Theorem 3.10. For all positive real numbers 𝑛, one has 𝑒(i)𝑏2𝜋𝑛/3=31/4𝑒𝜋𝑛/63𝑓2𝑒2𝜋𝑛/3𝑟3,𝑛,(ii)𝑏𝑒𝜋𝑛/3=31/4𝑒𝜋𝑛/123𝑓2𝑒𝜋𝑛/3𝑟3,𝑛,(3.16) where the parameters 𝑟3,𝑛 and 𝑟3,𝑛 are defined in (1.16) and (1.17), respectively.

Proof. We rewrite 𝑏(𝑞) in (1.7) as 𝑏(𝑞)=𝑓2(𝑞)𝑞1/12𝑓(𝑞)𝑞1/12𝑓𝑞3.(3.17) Setting 𝑞=𝑒2𝜋𝑛/3 and employing the definition of 𝑟3,𝑛, we arrive at (i). To prove (ii), we replace 𝑞 by 𝑞 in (3.17) and then use the definition of 𝑟3,𝑛.

Theorem 3.11. For all positive real number 𝑛, we have (i)𝑐(𝑒𝑛𝜋)=3𝑒𝑛𝜋/3𝑓3𝑒3𝑛𝜋𝑓(𝑒𝑛𝜋),(ii)𝑐(𝑒𝑛𝜋)=3𝑒𝑛𝜋/3𝑓3𝑒3𝑛𝜋𝑓(𝑒𝑛𝜋).(3.18)

Proof. It follows readily from (1.7) with 𝑞=𝑒𝑛𝜋 and 𝑞=𝑒𝑛𝜋.

Theorem 3.12. For all positive real number 𝑛, one has 𝑐𝑒2𝜋𝑛/3=33/4𝑒𝜋𝑛/23𝑓2𝑒2𝜋3𝑛𝑟3,𝑛.(3.19)

Proof. We set 𝑞=𝑒2𝜋𝑛/3 in (1.7) and then employ the definition of the parameter 𝑟𝑘,𝑛 to finish the proof.

4. Explicit Values of 𝑎(𝑞), 𝑏(𝑞), and 𝑐(𝑞)

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 𝑏𝑒(i)2𝜋/3𝑐𝑒2𝜋/3𝑏𝑒=1,(ii)2𝜋2/3𝑐𝑒2𝜋2/3=1+22/3,𝑏𝑒(iii)2𝜋𝑐𝑒2𝜋=31/21+32/321/3,𝑏𝑒(iv)4𝜋/3𝑐𝑒4𝜋/3=1+322,𝑏𝑒(v)2𝜋5/3𝑐𝑒2𝜋5/3=1+5210/3,𝑏𝑒(vi)2𝜋7/3𝑐𝑒2𝜋7/3=3+7223,𝑏𝑒(vii)4𝜋2/3𝑐𝑒4𝜋2/3=1+24/32+3,𝑏𝑒(viii)2𝜋3𝑐𝑒2𝜋3=32/321/314/3,(𝑏𝑒ix)6𝜋2/3𝑐𝑒6𝜋2/3=32/31+210/92+21+21/3+1+22/34/3,𝑏𝑒(x)2𝜋13/3𝑐𝑒2𝜋13/3=11+13+3+13224,𝑏𝑒(xi)10𝜋/3𝑐𝑒10𝜋/3=1161+310+5+2310+31024,𝑏𝑒(xi)14𝜋/3𝑐𝑒14𝜋/3=3+32237+32372+49+1332237+832372234.(4.1)

Proof. It follows directly from Theorem 3.1 and the corresponding values of 𝑟3,𝑛 listed in Lemma 2.1.

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

Theorem 4.2. One has 𝑎𝑒(i)2𝜋/3𝑐𝑒2𝜋/3=3𝑎𝑒2,(ii)2𝜋2/3𝑐𝑒2𝜋2/3=21/32+21/3,𝑎𝑒(iii)2𝜋𝑐𝑒2𝜋=33/21+322+11/3,𝑎𝑒(iv)4𝜋/3𝑐𝑒4𝜋/3=1+326+11/3,𝑎𝑒(v)2𝜋5/3𝑐𝑒2𝜋5/3=1+5210+11/3,𝑎𝑒(vi)2𝜋7/3𝑐𝑒2𝜋7/3=3+72233+11/3,𝑎𝑒(vii)4𝜋2/3𝑐𝑒4𝜋2/3=1+242+34+11/3,𝑎𝑒(viii)10𝜋/3𝑐𝑒10𝜋/3=1+316+5+2310+3102212+11/3,𝑎𝑒(ix)6𝜋/3𝑐𝑒6𝜋/3=921/414+11/3,𝑎𝑒(x)14𝜋/3𝑐𝑒14𝜋/3=3+3237+32372+49+133237+83237223+11/3,𝑎𝑒(xi)6𝜋𝑐𝑒6𝜋=341+210/32+21+24+1+28+11/3.(4.2)

Proof. It follows easily from (3.3) and the corresponding values of 𝑟3,𝑛 listed in Lemma 2.1.

Theorem 4.3. One has 𝑏𝑒(i)2𝜋𝑐𝑒2𝜋/3=13,𝑏𝑒(ii)2𝜋2𝑐𝑒2𝜋2/3=3+21/33,𝑏𝑒(iii)2𝜋3𝑐𝑒2𝜋/3=133211/3,𝑏𝑒(iv)4𝜋𝑐𝑒4𝜋/3=1231+231/4+3,𝑏𝑒(v)25𝜋𝑐𝑒25𝜋/3=13104+603+455+26151/6.(4.3)

Proof. It follows from Theorem 3.4 and the corresponding values of 𝑟9,𝑛 in listed in Lemma 2.1.

Theorem 4.4. One has 𝑐𝑒(i)8𝜋/3𝑐𝑒2𝜋/3=142+332,𝑐𝑒(ii)8𝜋𝑐𝑒2𝜋=141322/331/4311/31+3+24332/3.(4.4)

Proof. We set 𝑛=1 and 3 in Theorem 3.6 and then employ the values of 3,1 and 3,3 from Lemma 2.3(vii) and (iii), respectively, to finish the proof.

For the remaining part of this paper, we set 𝑎=𝜙(𝑒𝜋)=𝜋1/4/Γ(3/4).

Lemma 4.5. One has (i)𝑓(𝑒𝜋)=𝑎23/8𝑒𝜋/24,(ii)𝑓(𝑒𝜋)=𝑎21/4𝑒𝜋/24,(iii)𝑓𝑒2𝜋=𝑎21/2𝑒𝜋/12,(iv)𝑓𝑒3𝜋=𝑎𝑒𝜋/41+3+233/41/333/8217/241+31/6,(v)𝑓𝑒4𝜋=𝑎27/8𝑒𝜋/6,(vi)𝑓𝑒6𝜋=𝑎27/1233/8𝑒𝜋/4311/4,(vii)𝑓𝑒12𝜋=𝑎𝑒𝜋/225/2433/81+31+3+233/41/3,(viii)𝑓𝑒𝜋/3=𝑎27/2431/8𝑒𝜋/721+31+3+233/41/3,(ix)𝑓𝑒2𝜋/3=𝑎27/1231/8𝑒𝜋/36311/6,𝑒(x)𝑓2𝜋=𝑎213/16𝑒𝜋/122+11/4,𝑒(xi)𝑓3𝜋=𝑎21/333/8𝑒𝜋/83+11/6,𝑒(xii)𝑓6𝜋=𝑎𝑒𝜋/4232+35/4+33/41/3215/1633/8211/123+11/6.(4.5)

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

Theorem 4.6. One has (i)𝑏(𝑒𝜋𝑎)=233/81+31/625/121+3+233/41/3,𝑒(ii)𝑏2𝜋=𝑎233/8211/12311/6,𝑒(iii)𝑏4𝜋=𝑎2229/1233/81+31/21+3+233/41/3,𝑒(iv)𝑏𝜋/3=𝑎225/433/81+33/21+3+233/4,𝑒(v)𝑏2𝜋/3=𝑎231/8311/3213/123+11/6,(vi)𝑏(𝑒𝜋𝑎)=233/825/123+11/6,(vii)𝑏𝑒2𝜋=𝑎233/82+13/4211/123+11/623/2232+35/4+33/41/3.(4.6)

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

Theorem 4.7. One has 𝑒(i)𝑐4𝜋/3=𝑎237/81+31/6217/121+3+233/41/3,𝑒(ii)𝑐2𝜋/3=𝑎2213/1237/83+11/6,(𝑒iii)𝑐𝜋/3=217/1237/8𝑎21+31/21+3+233/41/3,𝑒(iv)𝑐4𝜋=𝑎233/821/41+33/21+3+233/4,𝑒(v)𝑐2𝜋=𝑎2311/333/8213/123+11/6,(vi)𝑐(𝑒𝜋𝑎)=21+3+233/431/827/41+31/2,𝑒(vii)𝑐8𝜋=141322/331/4311/31+3+2+4322/3×𝑎2311/333/8213/123+11/6.(4.7)

Proof. To prove (i)–(v), we set 𝑡=1/2,1,2,1/6, and 1/3, respectively, in (1.9) and then apply the corresponding values of 𝑏(𝑒𝑛𝜋) from Theorem 4.6.
To prove (vi), we set 𝑛=1 in Theorem 3.11 and use the corresponding values of 𝑓(𝑒𝑛𝜋) from Lemma 4.5. At last, (vii) follows from Theorems 4.7(v) and 4.4(ii).

Remark 4.8. Setting 𝑡=1/2 in (1.10) and then employing the value of 𝑐(𝑒𝜋) from Theorem 4.7(vi), we can also evaluate 𝑏(𝑒4𝜋/3).

Theorem 4.9. One has 𝑒(i)𝑎2𝜋=𝑎210+631/323+231/4,𝑒(ii)𝑎2𝜋/3=𝑎2310+631/323+231/4,𝑒(iii)𝑎6𝜋=𝑎2331/41+21/6+10+631/323+231/4,(𝑒iv)𝑎2𝜋/9=3𝑎231/41+21/6+10+631/323+231/4.(4.8)

Proof. To prove (i), we set 𝑛=3 in Theorem 3.7 and use 𝑓(𝑒2𝜋) from Lemma 4.5 and the values of 𝑟3,3 from Lemma 2.1.
To prove (ii), we set 𝑡=1 in (1.8) and then employ Theorem 4.9(i).
To prove (iii), we set 𝑛=2 in Theorem 3.8 and then employ the values of 𝑎(𝑒2𝜋) and 𝑏(𝑒2𝜋) from Theorems 4.9(i) and 4.6(ii), respectively.
To prove (iv), we set 𝑡=3 in (1.8) and use the value of 𝑎(𝑒6𝜋).

5. Theorems on Explicit Evaluations of 𝐴(𝑞), 𝐵(𝑞), and 𝐶(𝑞)

In this section, we use the parameters 𝑟𝑘,𝑛, 𝑘,𝑛, and 𝑔𝑘,𝑛 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 𝑛, one has 𝐵𝑒𝜋2𝑛𝐶𝑒𝜋2𝑛=𝑟122,𝑛=𝑔122𝑛.(5.1)

Proof. Employing the definition of 𝐵(𝑞) and 𝐶(𝑞) given in (1.14), we find that 𝐵(𝑞)𝐶=(𝑞)𝑓(𝑞)21/4𝑞1/24𝑓𝑞212.(5.2) Setting 𝑞=𝑒2𝜋𝑛/2 in (5.2) and then using the definition of 𝑟𝑘,𝑛, 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: 𝐵𝑒𝜋2/𝑛𝐶𝑒𝜋2/𝑛=𝐶𝑒𝜋2𝑛𝐵𝑒𝜋2𝑛.(5.3) 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 𝐵𝑒2𝜋𝑛𝐶𝑒𝜋𝑛=𝑟44,𝑛2.(5.4)

Proof. Theorem follows easily from (1.14) and the definition of 𝑟𝑘,𝑛 with 𝑘=4.

Remark 5.4. Using the fact that 𝑟4,1/𝑛=1/𝑟4,𝑛 in Theorem 5.3, we have the following transformation formula 𝑒4𝐵2𝜋/𝑛𝐶𝑒𝜋/𝑛=𝐶𝑒𝜋𝑛𝐵𝑒2𝜋𝑛.(5.5) Hence, if we know one quotient of (5.5) then the other quotient follows immediately.

Lemma 5.5. One has (i)𝜙𝑒2𝑛𝜋=𝑎21/8𝑛1/4𝑛,𝑛=𝑎𝑟2,2𝑛2𝑛1/421/4𝑟𝑛,𝑛,(ii)𝜙(𝑒𝑛𝜋𝑎)=𝑛1/4𝑛,𝑛=𝑎𝐺2𝑛2𝑛1/4𝑟𝑛,𝑛,(iii)𝜓(𝑒𝑛𝜋)=𝑎25/8𝑒𝑛𝜋/8𝑛1/4𝑔𝑛,𝑛=𝑎23/4𝑒𝑛𝜋/8𝑛1/4𝑟2,𝑛2/2𝑟𝑛,𝑛,𝑒(iv)𝜓𝜋/𝑛=𝑎𝑛1/423/4𝑟2,2𝑛2𝑒𝑛𝜋/8𝑟𝑛,𝑛,(5.6) where the parameters 𝑟𝑘,𝑛, 𝑘,𝑛, 𝑘,𝑛, 𝑔𝑘,𝑛, and 𝐺𝑛 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 𝑛, one has 𝑒(i)𝐵2𝜋𝑛=𝑎42𝑛4𝑛,𝑛=𝑎4𝑟42,2𝑛22𝑛𝑟4𝑛,𝑛,𝑒(ii)𝐵2𝜋/𝑛=𝑎4𝑛𝑟42,2/𝑛22𝑟4𝑛,𝑛,(5.7) where 𝑘,𝑛 is as defined in (1.21).

Proof. From [21, page 39, Entry 24(iii)], we note that 𝑓𝜙(𝑞)=2(𝑞)𝑓𝑞2.(5.8) Employing (5.8) in (1.14), we obtain 𝐵(𝑞)=𝜙4(𝑞).(5.9) Setting 𝑞=𝑒2𝜋𝑛 in (5.9) and then employing Lemma 5.5(i), we arrive at (i).
To prove (ii), we replace 𝑛 by 1/𝑛 in (i) and employ the result 𝑟1/𝑛,1/𝑛=𝑟𝑛,𝑛, which is easily derivable from (2.2).

Theorem 5.7. One has (i)𝐵(𝑒𝑛𝜋𝑎)=4𝑛4𝑛,𝑛=𝑎𝐺8𝑛2𝑛𝑟4𝑛,𝑛,(ii)𝐵𝑒𝜋/𝑛=𝑛𝑎44𝑛,𝑛=𝑎4𝑛𝐺8𝑛2𝑟4𝑛,𝑛,(5.10) where 𝑘,𝑛 is as defined in (1.21).

Proof. Replacing 𝑞 by 𝑞 in (5.9) and setting 𝑞=𝑒𝑛𝜋, we have 𝐵(𝑒𝑛𝜋)=𝜙4(𝑒𝑛𝜋),(5.11) Employing Lemma 5.5(ii) in (5.11), we finish the proof of (i).
To prove (ii), we replace 𝑛 by 1/𝑛 in (i) and use the results 𝑛,𝑛=1/𝑛,1/𝑛 [19] and 𝐺1/𝑛=𝐺𝑛.

Remark 5.8. The following transformation formula is apparent from Theorem 5.7(i) and (ii), 𝑛2𝐵(𝑒𝑛𝜋)=𝐵𝑒𝜋/𝑛.(5.12)

Theorem 5.9. For any positive real number 𝑛, one has 𝐶(𝑒𝑛𝜋)=2𝑎4𝑒𝑛𝜋/2𝑛𝑔4𝑛,𝑛,(5.13) where 𝑔𝑘,𝑛 is as defined in (1.23).

Proof. From [21, page 39, Entry 24(iii)], we notice that 𝑓𝜓(𝑞)=2𝑞2𝑓.(𝑞)(5.14) Thus, from (5.14) and (1.14), we find that 𝐶(𝑒𝑛𝜋)=8𝑒𝑛𝜋/2𝜓4(𝑒𝑛𝜋),(5.15) Setting 𝑞=𝑒𝑛𝜋 in (5.15) and employing Lemma 5.5(iii), we easily complete the proof.

Theorem 5.10. One has 𝐶𝑒𝜋/𝑛=𝑛𝑎4𝑟42,2𝑛2𝑟4𝑛,𝑛.(5.16)

Proof. Applying (5.14) in the definition of 𝐶(𝑞) given in (1.14) and setting 𝑞=𝑒𝜋/𝑛, we find that 𝐶𝑒𝜋/𝑛=8𝑒𝜋/2𝑛𝜓4𝑒𝜋/𝑛,(5.17) 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 𝐴(𝑞), 𝐵(𝑞), and 𝐶(𝑞), and also their quotients by using the results established in the previous section.

Theorem 6.1. One has 𝐵𝑒(i)𝜋2𝐶𝑒𝜋2𝐵𝑒=1,(ii)2𝜋𝐶𝑒2𝜋=23/2,𝐵𝑒(iii)𝜋6𝐶𝑒𝜋6=3+2𝐵𝑒2,(iv)𝜋22𝐶𝑒𝜋22=23/21+23/2,𝐵𝑒(v)𝜋10𝐶𝑒𝜋10=1+526,𝐵𝑒(vi)𝜋23𝐶𝑒𝜋33=23+13,𝐵𝑒(vii)𝜋14𝐶𝑒𝜋14=2+122126,𝐵𝑒(viii)4𝜋𝐶𝑒4𝜋=29/41+23,𝐵𝑒(ix)𝜋32𝐶𝑒𝜋32=3+24,𝐵𝑒(x)𝜋25𝐶𝑒𝜋25=181+535+1+23,𝐵