# Explicit Evaluations of Cubic and Quartic Theta-Functions

Authors: Nipen Saikia
Journal: ISRN Discrete Mathematics
 (2012)

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## 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.


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## Body

## 1. Introduction

For any complex numbera, define
(1.1)(a;q)n=(a;q)∞(aqn;q)∞,and(a;q)∞:=∏k=1∞(1-aqk-1).Ramanujan’s general theta-functionf(a,b) is given by
(1.2)f(a,b)=∑k=-∞∞ak(k+1)/2bk(k-1)/2,
where |ab|<1. If we set a=qe2iz, b=qe-2iz, 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 off(a,b):(1.3)ϕ(q):=f(q,q)=∑n=-∞∞qn2=(-q;q2)∞(q2;q2)∞(q;q2)∞(-q2;q2)∞,ψ(q):=f(q,q3)=∑k=0∞qk(k+1)/2=(q2;q2)∞(q;q2)∞,f(-q):=f(-q,-q2)=∑n=-∞∞(-1)nqn(3n-1)/2=(q;q)∞.Ifq=e2πiz with Im(z)>0, then f(-q)=q-1/24η(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:(1.4)qr∶=qr(x)=exp⁡⁡(-πcsc⁡(πr)F12(1/r,(r-1)/r,1,1-x)F12(1/r,(r-1)/r,1,x)),
where r= 3, 4, and 6, where F12 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-functionsa(q), b(q), and c(q) are defined by(1.5)a(q)=∑m,n=-∞∞qm2+mn+n2,b(q)=∑m,n=-∞∞wm-nqm2+mn+n2,c(q)=∑m,n=-∞∞q(m+1/3)2+(m+1/3)(n+1/3)+(n+1/3)2,
where w=exp(2πi/3). These theta-functions were first introduced by J. M. Borwein and P. B. Borwein [8], who also proved that(1.6)a3(q)=b3(q)+c3(q).
Cubic theta-functions b(q) and c(q) are related with the Dedekind eta-function by [7, page 109, Lemma 5.1]:(1.7)b(q)=f3(-q)f(-q3),c(q)=3q1/3f3(-q3)f(-q).The Borwein brothers [8, (2.2)] also established the following three transformation formulas:(1.8)a(e-2πt)=1t3a(e-2π/3t),(1.9)b(e-2πt)=1t3c(e-2π/3t),(1.10)c(e-2πt)=1t3b(e-2π/3t),
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
(1.11)A(q)=ϕ4(q)+16qψ4(q2),B(q)=ϕ4(q)-16qψ4(q2),C(q)=8qϕ2(q)ψ2(q2),
which also satisfy the equality:
(1.12)A2(q)=B2(q)+C2(q).
Berndt et al. [10] used (1.12) to establish the inversion formula:
(1.13)z4:=F12(14,34;1;x)=A(q),
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](1.14)B(q)=(f2(-q)f(-q2))4,C(q)=8q(f2(-q2)f(-q))4.While proving the explicit values ofϕ(q) and ψ(q) recorded by Ramanujan in his notebooks, Berndt [7], explicitly determined the value of cubic theta-function a(e-2π) [7, page 328, Corollary 3], namely,(1.15)a(e-2π)ϕ2(e-π)=1(12)1/83-1,
where ϕ(e-π)=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-functionf(q) and Ramanujan’s theta-functions ϕ(q) and ψ(q). For positive real numbers n and k, define(1.16)rk,n:=f(-q)k1/4q(k-1)/24f(-qk),q=e-2πn/k,(1.17)rk,n′:=f(q)k1/4q(k-1)/24f(qk),q=e-πn/k.
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 defined(1.18)λn=133f6(q)qf6(q3),q=e-πn/3.
Closely related to λn is the parameter μn defined by Ramanathan [13] as(1.19)μn=133f6(-q)qf6(-q3),q=e-2πn/3.From the definitions ofrk,n, μn, rk,n′, and μn, we note that r3,n′6=λn and r3,n6=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 hk,n and hk,n′ along with rk,n and rk,n′:
(1.20)hk,n:=ϕ(q)k1/4ϕ(qk),q=e-πn/k,(1.21)hk,n′:=ϕ(-q)k1/4ϕ(-qk),q=e-2πn/k,
where k and n are positive real numbers. Employing modular transformation formulas and theta-function identities, Yi evaluated several many explicit values of hk,n and hk,n′ to find explicit values of ϕ(q) and their quotients.Motivated by Yi’s work, for any positive real numbersk and n, Baruah and Saikia [17] defined the parameters gk,n and gk,n′ by(1.22)gk,n:=ψ(-q)k1/4q(k-1)/8ψ(-qk),q=e-πn/k,(1.23)gk,n′:=ψ(q)k1/4q(k-1)/8ψ(qk),q=e-πn/k.
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′, hk,n, hk,n′, and Weber-Ramanujan class-invariants Gn and gn, where Gn and gn defined by(1.24)Gn:=2-1/4q-1/24(-q;q2)∞,gn:=2-1/4q-1/24(q;q2)∞;q=e-πn.
They also found several values of the parameters gk,n and gk,n′.In Section2, we record some known values of above parameters, which will be used in this paper.In Sections3 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.
Ifrk,n is as defined in (1.16), then
(2.1)r1,1=1,r2,1=1,r2,2=21/8,r2,3=(1+2)1/6,r2,4=21/8(1+2)1/8,r2,5=1+52,r2,6=21/24(3+1)1/4,r2,7=(2+1+22-12)1/2,r2,8=23/16(1+2)1/4,r2,9=(2+3)1/3,r2,10=(12(1+5)(5+1+2))1/4,r2,12=(1+2)5/24(2(1+2+6))1/8,r2,16=21/8(1+2)1/4(4+2+102)1/8,r2,18=(1+3)1/3(1+3+2⋅33/4)1/3211/24,r2,20=(1+5)5/8(2+32+5)1/82,r2,27=(1+2)5/18{2+2(1+2)1/3+(1+2)2/3}1/3,r2,32=23/16(1+2)1/4(16+15⋅21/4+122+9⋅23/4)1/8,r2,36={2(1+352-283)}1/8(3-2)2/3,r2,49=1+7+21422+14+7+2142,r2,50=25/851/4-1,r2,72=(2+3)1/3(-2+4+23+33/4(3+1))1/3213/48(2-1)5/12,r2,3/2=(1+3)1/427/24,r2,5/2=(5+1+2)1/421/4,r2,7/2=(3+7)1/423/8,r2,9/2=(1+3+2⋅33/4)1/3213/24,r2,25/2=51/4+125/8,r2,27/2=(1+3)1/12(1-3+22/33)1/323/8(21/3-1)1/3,r2,63/2=(7-23+21+(3+3)3+1621-277)1/3213/24(3-1)2/3(3-7)1/12,r2,9/4=(-1+352+283)1/821/8(2+3)1/3,r2,9/8=25/48(2-1)5/12(3-1)1/3(3-2)1/3(-1-2+3+33/42-6)1/3,r3,3=31/12(3+23)1/12=31/8(1+3)1/621/12,r3,4=3+12,r3,5=(5+12)5/6=(11+552)1/6,r3,7=(3+72(2-3))1/4,r3,8=(2+1)1/3(2+3)1/4,r3,9=31/6(1+21/3+22/3)1/3=31/6(21/3-1)1/3,r3,18=31/6(1+2)5/18(2+2(1+2)1/3+(1+2)2/3)1/3,r3,25=12(1+103+5+2103+1023),r3,49=3+22373+23723+49+1322373+82372323,r4,4=25/16(1+2)1/4,r4,5=(1+5+2+1+52)1/2,r4,8=21/4(1+2)3/8(4+2+102)1/8,r4,9=12(1+234+3),r4,7=(8+37)1/4,r4,9=12+31/42+32,r4,25=12(3+54+5+534)=54+154-1,r4,49=14(4+7+21+87+7+21+87)2,r5,5=(25+105)1/6=5+52,r6,6=31/83+1(1+3+233/4)1/3213/24.
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(2.2)rk,1=1,rk,1/n=1rk,n,rk,n=rn,k.Lemma 2.2.
One has(2.3)(i)h1,1=1,(ii)h2,2=22-2,(iii)h3,3=(23-3)1/4=31/83-121/4,(iv)h4,4=23/424+1,(v)h5,5=5-25,(vi)h6,6=23/431/8((2-1)(3-1))1/6(-4+32+35/4+23-33/4+22⋅33/4)1/3.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(2.4)(i)h1,1′=1,(ii)h2,2′=21/16(2-1)1/4,(iii)h3,3′=21/331/8(3-1)1/6(1+3+2334)1/3,(iv)h4,4′=21/4(16+1524+122+9234)1/8,(v)h5,5′=12(54-1)5+5,(vi)h6,6′=21/431/8(2-1)1/12(3+1)1/6(-1-3+2⋅33/4)1/3(2-32+35/4+33/4)1/3,(vii)h3,1′=2-1/43-1.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(2.5)(i)g1,1′=1,(ii)g2,2′=23/8,(iii)g3,3′=31/3(1+3+2⋅33/4)1/3(1+3)1/62,(iv)g4,4′=23/8(1+2)1/2,(v)g5,5′=(5+5)1/2(51/4+1)2,(vi)g6,6′=31/8(1+3)5/6(1+3+2⋅33/4)2/3229/24,(vii)g9,9′=a+(2(b-2c))1/3+(2(b+2c))1/32,
where
(2.6)a=2+2⋅31/4+23+2⋅33/4,b=82+452+483+252⋅33/4,c=3(88+472⋅31/4+503+272⋅33/4).For proofs we refer to [17, page 1781, Theorem 6.7].


## 3. Theorems on Explicit Evaluation ofa(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 Section1. In the process, we also establish some transformation formulas of quotients of cubic theta-functions.Theorem 3.1.
For any positive real numbern, one has
(3.1)b(e-2πn/3)c(e-2πn/3)=r3,n4=μn2/3,
where rk,n and μn are as defined in (1.16) and (1.19), respectively.Proof.
Using the definitions ofb(q) and c(q) from (1.7), one has
(3.2)3b(q)c(q)=(f(-q)q1/12f(-q3))4.
Setting q=e-2πn/3 and then employing the definitions of rk,n and μn, we finish the proof.Remark 3.2.
Replacingn by 1/n in Theorem 3.1 and noting that r3,1/n=1/r3,n from (2.2), we also have
(3.3)b(e-2πn/3)c(e-2πn/3)=c(e-2π/3n)b(e-2π/3n).
Thus, if we know the value of one quotient of (3.3), then the other quotient follows readily.From Theorem3.1 and (1.6), the following theorem is apparent.Theorem 3.3.
One has(3.4)a(e-2πn/3)c(e-2πn/3)=(r3,n12+1)1/3.Theorem 3.4.
For any positive real numbern, one has
(3.5)b(e-2πn)c(e-2πn/3)=r9,n3.Proof.
From the definitionsb(q) and c(q) in (1.7), we observe that
(3.6)b(q3)c(q)=f(-q)3q1/3f(-q9).
Setting q=e-2πn/3 in (3.6) and then employing the definition of rk,n, we arrive at the desired result.Remark 3.5.
Noting thatr9,1/n=1/r9,n from (2.2) and using Theorem 3.4, we find that
(3.7)3b(e-2πn)c(e-2πn/3)=c(e-2π/3n)b(e-2π/n).
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 betweenc(q) and the parameter hk,n as defined in (1.21).Theorem 3.6.
For any positive real numbern, one has
(3.8)c(e-8πn/3)c(e-2πn/3)=14(1-3(h3,n′)2).Proof.
From [10, page 111, Lemma 5.5], we note that
(3.9)1-4c(q4)c(q)=(ϕ(-q)ϕ(-q3))2.
Now applying the definition of hk,n′, with k=3, in (3.9), we complete the proof.The next theorem connectsa(q) with the parameter rk,n defined in (1.16).Theorem 3.7.
For any positive real numbern, one has
(3.10)a12(e-2πn/3)=27(r3,n12+1)4e-2πn/3f24(-e-2πn/3)r3,n36.Proof.
From [20, page 196, (2.9)], we note that
(3.11)27qf24(-q)=a12(q)(1-α(q))3α(q),
where α(q)=c3(q)/a3(q).
Settingq=e-2πn/3 and then applying (3.3) in (3.11), we obtain
(3.12)27e-2πn/3f24(-e-2πn/3)=a12(e-2πn/3)(1-1r3,n12+1)3(1r3,n12+1),
which on simplification gives the required result.Theorem 3.8.
One has(3.13)a(e-3nπ)=13{a(e-nπ)+2b(e-nπ)}.Proof.
From [7, page 93, (2.8)], one has
(3.14)b(q)=12{3a(q3)-a(q)}.
Setting q=e-nπ in (3.14), we readily complete the proof.Theorem 3.9.
For any positive real numbern, one has
(3.15)(i)b(e-nπ)=f3(-e-nπ)f(-e-3nπ),(ii)b(-e-nπ)=f3(e-nπ)f(e-3nπ).Proof.
Settingq=e-nπ and q=-e-nπ in (1.7), we readily arrive at (i) and (ii), respectively.Theorem 3.10.
For all positive real numbersn, one has
(3.16)(i)b(e-2πn/3)=31/4e-πn/63f2(-e-2πn/3)r3,n,(ii)b(-e-πn/3)=31/4e-πn/123f2(e-πn/3)r3,n′,
where the parameters r3,n and r3,n′ are defined in (1.16) and (1.17), respectively.Proof.
We rewriteb(q) in (1.7) as
(3.17)b(q)=f2(-q)q1/12f(-q)q1/12f(-q3).
Setting q=e-2πn/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 numbern, we have
(3.18)(i)c(e-nπ)=3e-nπ/3f3(-e-3nπ)f(-e-nπ),(ii)c(-e-nπ)=-3e-nπ/3f3(e-3nπ)f(e-nπ).Proof.
It follows readily from (1.7) with q=e-nπ and q=-e-nπ.Theorem 3.12.
For all positive real numbern, one has
(3.19)c(e-2πn/3)=33/4e-πn/23f2(-e-2π3n)r3,n.Proof.
We setq=e-2πn/3 in (1.7) and then employ the definition of the parameter rk,n to finish the proof.


## 4. Explicit Values ofa(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(4.1)(i)b(e-2π/3)c(e-2π/3)=1,(ii)b(e-2π2/3)c(e-2π2/3)=(1+2)2/3,(iii)b(e-2π)c(e-2π)=31/2(1+3)2/321/3,(iv)b(e-4π/3)c(e-4π/3)=(1+32)2,(v)b(e-2π5/3)c(e-2π5/3)=(1+52)10/3,(vi)b(e-2π7/3)c(e-2π7/3)=3+72(2-3),(vii)b(e-4π2/3)c(e-4π2/3)=(1+2)4/3(2+3),(viii)b(e-2π3)c(e-2π3)=32/3(21/3-1)4/3,(ix)b(e-6π2/3)c(e-6π2/3)=32/3(1+2)10/9(2+2(1+2)1/3+(1+2)2/3)4/3,(x)b(e-2π13/3)c(e-2π13/3)=(11+13+3+1322)4,(xi)b(e-10π/3)c(e-10π/3)=116(1+103+5+2103+1023)4,(xi)b(e-14π/3)c(e-14π/3)=(3+22373+23723+49+1322373+82372323)4.Proof.
It follows directly from Theorem3.1 and the corresponding values of r3,n listed in Lemma 2.1.More values can be calculated by employing Theorem3.1 and the corresponding values of μn evaluated in [15, 16].Theorem 4.2.
One has(4.2)(i)a(e-2π/3)c(e-2π/3)=23,(ii)a(e-2π2/3)c(e-2π2/3)=21/3(2+2)1/3,(iii)a(e-2π)c(e-2π)=(33/2(1+3)22+1)1/3,(iv)a(e-4π/3)c(e-4π/3)=(((1+3)2)6+1)1/3,(v)a(e-2π5/3)c(e-2π5/3)=((1+52)10+1)1/3,(vi)a(e-2π7/3)c(e-2π7/3)=((3+72(2-3))3+1)1/3,(vii)a(e-4π2/3)c(e-4π2/3)=((1+2)4(2+3)4+1)1/3,(viii)a(e-10π/3)c(e-10π/3)=((1+163+5+2103+10232)12+1)1/3,(ix)a(e-6π/3)c(e-6π/3)=(9(21/4-1)4+1)1/3,(x)a(e-14π/3)c(e-14π/3)=((3+2373+23723+49+132373+82372323)+1)1/3,(xi)a(e-6π)c(e-6π)=(34(1+2)10/3(2+2(1+2)4+(1+2)8)+1)1/3.Proof.
It follows easily from (3.3) and the corresponding values of r3,n listed in Lemma 2.1.Theorem 4.3.
One has(4.3)(i)b(e-2π)c(e-2π/3)=13,(ii)b(e-2π2)c(e-2π2/3)=(3+2)1/33,(iii)b(e-2π3)c(e-2π/3)=(13(23-1))1/3,(iv)b(e-4π)c(e-4π/3)=123(1+2⋅31/4+3),(v)b(e-25π)c(e-25π/3)=13(104+603+455+2615)1/6.Proof.
It follows from Theorem3.4 and the corresponding values of r9,n in listed in Lemma 2.1.Theorem 4.4.
One has(4.4)(i)c(e-8π/3)c(e-2π/3)=14(2+3-32),(ii)c(e-8π)c(e-2π)=14(1-3(22/331/4(3-1)1/3(1+3+2334)2/3)).Proof.
We setn=1 and 3 in Theorem 3.6 and then employ the values of h3,1′ and h3,3′ from Lemma 2.3(vii) and (iii), respectively, to finish the proof.For the remaining part of this paper, we seta:=ϕ(e-π)=π1/4/Γ(3/4).Lemma 4.5.
One has(4.5)(i)f(-e-π)=a2-3/8eπ/24,(ii)f(e-π)=a2-1/4eπ/24,(iii)f(-e-2π)=a2-1/2eπ/12,(iv)f(-e-3π)=aeπ/4(1+3+2⋅33/4)1/333/8217/24(1+3)1/6,(v)f(-e-4π)=a2-7/8eπ/6,(vi)f(-e-6π)=a2-7/123-3/8eπ/4(3-1)1/4,(vii)f(-e-12π)=aeπ/225/2433/81+3(1+3+2⋅33/4)1/3,(viii)f(-e-π/3)=a27/2431/8eπ/721+3(1+3+2⋅33/4)1/3,(ix)f(-e-2π/3)=a2-7/1231/8eπ/36(3-1)1/6,(x)f(e-2π)=a2-13/16eπ/12(2+1)1/4,(xi)f(e-3π)=a2-1/33-3/8eπ/8(3+1)1/6,(xii)f(e-6π)=aeπ/4(2-32+35/4+33/4)1/3215/1633/8(2-1)1/12(3+1)1/6.For a proof of the lemma, we refer to [7, page 326, Entry 6] and [11, page 125–129].Theorem 4.6.
One has(4.6)(i)b(e-π)=a233/8(1+3)1/625/12(1+3+2⋅33/4)1/3,(ii)b(e-2π)=a233/8211/12(3-1)1/6,(iii)b(e-4π)=a22-29/1233/8(1+3)1/2(1+3+2⋅33/4)1/3,(iv)b(e-π/3)=a225/433/8(1+3)3/2(1+3+2⋅33/4),(v)b(e-2π/3)=a231/8(3-1)1/3213/12(3+1)1/6,(vi)b(-e-π)=a233/825/12(3+1)1/6,(vii)b(-e-2π)=a233/8(2+1)3/4(2-1)1/12(3+1)1/623/2(2-32+35/4+33/4)1/3.Proof.
To prove (i)–(v), we setn=1, 2, 4, 1/3, and 2/3, respectively, in Theorem 3.9(i) and use the corresponding values of f(±e-πn) from Lemma 4.5.
To prove (vi) and (vii), we setn= 1 and 2, respectively, in Theorem 3.9(ii) and then use the corresponding values f(±e-πn) from Lemma 4.5.Theorem 4.7.
One has(4.7)(i)c(e-4π/3)=a237/8(1+3)1/6217/12(1+3+2⋅33/4)1/3,(ii)c(e-2π/3)=a22-13/1237/8(3+1)1/6,(iii)c(e-π/3)=2-17/1237/8a2(1+3)1/2(1+3+2⋅33/4)1/3,(iv)c(e-4π)=a233/821/4(1+3)3/2(1+3+2⋅33/4),(v)c(e-2π)=a2(3-1)1/333/8213/12(3+1)1/6,(vi)c(e-π)=a2(1+3+2⋅33/4)31/827/4(1+3)1/2,(vii)c(e-8π)=14(1-3(22/3⋅31/4(3-1)1/3(1+3+2+324)2/3))×a2(3-1)1/333/8⋅213/12(3+1)1/6.Proof.
To prove (i)–(v), we sett=1/2,1,2,1/6, and 1/3, respectively, in (1.9) and then apply the corresponding values of b(e-nπ) from Theorem 4.6.
To prove (vi), we setn=1 in Theorem 3.11 and use the corresponding values of f(-e-nπ) from Lemma 4.5. At last, (vii) follows from Theorems 4.7(v) and 4.4(ii).Remark 4.8.
Settingt=1/2 in (1.10) and then employing the value of c(e-π) from Theorem 4.7(vi), we can also evaluate b(e-4π/3).Theorem 4.9.
One has(4.8)(i)a(e-2π)=a2(10+63)1/32(3+23)1/4,(ii)a(e-2π/3)=a23(10+63)1/32(3+23)1/4,(iii)a(e-6π)=a23(31/4(1+2)1/6+(10+63)1/32(3+23)1/4),(iv)a(e-2π/9)=3a2{(31/4(1+2)1/6+(10+63)1/32(3+23)1/4)}.Proof.
To prove (i), we setn=3 in Theorem 3.7 and use f(-e-2π) from Lemma 4.5 and the values of r3,3 from Lemma 2.1.
To prove (ii), we sett=1 in (1.8) and then employ Theorem 4.9(i).
To prove (iii), we setn=2 in Theorem 3.8 and then employ the values of a(e-2π) and b(e-2π) from Theorems 4.9(i) and 4.6(ii), respectively.
To prove (iv), we sett=3 in (1.8) and use the value of a(e-6π).


## 5. Theorems on Explicit Evaluations ofA(q), B(q), and C(q)

In this section, we use the parametersrk,n, hk,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 numbern, one has
(5.1)B(e-π2n)C(e-π2n)=r2,n12=g2n12.Proof.
Employing the definition ofB(q) and C(q) given in (1.14), we find that
(5.2)B(q)C(q)=(f(-q)21/4q1/24f(-q2))12.
Setting q=e-2πn/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 Theorem5.1 and (2.2), we have the following transformation formula:
(5.3)B(e-π2/n)C(e-π2/n)=C(e-π2n)B(e-π2n).
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(5.4)B(e-2πn)C(e-πn)=r4,n42.Proof.
Theorem follows easily from (1.14) and the definition of rk,n with k=4.Remark 5.4.
Using the fact thatr4,1/n=1/r4,n in Theorem 5.3, we have the following transformation formula
(5.5)4B(e-2π/n)C(e-π/n)=C(e-πn)B(e-2πn).
Hence, if we know one quotient of (5.5) then the other quotient follows immediately.Lemma 5.5.
One has(5.6)(i)ϕ(-e-2nπ)=a21/8n1/4hn,n′=ar2,2n2n1/421/4rn,n,(ii)ϕ(e-nπ)=an1/4hn,n=aGn22n1/4rn,n,(iii)ψ(e-nπ)=a2-5/8enπ/8n1/4gn,n′=a2-3/4enπ/8n1/4r2,n2/2rn,n,(iv)ψ(e-π/n)=an1/42-3/4r2,2n2enπ/8rn,n,
where the parameters rk,n, hk,n, hk,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 numbern, one has
(5.7)(i)B(e-2πn)=a42nhn,n′4=a4r2,2n242nrn,n4,(ii)B(e-2π/n)=a4nr2,2/n242rn,n4,
where hk,n′ is as defined in (1.21).Proof.
From [21, page 39, Entry 24(iii)], we note that
(5.8)ϕ(-q)=f2(-q)f(-q2).
Employing (5.8) in (1.14), we obtain
(5.9)B(q)=ϕ4(-q).
Setting q=e-2πn in (5.9) and then employing Lemma 5.5(i), we arrive at (i).
To prove (ii), we replacen 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(5.10)(i)B(-e-nπ)=a4nhn,n4=aGn28nrn,n4,(ii)B(-e-π/n)=na4hn,n4=a4nGn28rn,n4,
where hk,n′ is as defined in (1.21).Proof.
Replacingq by -q in (5.9) and setting q=e-nπ, we have
(5.11)B(-e-nπ)=ϕ4(e-nπ),
Employing Lemma 5.5(ii) in (5.11), we finish the proof of (i).
To prove (ii), we replacen by 1/n in (i) and use the results hn,n=h1/n,1/n [19] and G1/n=Gn.Remark 5.8.
The following transformation formula is apparent from Theorem5.7(i) and (ii),
(5.12)n2B(-e-nπ)=B(-e-π/n).Theorem 5.9.
For any positive real numbern, one has
(5.13)C(e-nπ)=2a4enπ/2ngn,n′4,
where gk,n′ is as defined in (1.23).Proof.
From [21, page 39, Entry 24(iii)], we notice that
(5.14)ψ(q)=f2(-q2)f(-q).
Thus, from (5.14) and (1.14), we find that
(5.15)C(e-nπ)=8e-nπ/2ψ4(e-nπ),
Setting q=e-nπ in (5.15) and employing Lemma 5.5(iii), we easily complete the proof.Theorem 5.10.
One has(5.16)C(e-π/n)=na4r2,2n24rn,n4.Proof.
Applying (5.14) in the definition of C(q) given in (1.14) and setting q=e-π/n, we find that
(5.17)C(e-π/n)=8e-π/2nψ4(e-π/n),
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-functionsA(q), B(q), and C(q), and also their quotients by using the results established in the previous section.Theorem 6.1.
One has(6.1)(i)B(e-π2)C(e-π2)=1,(ii)B(e-2π)C(e-2π)=23/2,(iii)B(e-π6)C(e-π6)=3+22,(iv)B(e-π22)C(e-π22)=23/2(1+2)3/2,(v)B(e-π10)C(e-π10)=(1+52)6,(vi)B(e-π23)C(e-π33)=2(3+1)3,(vii)B(e-π14)C(e-π14)=(2+122-12)6,(viii)B(e-4π)C(e-4π)=29/4(1+2)3,(ix)B(e-π32)C(e-π32)=(3+2)4,(x)B(e-π25)C(e-π25)=18(1+5)3(5+1+2)3,(xi)B(e-π26)C(e-π26)=(1+2)5/2(2(1+2+6))3/2,(xii)B(e-π42)C(e-π42)=23/2(1+2)3(4+2+102)3/2,(xiii)B(e-6π)C(e-6π)=2-11/2(1+3)4(1+3+2⋅33/4)4,(xiv)B(e-π210)C(e-π210)=2-6(1+5)15/2(2+32+5)3/2,(xv)B(e-5π)C(e-5π)=(51/4+1)12215/2,(xvi)B(e-π36)C(e-π66)=(1+2)10/3{2+2(1+2)1/3+(1+2)2/3}4,(xvii)B(e-8π)C(e-8π)=29/4(1+2)3(16+15⋅21/4+122+9⋅23/4)3/2,(xviii)B(e-π62)C(e-π62)=(2(1+352-283))3/2(3-2)8,(xix)B(e-π72)C(e-π72)=(1+7+21422+14+7+2142)12,(xx)B(e-10π)C(e-10π)=215/2(51/4-1)12,(xxi)B(e-12π)C(e-12π)=((2+3)(3+1)(1+2-3+2⋅33/46))4221/4(2-1)5,(xxii)B(e-3π)C(e-3π)=(1+3+2⋅31/4)4213/2,(xxiii)B(e-3π/2)C(e-3π/2)=(-1+352+283)3/223/2(2+3)4,(xxiv)B(e-π3)C(e-π3)=(3+1)327/2,(xxv)B(e-π32)C(e-π32)=(1+3)(1-3+22/3⋅3)429/2(21/3-1)4,(xxvi)B(e-π7)C(e-π7)=(3+7)329/2,(xxvii)B(e-π37)C(e-π37)=(7-23+21+(3+3)3+1621-277)4213/2(3-1)8(3-7).Proof.
We employ the values ofr2,n from Lemma 2.1 in Theorem 5.1 to finish the proof.Theorem 6.2.
One has(6.2)(i)B(e-2π)C(e-π)=12,(ii)B(e-π22)C(e-π2)=(2+12)1/2,(iii)B(e-2π3)C(e-π3)=12(2+3),(iv)B(e-4π)C(e-2π)=21/4(2+1),(v)B(e-2π5)C(e-π5)=18(1+5+2(1+5))2,(vi)B(e-2π7)C(e-π7)=(127+487)1/22,(vii)B(e-6π)C(e-3π)=12(12+31/42+32)4,(viii)B(e-10π)C(e-5π)=12(32+52+12(5+35))4,(ix)B(e-14π)C(e-7π)=12(4+7+21+87+7+21+872)8.Proof.
It follows easily from Theorem5.3 and the values of r4,n from Lemma 2.1.Theorem 6.3.
One has(6.3)(i)B(e-2π)=2-1/2a4,(ii)B(e-4π)=2-7/4a4(1+2),(iii)B(e-6π)=a4(1+3+2⋅334)4/3211/6⋅33/2(3-1)2/3,(iv)B(e-8π)=2-7/2a4(16+1524+122+9234)1/2,(v)B(e-10π)=a427/25(54-1)4(5+5)1/2,(vi)B(e-12π)=a4(2-32+35/4+33/4)4/333/2219/12(2-1)1/3(3+1)2/3(-1-3+2⋅33/4)4/3,(vii)B(e-3π)=a4(1+3+2⋅31/4)4/3(1+3+2⋅33/4)424⋅33/2(1+3)2/3,(viii)B(e-5π)=a4(51/4+1)4(3+2⋅51/4)52⋅22(1+5).Proof.
(i)–(vi) follow readily from the first equality of Theorem5.6(i) and the corresponding values of hn,n′ in Lemma 2.3(i)–(vi), respectively. To prove (vii) and (viii), we employ the corresponding values of rk,n listed in Lemma 2.1 to the second equality of Theorem 5.6(i).Theorem 6.4.
One has(6.4)(i)B(e-π)=a42,(ii)B(e-π/2)=a42(1+2)2,(iii)B(e-π/3)=243a4(1+3)10/3(1+3+2⋅33/4)8/3.Proof.
We setn= 2, 3, and 6, respectively, in Theorem 5.6(ii) and then use the corresponding values of rk,n from Lemma 2.1 to complete the proofs.Theorem 6.5.
One has(6.5)(i)B(-e-π)=a4,(ii)B(-e-2π)=a48(2-2)2,(iii)B(-e-3π)=2a433/2(3-1)2,(iv)B(-e-4π)=a4(24+1)432,(v)B(-e-5π)=a45(5-25)2,(vi)B(-e-6π)=a4(-4+32+35/4+23-33/4+22⋅33/4)4/324⋅33((2-1)(3-1))2/3.Proof.
We employ the values ofhn,n given in Lemma 2.2 in Theorem 5.7(i) to finish the proof.Theorem 6.6.
One has(6.6)(i)B(-e-π/2)=a42(2-2)2,(ii)B(-e-π/3)=3a423-3,(iii)B(-e-π/4)=a42(24+1)4,(iv)B(-e-π/5)=5a4(5-25)2,(v)B(-e-π/6)=3a4(-4+32+35/4+23-33/4+22⋅33/4)4/3223((2-1)(3-1))2/3.Proof.
We use the values ofhn,n from Lemma 2.2 in Theorem 5.7(ii).Theorem 6.7.
One has(6.7)(i)C(e-π)=2a4,(ii)C(e-2π)=a44,(iii)C(e-3π)=25/2a437/3(1+3+2⋅33/4)4/3(1+3)2/3,(iv)C(e-4π)=a423(1+2)2,(v)C(e-5π)=29/2a45(5+5)2(51/4+1)4,(vi)C(e-6π)=213/3a433/2(1+3)10/3(1+3+2⋅33/4)8/3,(vii)C(e-9π)=a429g9,9′4,
where g9,9′ is given in Lemma 2.4(vii).Proof.
The proof of the theorem follows from Theorem5.9 and the values of gn,n′ from Lemma 2.4.Theorem 6.8.
One has(6.8)(i)C(e-π/2)=25/4a4(1+2),(ii)C(e-π/3)=2-3/23a4(1+3)2/3(2⋅33/4+3+1)4/3,(iii)C(e-π/4)=23/2a4(16+15⋅21/4+122+9⋅23/4)1/2,(iv)C(e-π/5)=5⋅29/2a4(5+5)2(51/4-1)4,(v)C(e-π/6)=217/331/2a4((2+3)(1+3)(1+2-3+12⋅33/4))4/3(1+3)2(1+3+2⋅33/4)4/3(2-1)5/3.Proof.
We setn=2,3,4,5, and 6 in Theorem 5.10 and then employ the corresponding values of rk,n listed in Lemma 2.1.



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