International Scholarly Research Network ISRN Discrete Mathematics Volume 2012, Article ID 956594, 29 pages doi: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 Correspondence should be addressed to Nipen Saikia,
[email protected] Received 24 February 2012; Accepted 8 April 2012 Academic Editors: H.-J. Kreowski and W. Liu Copyright q 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. 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
∞
, ∞
∞ 1 − aqk−1 . and a; q ∞ :
1.1
k1
Ramanujan’s general theta-function fa, b is given by
fa, b
∞
akk1/2 bkk−1/2 ,
1.2
k−∞
where |ab| < 1. If we set a qe2iz , b qe−2iz , and q eπiτ , where z is complex and Imτ > 0, then fa, b ϑ3 z, τ, where ϑ3 z, τ 1, page 464 denotes one of the classical thetafunctions in its standard notation.
2
ISRN Discrete Mathematics We also define the following three special cases of fa, b: ∞ −q; q2 ∞ q2 ; q2 ∞ n2 φ q : f q, q q 2 2 2 , q; q ∞ −q ; q ∞ n−∞ 2 2 ∞ q ;q 3 kk1/2 ψ q : f q, q q 2 ∞ , q; q ∞ k0
1.3
∞ f −q : f −q, −q2 −1n qn3n−1/2 q; q ∞ . n−∞
If q e2πiz with Imz > 0, then f−q q−1/24 ηz, where ηz 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: π 2 F1 1/r, r − 1/r, 1, 1 − x , qr : qr x exp −πcsc r 2 F1 1/r, r − 1/r, 1, x
1.4
where r 3, 4, and 6, where 2 F1 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 aq, bq, and cq are defined by a q
∞
qm
2
mnn2
,
b q
m,n−∞
c q
∞
wm−n qm mnn , 2
m,n−∞ ∞
2
2
1.5 2
qm1/3 m1/3n1/3n1/3 ,
m,n−∞
where w exp2πi/3. These theta-functions were first introduced by J. M. Borwein and P. B. Borwein 8, who also proved that a3 q b3 q c3 q .
1.6
ISRN Discrete Mathematics
3
Cubic theta-functions bq and cq are related with the Dedekind eta-function by 7, page 109, Lemma 5.1: f 3 −q b q 3 , f −q
3q1/3 f 3 −q3 . c q f −q
1.7
The Borwein brothers 8, 2.2 also established the following three transformation formulas: 1 a e−2πt √ a e−2π/3t , t 3 1 b e−2πt √ c e−2π/3t , t 3 1 c e−2πt √ b e−2π/3t , t 3
1.8 1.9 1.10
where Ret > 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 aq, bq, and cq in cubic theory is the functions Aq, Bq, and Cq 10, defined by A q φ4 q 16qψ 4 q2 ,
B q φ4 q − 16qψ 4 q2 , 1.11
C q 8 qφ2 q ψ 2 q2 , 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 : 2 F1
1 3 , ; 1; x 4 4
A 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 B q
4 f 2 −q , f −q2
C q 8 q
4 f 2 −q2 . f −q
1.14
4
ISRN Discrete Mathematics
While proving the explicit values of φq and ψq recorded by Ramanujan in his notebooks, Berndt 7, explicitly determined the value of cubic theta-function ae−2π 7, page 328, Corollary 3, namely, a e−2π 1 ,
√ φ2 e−π 121/8 3−1
1.15
where φe−π π 1/4 /Γ3/4 is classical 1. Certain quotients of Aq, Bq, and Cq 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 fq and Ramanujan’s theta-functions φq and ψq. For positive real numbers n and k, define
rk,n :
f −q k1/4 qk−1/24 f
rk,n :
−qk
f q
,
, k1/4 qk−1/24 f qk
q e−2π
q e−π
√
n/k
√
n/k
,
1.16
.
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 defined
f6 q 1 λn √ √ 6 3 , 3 3 qf q
q e−π
√
n/3
1.18
.
Closely related to λn is the parameter μn defined by Ramanathan 13 as f 6 −q 1 μn √ √ 6 3 , 3 3 qf −q
q e−2π
√
n/3
1.19
.
6 6 , and μn , we note that r3,n λn and r3,n μn . From the definitions of rk,n , μn , rk,n 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 .
ISRN Discrete Mathematics
5
In 11, Yi also introduced the following two parameterizations hk,n and hk,n along : with rk,n and rk,n φ q hk,n : 1/4 k , k φ q φ −q hk,n : 1/4 k , k φ −q
q e−π
√
q e−2π
n/k
,
1.20
√
n/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 hk,n and hk,n to find explicit values of φq and their quotients. Motivated by Yi’s work, for any positive real numbers k and n, Baruah and Saikia 17 by defined the parameters gk,n and gk,n ψ −q
gk,n : gk,n
, k
k1/4 qk−1/8 ψ −q ψ q : 1/4 k−1/8 k , k q ψ q
q e−π q e−π
√
n/k
√
n/k
,
1.22
.
1.23
and established In 17, they proved many properties of the parameterizations gk,n and gk,n their relationship with Yi’s parameters rk,n , rk,n , hk,n , hk,n , and Weber-Ramanujan classinvariants Gn and gn , where Gn and gn defined by
Gn : 2−1/4 q−1/24 −q; q2 ,
gn : 2−1/4 q−1/24 q; q2 ;
∞
∞
q e−π
√ n
.
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, r2,5
r2,1 1, √ 1 5 , 2
r2,2 21/8 ,
r2,6 2
1/24
√
√ 1/6 r2,3 1 2 ,
1/4 31 ,
r2,7
√ 1/8 r2,4 21/8 1 2 ,
√
1/2 √ 21 2 2−1 , 2
6
ISRN Discrete Mathematics √ 1/4 , r2,8 23/16 1 2 r2,10 r2,12
√ 1/4 √ √ 1 1 5 51 2 , 2 √ √ 5/24 √ 1/8 2 1 2 6 1 2 ,
r2,16 2
1/8
r2,18
r2,20 r2,27
√ √ 1/3 2 3 ,
r2,9
1
√ 1/4 √ 1/8 1 2 4 2 10 2 ,
1/3 √ √ 1/3 √ 1 3 2 · 33/4 3
, 211/24 √ √ 5/8 √ 1/8 1 5 23 2 5 , √ 2 √ 5/18 √ √ √ 1/3 √ 2/3 1/3 1 2 2 2 1 2 1 2 ,
1/8 √ 1/4 √ r2,32 23/16 1 2 16 15 · 21/4 12 2 9 · 23/4 ,
r2,36
r2,49
r2,72
√ √ 1/8 2 1 35 2 − 28 3 , √ √ 2/3 3− 2
√ √ √ 14 7 2 14 7 2 14 25/8 , r2,50 1/4 , √ 2 5 −1 2 2 √ √ 1/3 √ 1/4 √ 1/3 √ √ 1 3 − 2 4 2 3 33/4 3 1 2 3 , r2,3/2 , √ 5/12 27/24 13/48 2 2−1 1
r2,5/2
r2,25/2
√ √ 1/4 51 2 21/4 51/4 1 , 25/8
r2,27/2
,
r2,63/2
√ √ √ 1/3 √ √
7 − 2 3 21 3 3 3 16 21 − 27 7 , √ 2/3 √ 1/12 3− 7 213/24 3 − 1
r2,9/4
√ 1/4 7
, r2,9/2 23/8 √ 1/12 √ √ 1/3 1 3 1 − 3 22/3 3 , 1/3 23/8 21/3 − 1
r2,7/2
3
√ √ 1/8 −1 35 2 28 3 √ √ 1/3 , 21/8 2 3
1
√
3
√
2 · 33/4
213/24
1/3 ,
ISRN Discrete Mathematics 25/48
r2,9/8
r3,3 3
1/12
√ r3,5
r3,8
√
√
5/12 √ 1/3 √ √ 1/3 3− 2 2−1 3−1 , √ √ √ √ 1/3 3/4 −1 − 2 3 3 2− 6
√ 1/6 1/8 1 3 √ 1/12 3 32 3 , 21/12
51 2
5/6
r3,18 3
√
3
r4,4 2
5/16
⎞1/4
⎜ 3 7 ⎟ r3,7 ⎝ √ ⎠ 2 2− 3
,
1/3 r3,9 31/6 1 21/3 22/3
31/6
1/3 , 21/3 − 1
√ 5/18 √ 1/3 √ 2/3 1/3 √ 1 2 1 2 , 2 2 1 2
√ √ √ √ √ √ √ √ 3 3 3 3 22 3 7 3 2 72 49 13 22 3 7 8 3 2 72 , √ 2 3
√ 1/4 1 2 ,
r4,5
1
√
√
5
√ 3/8 √ 1/8 r4,8 21/4 1 2 , 4 2 10 2 √ 1/4 r4,7 8 3 7 ,
r5,5
√
√
√ √ 1 3 3 3 1 10 5 2 10 102 , 2
r3,49
r4,49
31 √ , 2
r3,4 ⎛
√ 1/6 11 5 5 , 2
1/3 √ √ 1/4 21 2 3 ,
1/6
r3,25
7
r4,9
2 2
√ 1/2 1 5
r4,9
√ 3 1 31/4 √ , 2 2 2
,
√ √ √ 1 4 1 2 3 3 , 2
r4,25
4 √
√ √ 51 1 4 4 3 5 5 53 √ , 4 2 5−1
⎞2 ⎛ √ √ √ √ 1⎝ 4 7 21 8 7 7 21 8 7⎠ , 4
√ 1/6 25 10 5
√ 5 5 , 2
r6,6
31/8
1/3
√ √ √ 3 1 1 3 2 33/4 213/24
. 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
1 , rk,n
rk,n rn,k .
2.2
8
ISRN Discrete Mathematics
Lemma 2.2. One has h1,1 1, √ ii h2,2 2 2 − 2,
i
iii iv v vi
h3,3
√ 1/4 31/8 √3 − 1 2 3−3 , 21/4
23/4 , h4,4 √ 4 21 √ h5,5 5 − 2 5, h6,6
23/4 31/8
2.3
√ √ 1/6 2−1 3−1
1/3 . √ √ √ −4 3 2 35/4 2 3 − 33/4 2 2 · 33/4
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
h1,1 1,
ii
h2,2 21/16
1/4 2−1 ,
√ 1/6 21/3 31/8 3 − 1 1/3 , √ √ √ 4 1 3 2 33
iii
h3,3
iv
h4,4
v
h5,5
vi
√
21/4 1/8 , √ √ √ 4 16 15 4 2 12 2 9 23
2.4
√ 1 √ 4 5−1 5 5, 2 √ 1/3 1/6 1/12 √ √ √ −1 − 3 2 · 33/4 31 21/4 31/8 2 − 1 , h6,6 1/3 √ 2 − 3 2 35/4 33/4
vii h3,1 2−1/4
√ 3 − 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.
ISRN Discrete Mathematics
9
Lemma 2.4. One has g1,1 1,
i ii
iii
23/8 , g2,2
g3,3
1/3 √ √ √ 1/6 1 3 31/3 1 3 2 · 33/4 , √ 2
√ 1/2 23/8 1 2 , iv g4,4 v
g5,5
vi g6,6
vii g9,9
√ 1/2 1/4 5 5 5 1 2
2.5 ,
2/3 √ √ 5/6 √ 1 3 2 · 33/4 31/8 1 3 229/24
,
a 2b − 2c1/3 2b 2c1/3 , 2
where a2
√
√ √ 2 · 31/4 2 3 2 · 33/4 ,
√ √ √ b 82 45 2 48 3 25 2 · 33/4 ,
√ √ √ c 3 88 47 2 · 31/4 50 3 27 2 · 33/4 .
2.6
For proofs we refer to 17, page 1781, Theorem 6.7.
3. Theorems on Explicit Evaluation of aq, bq, and cq In this section, we present some general formulas for the explicit evaluations of cubic thetafunctions 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 √ b e−2π n/3 4 μ2/3 √ r3,n n , −2π n/3 c e
where rk,n and μn are as defined in 1.16 and 1.19, respectively.
3.1
10
ISRN Discrete Mathematics
Proof. Using the definitions of bq and cq from 1.7, one has 4 f −q 3b q . c q q1/12 f −q3 Setting q e−2π
√
n/3
3.2
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 √ √ b e−2π n/3 c e−2π/ 3n √ √ . b e−2π/ 3n c e−2π n/3
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 √ a e−2π n/3 1/3 12 1 . √ r3,n c e−2π n/3
3.4
Theorem 3.4. For any positive real number n, one has √ b e−2π n r9,n −2π √n/3 √ . c e 3
3.5
Proof. From the definitions bq and cq in 1.7, we observe that b q3 f −q 1/3 9 . c q 3q f −q Setting q e−2π result.
√ n/3
3.6
in 3.6 and then employing the definition of rk,n , we arrive at the desired
Remark 3.5. Noting that r9,1/n 1/r9,n from 2.2 and using Theorem 3.4, we find that √ √ 3b e−2π n c e−2π/3 n √ −2π/√n . c e−2π n/3 b e
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.
ISRN Discrete Mathematics
11
In the next theorem, we give a relation between cq and the parameter hk,n as defined in 1.21. Theorem 3.6. For any positive real number n, one has √ c e−8π n/3 √ 2 1 . 1 − 3 h3,n √ 4 c e−2π n/3
3.8
Proof. From 10, page 111, Lemma 5.5, we note that 2 c q4 φ −q 1−4 . c q φ −q3
3.9
Now applying the definition of hk,n , with k 3, in 3.9, we complete the proof. The next theorem connects aq with the parameter rk,n defined in 1.16. Theorem 3.7. For any positive real number n, one has
a12 e−2π
√
n/3
√ √ 4 12 1 e−2π n/3 f 24 −e−2π n/3 27 r3,n 36 r3,n
3.10
.
Proof. From 20, page 196, 2.9, we note that 3 27qf 24 −q a12 q 1 − α q α q , where αq c3 q/a3 √ q. Setting q e−2π
27e
−2π
√
n/3 24
f
n/3
−e
3.11
and then applying 3.3 in 3.11, we obtain
−2π
√
n/3
a
12
e
−2π
√
n/3
1 1 − 12 r3,n 1
3
1 12 r3,n 1
,
3.12
which on simplification gives the required result. Theorem 3.8. One has 1 a e−3nπ a e−nπ 2b e−nπ . 3
3.13
Proof. From 7, page 93, 2.8, one has 1 3 3a q − a q . b q 2 Setting q e−nπ in 3.14, we readily complete the proof.
3.14
12
ISRN Discrete Mathematics
Theorem 3.9. For any positive real number n, one has i
f 3 −e−nπ , b e−nπ f−e−3nπ
b −e
ii
−nπ
3.15
f 3 e−nπ . fe−3nπ
Proof. Setting q e−nπ and q −e−nπ in 1.7, we readily arrive at i and ii, respectively. Theorem 3.10. For all positive real numbers n, one has √ √ √ √ b e−2π n/3 31/4 e−π n/6 3 f 2 −e−2π n/3 r3,n ,
i
b −e
ii
−π
√
n/3
√ √ √ 31/4 e−π n/12 3 f 2 e−π n/3 r3,n ,
3.16
where the parameters r3,n and r3,n are defined in 1.16 and 1.17, respectively.
Proof. We rewrite bq in 1.7 as b q f 2 −q q1/12
f −q . q1/12 f −q3
3.17
√ Setting q e−2π n/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
f 3 −e−3nπ −nπ −nπ/3 , c e 3e f−e−nπ
ii c −e
−nπ
3.18
f 3 e−3nπ −nπ/3 . −3e fe−nπ
Proof. It follows readily from 1.7 with q e−nπ and q −e−nπ . Theorem 3.12. For all positive real number n, one has √ 33/4 e−π c e−2π n/3
Proof. We set q e−2π finish the proof.
√
n/3
√ √ n/2 3 2 f
r3,n
−e−2π
√ 3n
.
3.19
in 1.7 and then employ the definition of the parameter rk,n to
ISRN Discrete Mathematics
13
4. Explicit Values of aq, bq, and cq 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
i
√ b e−2π/ 3 √ 1, c e−2π/ 3
ii
iii
iv
v
vi
vii
viii
ix
x
√ b e−2π 2/3 √ 2/3 , √ 1 2 c e−2π 2/3 31/2 1 √3 2/3 b e−2π , ce−2π 21/3 √ √ 2 b e−4π/ 3 1 3 , √ √ 2 c e−4π/ 3 √ √ 10/3 b e−2π 5/3 1 5 , √ 2 c e−2π 5/3 √ √ √ b e−2π 7/3 3 7 √ √ , 2 2− 3 c e−2π 7/3 √ b e−4π 2/3 √ 4/3 √ √ 2 3 , √ 1 2 c e−4π 2/3 √ b e−2π 3 32/3 √ 4/3 , 21/3 − 1 c e−2π 3 √ b e−6π 2/3 √ 10/9 √ √ 1/3 √ 2/3 4/3 2/3 1 2 2 2 1 2 1 2 , √ 3 c e−6π 2/3 √
√ √ 4 b e−2π 13/3 11 13 3 13 , √ √ 2 2 c e−2π 13/3
14
ISRN Discrete Mathematics xi
xi
√ 4 b e−10π/ 3
√ √ 1 3 3 3 1 10 5 2 10 102 , √ 16 c e−10π/ 3 ⎞4 ⎛ √ √ √ √ √ √ √ √ √ 3 3 2 3 7 3 2 3 72 2 3 7 8 3 2 3 72 b e−14π/ 3 2 2 49 13 3 ⎟ ⎜ √ ⎠. √ ⎝ −14π/ 3 2 3 c e 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
i
ii
iii
iv
v
vi
vii
√ a e−2π/ 3 √ 3 2, √ −2π/ 3 c e √ a e−2π 2/3 √ 1/3 , √ 21/3 2 2 c e−2π 2/3 ⎞1/3 ⎛ √ 2 33/2 1 3 a e−2π ⎟ ⎜ 1⎠ , ⎝ 2 ce−2π ⎞1/3 ⎛⎛ √ √ ⎞6 a e−4π/ 3 1 3 ⎟ ⎟ ⎜⎜ √ ⎠ 1⎠ , √ ⎝⎝ −4π/ 3 2 c e √ ⎛ ⎞1/3 √ 10 a e−2π 5/3 5 1 1⎠ , √ ⎝ 2 c e−2π 5/3 ⎞1/3 ⎞3 √ ⎛⎛ √ √ a e−2π 7/3 ⎟ ⎜⎜ 3 7 ⎟ √ ⎝⎝ √ ⎠ 1⎠ , −2π 7/3 2 2− 3 c e √ 1/3 a e−4π 2/3 √ 4 √ √ 4 2 3 1 , 1 2 √ c e−4π 2/3
ISRN Discrete Mathematics
viii
ix
x
xi
15
⎞12 ⎞1/3 ⎛⎛ √ √ √ √ 3 3 3 2 a e−10π/ 3 16 5 2 10 10 1 ⎟ ⎟ ⎜⎜ ⎠ 1⎠ , √ ⎝⎝ 2 −10π/ 3 c e √
1/3 a e−6π/ 3 9 , √ 1/4 4 1 2 −1 c e−6π/ 3 ⎞ ⎞1/3 ⎛⎛ √ √ √ √ √ √ √ √ √ 3 3 3 3 3 3 3 3 2 2 a e−14π/ 3 2 7 2 7 49 13 2 7 8 2 7 3 ⎟ ⎟ ⎜⎜ √ ⎠ 1⎠ , √ ⎝⎝ −14π/ 3 2 3 c e 1/3 √ 10/3 √ 4 √ 8 √ a e−6π 4 1 2 2 1 2 1 2 . 3 2 1 ce−6π 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
i
ii
iii
iv
v
b e−2π 1 √ , c e−2π/3 3 √ √ √ 1/3 b e−2π 2 3 2 , √ √ 3 c e−2π 2/3 ⎛ ⎞1/3 √ b e−2π 3 1 ⎜ ⎟ ⎠ , √ ⎝ √ 3 −2π/ 3 c e 3 2−1
4.3
√ √ b e−4π 1 −4π/3 √ 1 2 · 31/4 3 , c e 2 3 √ b e−2 5 π √ 1/6 √ √ 1 . √ 104 60 3 45 5 26 15 √ 3 c e−2 5 π/3
Proof. It follows from Theorem 3.4 and the corresponding values of r9,n in listed in Lemma 2.1.
16
ISRN Discrete Mathematics
Theorem 4.4. One has √ √
√ c e−8π/ 3 2 3−3 1 , i √ √ 4 2 c e−2π/ 3 ⎛ ⎛ √ 1/3 ⎞⎞ 2/3 1/4 3 3 − 1 2 √ c e−8π 1⎜ ⎟⎟ ⎜ ⎝1 − 3⎝ ii 2/3 ⎠⎠. √ √ √ ce−2π 4 4 1 3 2 33
4.4
Proof. We set n 1 and 3 in Theorem 3.6 and then employ the values of h3,1 and h3,3 from Lemma 2.3vii and iii, respectively, to finish the proof. For the remaining part of this paper, we set a : φe−π π 1/4 /Γ3/4. Lemma 4.5. One has f −e−π a2−3/8 eπ/24 , ii f e−π a2−1/4 eπ/24 , iii f −e−2π a2−1/2 eπ/12 , i
1/3 √ √ aeπ/4 1 3 2 · 33/4 , √ 1/6 33/8 217/24 1 3
iv
f −e−3π
v
f −e−4π a2−7/8 eπ/6 ,
vi
√ 1/4 , f −e−6π a2−7/12 3−3/8 eπ/4 3 − 1
vii f −e−12π
a eπ/2 1/3 ,
√ √ √ 25/24 33/8 1 3 1 3 2 · 33/4
a 27/24 31/8 eπ/72 viii f −e−π/3
1/3 , √ √ √ 1 3 1 3 2 · 33/4 ix
√ 1/6 f −e−2π/3 a2−7/12 31/8 eπ/36 3 − 1 ,
x
√ 1/4 , f e−2π a2−13/16 eπ/12 2 1
xi
√ 1/6 , f e−3π a2−1/3 3−3/8 eπ/8 3 1
xii
f e−6π
1/3 √ aeπ/4 2 − 3 2 35/4 33/4 √ 1/6 . 1/12 √ 31 215/16 33/8 2 − 1
4.5
ISRN Discrete Mathematics
17
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 e
−π
√ 1/6 a2 33/8 1 3 1/3 , √ √ 25/12 1 3 2 · 33/4
ii
b e−2π
iii
1/3 √ √ 1/2 √ 1 3 2 · 33/4 b e−4π a2 2−29/12 33/8 1 3 ,
iv
b e−π/3
a2 33/8 √ 1/6 , 211/12 3 − 1
a2 25/4 33/8 , √ √ 3/2 √ 1 3 1 3 2 · 33/4
v
√ 1/3 a2 31/8 3 − 1 b e−2π/3 √ 1/6 , 213/12 3 1
vi
b −e−π
vii
4.6
a2 33/8 √ 1/6 , 25/12 3 1
b −e−2π
a2 33/8
√
1/6 3/4 √ 1/12 √ 31 21 2−1 . 1/3 √ 23/2 2 − 3 2 35/4 33/4
Proof. To prove i–v, we set n 1, 2, 4, 1/3, and 2/3, respectively, in Theorem 3.9i and use the corresponding values of f±e−πn from Lemma 4.5. To prove vi and vii, we set n 1 and 2, respectively, in Theorem 3.9ii and then use the corresponding values f±e−πn from Lemma 4.5. Theorem 4.7. One has √ 1/6 a2 37/8 1 3 1/3 , √ √ 217/12 1 3 2 · 33/4
i
c e−4π/3
ii
√ 1/6 c e−2π/3 a2 2−13/12 37/8 3 1 ,
1/3 √ √ 1/2 √ 1 3 2 · 33/4 , iii c e−π/3 2−17/12 37/8 a2 1 3 iv
c e−4π 21/4
a2 33/8 , √ √ 3/2 √ 1 3 1 3 2 · 33/4
18
ISRN Discrete Mathematics
√
1/3 3−1 √ 1/6 , 3/8 13/12 3 2 31 a2
v
c e−2π
vi
√ √ 2 3/4 −π a 1 3 2 · 3 c e √ 1/2 , 31/8 27/4 1 3
vii
⎛ ⎛ √ √ 1/3 ⎞⎞ 1/3 2/3 1/4 · 3 3 − 1 2 a2 3 − 1 1 √ ⎜ ⎟ ⎟ ⎜ c e−8π ⎝1 − 3⎝ 2/3 ⎠⎠ × √ 1/6 . √ √ √ 4 4 1 3 2 32 33/8 · 213/12 3 1 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 be−nπ from Theorem 4.6. To prove vi, we set n 1 in Theorem 3.11 and use the corresponding values of f−e−nπ from Lemma 4.5. At last, vii follows from Theorems 4.7v and 4.4ii. Remark 4.8. Setting t 1/2 in 1.10 and then employing the value of ce−π from Theorem 4.7vi, we can also evaluate be−4π/3 . Theorem 4.9. One has
i
√ 1/3 a2 10 6 3 a e−2π √ 1/4 , 2 32 3
ii
√ 1/3 √ a2 3 10 6 3 a e−2π/3 √ 1/4 , 2 32 3 ⎛
iii
√ 1/3 ⎞ 10 6 3 a2 √ 1/6 ⎜ ⎟ a e−6π ⎝31/4 1 2 ⎠, 1/4 √ 3 2 32 3
⎧⎛ ⎨ √ ⎪ √ 1/6 ⎜ iv a e−2π/9 3a2 ⎝31/4 1 2 ⎪ ⎩
4.8
√ 1/3 ⎞⎫ ⎪ ⎬ 10 6 3 ⎟ . ⎠ √ 1/4 ⎪ ⎭ 2 32 3
Proof. To prove i, we set n 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 set t 1 in 1.8 and then employ Theorem 4.9i. To prove iii, we set n 2 in Theorem 3.8 and then employ the values of ae−2π and −2π be from Theorems 4.9i and 4.6ii, respectively. To prove iv, we set t 3 in 1.8 and use the value of ae−6π .
ISRN Discrete Mathematics
19
5. Theorems on Explicit Evaluations of Aq, Bq, and Cq In this section, we use the parameters rk,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 number n, one has √ B e−π 2n 12 12 g2n . √ r2,n C e−π 2n
5.1
Proof. Employing the definition of Bq and Cq given in 1.14, we find that
12 B q f −q . C q 21/4 q1/24 f −q2
5.2
√ Setting q e−2π n/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: √ √ B e−π 2/n C e−π 2n √ √ . B e−π 2n C e−π 2/n
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 √ 4 B e−2π n r4,n . −π √n 2 C e
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 √ √ C e−π n 4B e−2π/ n √ √ . C e−π/ n B e−2π n Hence, if we know one quotient of 5.5 then the other quotient follows immediately.
5.5
20
ISRN Discrete Mathematics
Lemma 5.5. One has i
φ −e−2nπ
ii
φ e−nπ
−nπ
ar2,2n2 a , 21/8 n1/4 hn,n n1/4 21/4 rn,n
aG2n2 a , n1/4 hn,n n1/4 rn,n
a2−3/4 enπ/8 a2−5/8 enπ/8 , n1/4 gn,n n1/4 r2,n2 /2 rn,n
iii
ψ e
iv
an1/4 2−3/4 r2,2n2 enπ/8 , ψ e−π/n rn,n
5.6
, and Gn are as defined in 1.16, 1.20, 1.21, 1.23, and where the parameters rk,n , hk,n , hk,n , gk,n 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 6.2ii and 17, Theorem 6.3ii, respectively. Theorem 5.6. For any positive real number n, one has
i
ii
a4 r2,4 2n2 a4 , B e−2πn √ 4 4 2nrn,n 2 n hn,n
B e−2π/n
4 a4 n r2,2/n 2 4 2rn,n
5.7
,
where hk,n is as defined in 1.21. Proof. From 21, page 39, Entry 24iii, we note that f 2 −q φ −q 2 . f −q
5.8
B q φ4 −q .
5.9
Employing 5.8 in 1.14, we obtain
Setting q e−2πn in 5.9 and then employing Lemma 5.5i, 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.
ISRN Discrete Mathematics
21
Theorem 5.7. One has i
B −e−nπ
ii B −e
−π/n
aG8n2 a4 , 4 nh4n,n nrn,n 8 4 na4 a nGn2 4 , 4 hn,n rn,n
5.10
where hk,n is as defined in 1.21. Proof. Replacing q by −q in 5.9 and setting q e−nπ , we have B −e−nπ φ4 e−nπ ,
5.11
Employing Lemma 5.5ii in 5.11, we finish the proof of i. To prove ii, we replace n 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 Theorem 5.7i and ii, n2 B −e−nπ B −e−π/n .
5.12
Theorem 5.9. For any positive real number n, one has
C e
−nπ
√
2a4 enπ/2 , 4 ngn,n
5.13
where gk,n is as defined in 1.23.
Proof. From 21, page 39, Entry 24iii, we notice that f 2 −q2 ψ q . f −q
5.14
Thus, from 5.14 and 1.14, we find that C e−nπ 8e−nπ/2 ψ 4 e−nπ ,
5.15
Setting q e−nπ in 5.15 and employing Lemma 5.5iii, we easily complete the proof. Theorem 5.10. One has 4 na4 r2,2n 2 . C e−π/n 4 rn,n
5.16
22
ISRN Discrete Mathematics
Proof. Applying 5.14 in the definition of Cq given in 1.14 and setting q e−π/n , we find that C e−π/n 8e−π/2n ψ 4 e−π/n ,
5.17
Now, employing Lemma 5.5iv 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 Aq, Bq, and Cq, and also their quotients by using the results established in the previous section. Theorem 6.1. One has
i
√ B e−π 2 √ 1, C e−π 2
B e−2π 23/2 , ii Ce−2π √ B e−π 6 √ √ 3 2 2, iii C e−π 6 iv
v
vi
vii
viii
ix
√ B e−π2 2 √ 3/2 3/2 1 2 , √ 2 C e−π2 2 √ √ 6 B e−π 10 1 5 , √ 2 C e−π 10 √ B e−π2 3 3 √ √ 2 31 , √ C e−π3 3 √
√
6 √ B e−π 14 21 2 2−1 , √ 2 C e−π 14 √ 3 B e−4π −4π 29/4 1 2 , C e √ B e−π3 2 √ √ 4 3 2 , √ C e−π3 2
ISRN Discrete Mathematics √ B e−π2 5 √ 3 √ 3 √ 1 1 5 51 2 , x √ 8 C e−π2 5 √ B e−π2 6 √ √ 5/2 √ 3/2 21 2 6 , xi √ 1 2 C e−π2 6 √ B e−π4 2 √ 3 √ 3/2 3/2 1 2 , 4 2 10 2 xii √ 2 C e−π4 2 4 √ √ 4 √ B e−6π −11/2 3/4 1 1 2 3 3 2 · 3 , xiii Ce−6π √ B e−π2 10 √ √ 15/2 √ 3/2 −6 1 5 23 2 5 , xiv √ 2 C e−π2 10 12 1/4 B e−5π 5 1 , xv Ce−5π 215/2 √ B e−π3 6 √ √ 2/3 4 √ 1/3 √ 10/3 √ 2 1 , 2 2 1 2 2 1 xvi √ C e−π6 6 3/2 √ 3 √ B e−8π 9/4 1/4 3/4 1 16 15 · 2 2 2 12 2 9 · 2 , xvii Ce−8π √ √ √ 3/2 2 1 35 2 − 28 3 B e−π6 2 , xviii √ √ √ 8 C e−π6 2 3− 2
xix
⎞12 ⎛ √
√ √ √ B e−π7 2 14 7 2 14 ⎟ ⎜ 1 7 2 14 √ ⎠ , √ ⎝ 2 −π7 2 2 2 C e
B e−10π 215/2 −10π xx 12 , C e 51/4 − 1 √ √ √ √ √ √ √ 4 −12π 3/4 2 3 3 1 1 2 − 3 2 · 3 6 B e , −12π xxi √ 5 C e 221/4 2 − 1 4 √ √ 1 3 2 · 31/4 B e−3π , xxii Ce−3π 213/2 √ √ √ 3/2 B e−3π/ 2 −1 35 2 28 3 , xxiii √ √ √ 4 C e−3π/ 2 23/2 2 3
23
24
ISRN Discrete Mathematics
xxiv
xxv
xxvi
xxvii
√ √ 3 B e−π 3 31 , √ 27/2 C e−π 3 √ √ √ √ 4 B e−π3 2 1 3 1 − 3 22/3 · 3 , √ 4 29/2 21/3 − 1 C e−π3 2 √ √ 3 3 7 B e−π 7 , √ 29/2 C e−π 7 √ √ √ √ 4 √ √
B e−π3 7 7 − 2 3 21 3 3 3 16 21 − 27 7 . √ √ 8 √ C e−π3 7 3− 7 213/2 3 − 1 6.1
Proof. We employ the values of r2,n from Lemma 2.1 in Theorem 5.1 to finish the proof. Theorem 6.2. One has B e−2π 1 , i Ce−π 2 √
1/2 √ B e−π2 2 21 , √ ii 2 C e−π 2
iii
√ B e−2π 3 √ 1 √ 2 3 , √ 2 C e−π 3
√ B e−4π 1/4 2 2 1 , iv Ce−2π √
2 B e−2π 5 √ √ 1 1 5 2 1 5 , √ v 8 C e−π 5
vi
vii
√ √ 1/2 B e−2π 7 127 48 7 , √ 2 C e−π 7 √ 4 B e−6π 3 1 1 31/4 √ , −3π 2 Ce 2 2 2
ISRN Discrete Mathematics ⎛ ⎞4 √ √ B e−10π 5 1 1⎝3 53 5 ⎠ , 2 2 2 2 Ce−5π
viii
ix
25
−14π
⎛
B e 1⎜ ⎝ −7π 2 Ce
4
√
7
√ 21 8 7 2
⎞8
√ √ 7 21 8 7 ⎟ ⎠. 6.2
Proof. It follows easily from Theorem 5.3 and the values of r4,n from Lemma 2.1. Theorem 6.3. One has i
B e−2π 2−1/2 a4 ,
ii
√ B e−4π 2−7/4 a4 1 2 ,
iii B e−6π
iv v
vi
vii
viii
4/3 √ √ √ 4 a4 1 3 2 · 33 √ 2/3 , 211/6 · 33/2 3 − 1
1/2
√ √ 4 4 B e−8π 2−7/2 a4 16 15 2 12 2 9 23 , a4 27/2 B e−10π √ 4 √ 1/2 , 5 5 5 45−1
B e−12π
B e−3π
B e
−5π
6.3
4/3 √ a4 2 − 3 2 35/4 33/4 √ 4/3 , 1/3 √ 2/3 √ √ −1 − 3 2 · 33/4 33/2 219/12 2 − 1 31 4/3 4 √ √ √ √ 1 3 2 · 33/4 a4 1 3 2 · 31/4 , √ 2/3 4 3/2 1 3 2 ·3
4 a4 51/4 1 3 2 · 51/4 . √ 52 · 22 1 5
Proof. i–vi follow readily from the first equality of Theorem 5.6i and the corresponding values of hn,n in Lemma 2.3i–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 5.6i.
26
ISRN Discrete Mathematics
Theorem 6.4. One has a4 B e−π , 2 a4 ii B e−π/2 √ 2 , 2 1 2
i
iii
B e
−π/3
6.4
√ 24 3a4 8/3 . √ √ 10/3 √ 1 3 1 3 2 · 33/4
Proof. We set n 2, 3, and 6, respectively, in Theorem 5.6ii and then use the corresponding values of rk,n from Lemma 2.1 to complete the proofs. Theorem 6.5. One has i
B −e−π a4 ,
ii
a4 B −e−2π √ 2 , 8 2−2
iii
B −e−3π
iv B −e−4π v
2a4 √ 2 , 33/2 3 − 1 a4
√ 4
4 21
32
,
6.5
a4 B −e−5π √ 2 , 5 5−2 5
vi B −e−6π
4/3 √ √ √ a4 −4 3 2 35/4 2 3 − 33/4 2 2 · 33/4 . √ √ 2/3 24 · 33 2−1 3−1
Proof. We employ the values of hn,n given in Lemma 2.2 in Theorem 5.7i to finish the proof. Theorem 6.6. One has a4 i B −e−π/2 √ 2 , 2 2−2 ii
3a4 , B −e−π/3 √ 2 3−3
ISRN Discrete Mathematics
27
iii
a4 √ 4 4 21 , B −e−π/4 2
iv
B −e−π/5
v
5a4 √ 2 , 5−2 5
√
B −e−π/6
4/3 √ √ √ 3 a4 −4 3 2 35/4 2 3 − 33/4 2 2 · 33/4 . √ 2/3 √ √ 2 2 3 2−1 3−1 6.6
Proof. We use the values of hn,n from Lemma 2.2 in Theorem 5.7ii. Theorem 6.7. One has
i
√ C e−π 2a4 ,
a4 ii C e−2π , 4 iii
C e−3π
iv
C e−4π
v
vi
25/2 a4
4/3 √ √ √ 2/3 , 1 3 37/3 1 3 2 · 33/4 a4 √ 2 , 23 1 2
29/2 a4 C e−5π √ 2 4 , 5 5 5 51/4 1 C e−6π 33/2
vii
6.7
C e
−9π
213/3 a4 8/3 , √ √ 10/3 √ 1 3 1 3 2 · 33/4
√ a4 2 4 , 9g9,9
where g9,9 is given in Lemma 2.4(vii). Proof. The proof of the theorem follows from Theorem 5.9 and the values of gn,n from Lemma 2.4.
28
ISRN Discrete Mathematics
Theorem 6.8. One has i
√ C e−π/2 25/4 a4 1 2 ,
ii
4/3 √ 2/3 √ √ √ 2 · 33/4 3 1 , C e−π/3 2−3/2 3a4 1 3
iii
1/2 √ , C e−π/4 23/2 a4 16 15 · 21/4 12 2 9 · 23/4
iv C e−π/5
5 · 29/2 a4 √ 2 4 , 1/4 5 5 5 −1
v
C e−π/6
6.8
√ 4/3 √ √ √ √ √ 2 3 1 3 1 2 − 3 12 · 33/4 217/3 31/2 a4 . 4/3 √ 5/3 √ √ 2 √ 3/4 1 3 1 3 2·3 2−1
Proof. We set n 2, 3, 4, 5, and 6 in Theorem 5.10 and then employ the corresponding values of rk,n listed in Lemma 2.1.
References 1 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, UK, 1996, Indian edition is published by Universal Book Stall, New Delhi, 1991. 2 S. Ramanujan, “Modular equations and approximations to π,” Quarterly Journal of Mathematics, vol. 45, pp. 350–372, 1914. 3 S. Ramanujan, Collected Papers, Chelsa, New York, NY, USA, 1962. 4 J. M. Borwein and P. B. Borwein, Pi and the AGM, John Wiley & Sons, New York, NY, USA, 1987. 5 S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, India, 1957. 6 B. C. Berndt, S. Bhargava, and F. G. Garvan, “Ramanujan’s theories of elliptic functions to alternative bases,” Transactions of the American Mathematical Society, vol. 347, no. 11, pp. 4163–4244, 1995. 7 B. C. Berndt, Ramanujan’s Notebooks. Part V, Springer-Verlag, New York, NY, USA, 1998. 8 J. M. Borwein and P. B. Borwein, “A cubic counterpart of Jacobi’s identity and the AGM,” Transactions of the American Mathematical Society, vol. 323, no. 2, pp. 691–701, 1991. 9 S. Cooper, “Cubic theta functions,” Journal of Computational and Applied Mathematics, vol. 160, no. 1-2, pp. 77–94, 2003. 10 B. C. Berndt, H. H. Chan, and W.-C. Liaw, “On Ramanujan’s quartic theory of elliptic functions,” Journal of Number Theory, vol. 88, no. 1, pp. 129–156, 2001. 11 J. Yi, Construction and application of modular equation, Ph.D. thesis, University of Illionis, Urbana, Ill, USA, 2001. 12 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, India, 1988. 13 K. G. Ramanathan, “On some theorems stated by Ramanujan,” in Number Theory and Related Topics (Bombay, 1988), pp. 151–160, Oxford University, Bombay, India, 1989. 14 B. C. Berndt, H. H. Chan, S.-Y. Kang, and L.-C. Zhang, “A certain quotient of eta-functions found in Ramanujan’s lost notebook,” Pacific Journal of Mathematics, vol. 202, no. 2, pp. 267–304, 2002. 15 N. D. Baruah and N. Saikia, “Some general theorems on the explicit evaluations of Ramanujan’s cubic continued fraction,” Journal of Computational and Applied Mathematics, vol. 160, no. 1-2, pp. 37–51, 2003. 16 N. D. Baruah and N. Saikia, “Some new explicit values of Ramanujan’s continued fractions,” Indian Journal of Mathematics, vol. 46, no. 2-3, pp. 197–222, 2004. 17 N. D. Baruah and N. Saikia, “Two parameters for Ramanujan’s theta-functions and their explicit values,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 6, pp. 1747–1790, 2007.
ISRN Discrete Mathematics
29
18 N. D. Baruah and N. Saikia, “Modular relations and explicit values of Ramanujan-Selberg continued fractions,” International Journal of Mathematics and Mathematical Sciences, Article ID 54901, 15 pages, 2006. 19 J. Yi, “Theta-function identities and the explicit formulas for theta-function and their applications,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 381–400, 2004. 20 H. H. Chan, “On Ramanujan’s cubic transformation formula for 2 F1 1/2, 2/3; 1; z,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 124, no. 2, pp. 193–204, 1998. 21 B. C. Berndt, Ramanujan’s Notebooks. Part III, Springer-Verlag, New York, NY, USA, 1991.
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