J. Comp. & Math. Sci. Vol.2 (1), 129-134 (2011)
Evaluation of Cubic Theta Functions MANOJ KUMAR PATHAK and H. S. SHUKLA Department of Mathematics, R. S. K. D. P. G. College, Jaunpur, (U.P.) India ABSTRACT In this paper, making use of Ramanujan’s modular equations, we have evaluated cubic theta functions for some fixed values of q. Keywords:- modular equation/ modular identity/ cubic theta function. ∞
c(q, z ) = q1 / 3 ∑ qm
1. INTRODUCTION Borweins discovered cubic analogue of Jacobi’s theta functions as,
a (q ) =
b (q ) =
c (q ) =
∞
∑
m , n = −∞ ∞
∑
2 2 q m + mn+ n ,
ω m −n q m
2
+ mn + n 2
,
m , n = −∞
(1.1)
(ω = e2π i /3 )
,
(1.2) 2 ∞ (m + 1 / 3) + (m + 1 / 3)(n + 1/ 3 ) + (n +1/ 3 ) , ∑q 2
where IqI<1. Hirschhorn et al generalized these functions for IqI<1 and z ≠ 0 as;
a (q , z ) = b (q , z ) =
∞
∑q
m
2
+ mn + n
m , n = −∞ ∞
m ∑q
m , n = −∞
2
zn − m ,
+ mn + n
2
zn (1.5)
and
zn− m
(1.6) respectively and gave simple proof of several elegant identities by employing Jacobi’s triple product identity. The functions a(q) = a(q,1), b(q) = b(q,1) and c(q) = c(q,1) Bhargava further generalized them for IqI<1, ε, z ≠ 0 as; ∞
a (q, ε , z ) = ∑ qm
2
+ mn + n
m ,n =−∞
b (q , ε , z ) =
∞
∑ω
m − n m
q
m , n = −∞
2
2
ε m + n zn − m ,
+ mn + n
(ω = e 2 π i / 3 )
(1.7) ε zn ,
2
m
(1.8)
and
(1.4) 2
2 + mn + n + m + n
m,n =−∞
m, n = −∞
(1.3)
2
∞
c(q,ε , z ) = q1 / 3 ∑ qm m ,n= −∞
2
2 + mn + n + m + n m + n n − m
ε
z
(1.9) and made an attempt to unify these functions. In chapter 16 of his second notebook, Ramanujan7 defines the function:
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
Manoj Kumar Pathak, et al., J. Comp. & Math. Sci. Vol.2 (1), 129-134 (2011)
130
∞
f (a,b) =
a n ( n + 1) 2 b n ( n − 1) 2 ,
∑
n = −∞
|ab|<1 (1.10) which is equivalent to Jacobi’s thetafunction θ 3 (z , τ ) =
∞
∑
qn
2
cos 2 nz ,
q = e π i τ , Im τ > 0 .
(1.11)
In fact
f (a,b) = θ3(z,τ ), a = qe 2iz , b = qe −2iz (1.12) The following special cases of f(a,b) namely, ∞ 2 [− q;−q]∞ [− q; q 2 ]∞ [q 2 ; q 2 ]∞ φ (q) = f (q, q ) = ∑ q n = = [q;−q]∞ [− q 2 ; q 2 ]∞ [q; q 2 ]∞ n = −∞ (1.13) ∞
∑ q n ( n + 1)
2
n=0
=
[q ; q ] [q ; q ] 2
2
2
∞
∞
(1.14) and
f (−q) = f (−q, −q ) = 2
∞
∑
n =−∞
(−) q
n (3 n −1) 2
n
= [ q; q]
(1.15)
∞
are important for Ramanujan’s development of his theories of theta and elliptic functions. Here
the
expression
[a ; q ]n
denotes the q-Pochhammer symbols,
[a ; q ]n
[a ; q ]0
= ( 1 − a )( 1 − aq )...( 1 − aq n − 1 ),
=1
and
[a ; q ]∞
=
∞
∏ (1 − aq r ) .
r =0
In this present paper, we have evaluated cubic theta functions b ( q ) ,
a (q )
and c ( q )
for some particular
(1.16) [Berndt2;chapter 33, Lemma(2.1) p.94]
3( − q) , f (− q 3 ) f
b (q ) =
n = −∞
ψ (q ) = f (q, q 3 ) =
values of q. we shall make use of following known results in our analysis. 1 + 4 αβ , z1 z3 a (q ) =
c (q )
= 3 q1/3
f
3
(1.17)
(−
q 3 f (− q )
),
(1.18)
φ (− q ) , c (q 2 ) (1.19) = c 2 (q ) φ 3 (− q 3 ) ψ 3 (q 6 ) , c 2 (q 4 ) (1.20) = q2 2 2 ψ (q ) 3 c (q ) c 2 (q 4 ) ψ 3 (q 6 )φ (− q ) .(1.21) = q2 c 2 (q ) ψ (q 2 )φ 3 (− q 3 ) [Berndt 2; chapter 33, p. 109] −π 1 If λ = π / 4 , then φ (e )= λ . Γ [3 / 4 ] (1.22) φ (e − π ) . (1.23) 4 6 3 − 9 = φ (e − 3 π ) [Berndt 2; chapter 35, Entry 10 (i), p. 325 and Entry 4, p. 327] 3
Following modular equations are also needed.
m 2 + 2m − 3 , 4m 1/ 8 m −1, 3 (β / α ) = 2 1 / 8 3 (1 − β ) m +1 , = (1 − α ) 2
(αβ )1 / 4
(1 − α ) (1 − β )
=
3 1 / 8
=
3−m , 2m
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
(1.24) (1.25) (1.26) (1.27)
Manoj Kumar Pathak, et al., J. Comp. & Math. Sci. Vol.2 (1), 129-134 (2011)
α 3 β
1/8
3 + m . (1.28) = 2m [Berndt 1; chapter 19, (5.10) & (5.3) p. 232233]
For details about modular equations, one is referred Berndt, chapter 33. Following are the results due to Ramanujan [7: chapter 17, entries 10, 11 and 12]:
φ (q ) =
z
,
(1.29)
φ (- q) = √ 1 1/4 ,
(1.30)
φ (- q²) = √ 1 1/8 ,
(1.31)
z x 1 / 4 , 2 2
ψ (q 2 ) = f (− q ) =
z 62
f (− q 2 ) =
(1.32)
1 / 24
(1 − x )1 / 6 x
z 32
, (1.33)
q
1 /1 2 x (1 − x ) , (1.34) q
1 / 24 f (− q 4 ) = z (1 − x ) 34
x q
1/ 6
. (1.35)
multiplier associated with α and β , so using equations (1.29)-(1.35), let us define;
φ (q ) = φ (q 3 )
φ (− q 3 ) = φ (− q ) 3
Q=
φ 3(− q 6 ) Q = = 1 φ (− q 2 )
z1 = z3
(2.6) 1 3 /12 , 3 1 z13 α (1 − α ) = Q = 5 f (− q 6 ) 3 4 z3 β (1 − β ) (2.7) 3 f (− q6) 1 z3 β 3(1 − β )3 1/12 , Q = q2 / 3 = 3 6 f (− q 2) 3 4 z1 α (1 − α ) (2.8) 1/ 24 3 1/ 6 , 3 3 4 α f (− q ) 1 3 (1 − α )
f 3 (− q 2)
Q = = 7 f (− q12) 3 16
z1 z3 (1 − β )
β
(2.9) 3 f (− q12) 1 Q = q4/ 3 = 8 3 4 16 f (− q )
z 33 (1 − β ) z1 (1 − α )
3 1/ 24
3 1/ 6
β α
3. EVALUATION OF THETA FUNCTIONS
In modular equations (1.24)-(1.28), is of degree three over α and m is the
P =
3 1 / 4 , (2.4) z 33 β α z1 3 1/ 6 3 1/ 24 , z13 (1 − α ) α z3 (1 − β ) β (2.5) 3 1/6 3 1/ 24 3 3 , f (− q ) 1 z3 (1− β ) β Q = q1/3 = 3 3 4 α f (− q) 2 z1 (1−α ) 1 4
(2.10)
2. MODULAR IDENTITIES
β
ψ 3(q 6 ) Q = q2 = 2 ψ (q 2 ) f 3(− q ) 1 Q = = 3 f (− q3) 3 2
131
m , 3 1/ 4
P=
m =
(2.1)
z 33 z1
(1 − β ) (1 − α )
, (2.2)
z 33 z1
(1 − β ) (1 − α )
, (2.3)
3 1/ 8
Taking q = e − π in equation (2.1) and using equation (1.23) we get, z1 φ (e − π ) 4 = = 6 3 −9 z3 φ (e − 3π ) (3.1)
Putting this value of m in equations (4.1.24)(4.1.28) we get, (αβ )1 / 4 = 3 + 1 − 2 3 , (3.2) 2
β 3 1 / 8 = α
6
3 − 9 −1 , 2
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
(3.3)
Manoj Kumar Pathak, et al., J. Comp. & Math. Sci. Vol.2 (1), 129-134 (2011)
132
(1 − β )3 1 / 8 (1 − α )
(1 − α )3 (1 − β )
α 3 1/ 8 β
1/8
3 − 9 + 1 , (3.4) 2
6
=
3 −
=
3 − 9
6
2
,
(3.5)
3+ 6 3−9 . = 2 6 3−9
(3.6)
4 6 3 −9
Using equation (3.7) and equation (3.8) we obtain,
z1 z 3 = z 13 z3
z 33 z1
λ
2
,
(3.9)
46 3 −9
2 = λ 4 6
3 − 9
=
(6
,
)
2 1 + 6 3 − 9 3 − 3 π ) = λ2 4 6 3 − 9 φ (− e , Q= φ (− e−π ) 4 6 3 −9
)
(
(3.12) 24
6 3 − 9 1 + 6 3 − 9
(
)
)
(
Q = e−π /3 4
24
λ 3 f (− e−3π ) = π − f (− e )
3 − 9 1 +
6
6
2 6 3 −9
)
,
6 3 − 9 ,
3 ( 3 − 1)
(3.16) f (− e − 2π Q = 5 f (− e − 6π
)= λ ) {2 (
3
Q = 6
24
6 3 − 9 , (3.17) 2 /3 3 −1
)}
3 2 f (− e−6π ) λ 4 6 3 − 9 , = 6 3 f (− e−2π )
e −2π /3
(3.18) 3 f (− e − 4π ) Q = = 7 f (− e−12π )
λ2 4 6 3 − 9 3 + 6 3 − 9 ,
(
)
4 6 3 − 9 3 2 3 −1
(3.19) e − 4π /3
2 3 f (− e−12π ) λ 4 6 3 − 9 = 6 3 − 9 −1 f (− e−4π ) 12 3 3 −1
)
(
(3.20)
(3.11)
Now, using equations (3.2)-(3.11) we have following evaluations from equations (2.2)(2.10) for q = e − π :
3 φ (− e− 6π ) = λ Q = 1 φ (− e− 2π )
16 6 3 − 9
(3.10)
3 − 9 .
3 − 9
(
2 3 f (− e −π ) λ 4 6 3 − 9 3 − 6 3 − 9 , Q = = 3 2 3 − 1 2 6 3 − 9 3 f (− e−3π )
Q = 8
λ2 4 6
)
)=
(3.15)
From equation (1.22) and equation (1.29) we have, z1 = φ (e −π ) = λ (3.7) From (3.7) and (3.2) we find λ (3.8) z = φ (e −3π ) = 3
ψ
( e−2π
2 2 λ 4 6 3 − 9 6 3 − 9 −1 ,
(3.14)
3 − 9
6
Q = 2
−2π ψ 3(e − 6π e
4. EVALUATION OF CUBIC THETA FUNCTIONS Putting q = e − π in equation (1.16) and using equation (3.2) and equation (3.8) we get, λ2 3+ 3 − 2 3 , a ( e− π ) = 4 6 3−9
2
(4.1) − π Putting q = e in equation (1.17) and using equation (3.15) we get, 2 λ 4 6 3 − 9 3 − 6 3 − 9 , b ( e− π ) = 32
(
3 −1
)
2 6 3 −9
(3.13) Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
(4.2)
Manoj Kumar Pathak, et al., J. Comp. & Math. Sci. Vol.2 (1), 129-134 (2011)
Again, putting q = e − π in equation (1.18) and using equation (3.16) we find 2 λ 4 6 3 − 9 1 +
c ( e− π ) =
(
2 3
6 3−9
3 −1
)
,
(4.3) Putting q = e − 2 π in equation (1.17) and using equation (3.17) we have 2 λ 4 6 3 − 9 , (4.4) b (e− 2 π ) = 2 /3 2 3 −1 Again, putting q = e − 4 π in equation (1.17) and using equation (3.19) we find,
{(
b (e − 4 π
)
=
)}
λ 2 4 6 3 − 9 3 +
4 6 3 −9 3 2
6 3 − 9
(
3 −1
)
.
(4.5) − π Putting q = e 2 in equation (1.18) and using equation (3.18) we get,
c (e − 2 π
24 )= λ
6 3−9 . 2 3
(4.6)
q = e − 4π
Taking in equation (1.18) and using equation (3.20) we obtain,
λ2 4 6 3 − 9 c (e−4π ) =
(
)
4 3 3 −1
6
3 − 9 −1 . (4.7)
q = e −π
Putting in equation (1.19) and using equation (3.12) we obtain,
(
)
c (e − 2π ) = 4 3 2− 3 . π − 2 2 e c ( ) λ2 4 6 3 − 9 1+ 6 3 − 9
(4.8)
133
Putting q = e − 2 π in equation (1.19) and using equation (3.13) we find,
(
)
c (e− 4π ) = 2 3 2− 3 , c 2(e− 2π ) λ2 4 6 3 − 9 1 + 6 3 − 9
(4.9) − π in equation (1.20) Putting q = e and using equation (3.14) we get,
c 2(e − 4π ) = c (e − 2π )
2 2 λ 4 6 3 − 9 6 3 − 9 −1
(
16 3 2 − 3
)
.
(4.10) From equation (4.9) and equation (4.10) we get,
(e − 4 π ) = c (e − 2 π ) c
6 2
(
3 − 9 − 1 . 3− 1
)
(4.11)
Putting q = e − π in equation (1.21) and using equation (3.14) and equation (3.12) we get,
2 c2(e −6π ) = 1 6 3 − 9 −1 . c (e−π ) 4 6 3 − 9 + 1
(4.12)
We are thankful to Dr. S. N. Singh, Department of Mathematics, T.D.P.G. College Jaunpur and Dr. H. S. Shukla, Department of Mathematics, R. S. K. D. P. G. College, Jaunpur for his able guidance in the preparation of this paper.
REFERENCES 1. Berndt, B. C., Ramanujan’s Notebook, Part III, Springer – Verlag, New York (1991).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
134
Manoj Kumar Pathak, et al., J. Comp. & Math. Sci. Vol.2 (1), 129-134 (2011)
2. Berndt, B. C., Ramanujan’s Notebook, Part V, Springer – Verlag, New York (1998). 3. Bhargava, S., J. Mathematical Analysis and Applications, 193: 543 –558 (1995). 4. Borwein, J. M. and Borwein, P. B., Trans. Amer. Math. Soc., 323: 691 – 701 (1991). 5. Denis, R.Y. and Singh, S.N., on certain modular identities and evaluation of
theta functions with application, Journal of Indian Acad. Math. Vol. 31, No.1, pp. 277 – 287 (2009). 6. Hirschhorn, M. D., Garvan, F. G. and Borwein, J. M., Canadian J. Math. 45 : 673 – 694 (1993). 7. Ramanujan, S. (1957), Notebooks of Srinivasa Ramanujan, Vol. III, Tata Institute of Fundamental Research, Bombay.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)