J. Comp. & Math. Sci. Vol.2 (3), 557-565 (2011)
Mock Theta Functions of Order Six and Eight SHIKHA SRIVASTAVA1 and R. P. SINGH2 1
Dept. of Mathematics, G.L.A. University, Mathura (U.P.) India Dept. of Mathematics, T.D.P.G. College, Jaunpur (U.P.) India
2
ABSTRACT In this paper, making use of a known result we have established continued fraction representations of some mock theta functions of order six and eight. In the last section of this paper, relations among mock theta functions of order six and eight have also been established. Keywords: Mock theta functions, order, continued fraction, identity. AMS subject classification code: 33A15.
1. INTRODUCTION, NOTATION AND DEFINITIONS For real or complex ( q < 1) , let
(
α and q
[α ; q ]n = (1 − α )(1 − α q ) ... 1 − α q n−1 [α ; q ]
0
=1
∞ [ α ; q ] ∞ = ∏ (1 − α q r r=0
and
)
) , n ≥ 1,
[α 1 , α 2 ,..., α r ; q ]n = [α 1 ; q ]n [α 2 ; q ]n ...[α r ; q ]n . From the above notations, a generalized basic hypergeometric series is defined as n a1 , a 2 ,..., a r ; q ; z ∞ a1 , a 2 ,..., a r ; q n z Φ = , ∑ r s b , b ,..., b 1 2 n = 0 q , b , b ,..., bs ; q s 1 2 n
(1.1) where max. ( q , z ) < 1 for the convergence of the series (1.1). We shall make use of following known result due to Singh, S.N.3 in our analysis.
aq , b , c ; q ; de / abc de Φ ( a − d )(1 − b )(1 − c ) / (1 − e )(1 − d )(1 − dq ) 2 dq , e = 1 abc e 1− a , b , c ; q ; de / abc 1− Φ 3 2 d , e aq + 1− e
3
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
558
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011)
)
(
(1 − aq ) 1 −
dq dq e 1− / (1 − e )(1 − d q ) 1 − dq 2 b c a q 1− deq ( a − d q )(1 − b q )(1 − c q ) / (1 − e q ) 1 − d q 2 1 − d q 3 abc e 1 − aq + ... . 1 − eq
)(
(
)
(1.2)
Putting a=1 in (1.2) we get de
q , b, c; q; d e / bc 1 b c (1 − b )(1 − c ) / (1 − e )(1 − d q ) Φ = 3 2 dq,e e 1− 1− q + 1− e dq dq e 1− / (1 − e )(1 − d q ) 1 − d q 2 (1 − q ) 1 − b c q
)
(
1− deq (1 − d q bc
(
) (1 − b q ) (1 − c q ) / (1 − e q ) 1 − d q 2
)(1 − d q 3 )
e 1− q + ... . 1 − eq
(1.3)
In this paper, we shall discuss about the mock theta functions of order six and eight. We shall also try to give continued fraction representation of some of them.
whose form slightly differs from Φ taken
2. MOCK THETA FUNCTIONS OF ORDER SIX
(ii)
Andrews and Hickerson1 defined following mock theta functions of order six, n n2 2 (i) Φ q = ∞ ( − ) q q ; q n . (2.1) ( ) ∑ [ − q ; q ]2 n n=0 We can write it as, α , q , q 2 ; q 2 ; q / α (2.2) , Φ ( q ) = Lim 3Φ 2 α →∞ − q, − q 2
3 2
in (1.3) so it is not expressible in continued fraction. n n 2 + 2 n +1 q ; q 2 ∞ (− ) q n Ψ (q ) = ∑ . [ − q ; q ]2 n +1 n=0 α , q , q 2 ; q 2 ; q 3 / α q . = Φ αLim 3 2 2 3 1 + q →∞ − q , − q
(2.3)
It is not possible to express Ψ ( q ) in continued fraction. (iii)
∞ q ρ (q) = ∑ n =0
n n+1 /2
[ −q; q ]n
q; q 2 n+1
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011) =
α , − q , q; q ; − q / α 1 Lim Φ . (1 − q ) α →∞ 3 2 − q 3/2 , − q 3/2
ρ ( q ) is also not expressible in continued fraction.
(iv)
∞ q σ (q ) = ∑ n=0
n +1 ( n + 2 )/2
α , − q , q; q; − q 2 / α q . Φ αLim 3 2 3/2 1 − q →∞ , − q 3/2 q
(2.4)
=
559 (2.5)
n
( − ) q n q ; q 2 n (v) λ ( q ) = ∞ ∑
[ − q ; q ]n
n=0
[ − q; q ]n
= 3Φ 1
q; q 2 n +1
q,−
q ; q; q; − q
.
− q
(2.6)
n n +1 (1 + q n ) q ; q 2 n ∞ (− ) q (vi) 2 µ ( q ) = ∑ [ − q ; q ]n +1 n =0 α , q α , − q α , q , − q ; q ; − q q . = Lim Φ3 α →1 1 + q 5 α , − α , − q 2 α , q α − q α , q , − q ; q ; − q q . Lim 5Φ 3 α , − α , −q2 1 + q α →1
=
q = 1+ q
(vii)
(2.7)
q , − q , q; q; − q 2 q , − q , q; q; − q . + Φ 3 Φ1 3 1 2 −q2 −q
2 ∞ q n [ q ; q ]n γ (q ) = ∑ n =0 q , ω q , ω 2 q; q
α , β , q ; q ; q αβ = L im α ,β → ∞ ω q ; ω 2 q n
.
(2.8)
(2.9)
Making use of (1.3) we can express γ ( q ) in the form of continued fraction as shown below. 2 1 q α β (1 − α L im ϑ (q ) = α ,β → ∞
)(
(
)(1 − β ) / 1 − ω 2 q 2 1 − ω q 2 1−ω 2 1− ω 2q2
+
)
.
(1 − q 2 )(1 − ω q 2 / α )(1 − ω q 2 / β ) ω 2 / (1 − ω 2 q 2 )(1 − ω q 2 )(1 − ω q 4 ) 1−
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
560 q4
αβ
=
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011)
(1 − α q 2 )(1 − β q 2 )(1 − ω q 2 ) / (1 − ω 2 q 4 )(1 − ω q 4 )(1 − ω q 6 )
(
1−ω2 1− ω 2q4
+ ... .
) (1 − q 2 ) ω 2 / (1 − ω q 2 )(1 − ω q 4 ) q 6 (1 − ω q 2 ) / (1 − ω q 4 )(1 − ω q 6 ) 1− ) (1 − ω 2 ) + ... .
2 2 1 q / 1−ωq 1− 1− ω 2 +
(
(2.10)
Out of these seven mock theta function of order six only one i.e. γ ( q ) in expressible in continued fraction. 3. MOCK THETA FUNCTIONS OF ORDER EIGHT Gordon and McIntosh2 found following eight mock theta functions of order eight,
(i)
(ii)
(iii)
2 q n − q ; q 2 α , − q , q 2 ; q 2 ; − q / α ∞ n S0 ( q ) = ∑ (3.1) = L im Φ α → ∞ 3 1 −q2 n=0 − q 2 ; q 2 n n2 + 2n − q; q 2 α , − q , q 2 ; q 2 ; − q 3 / α ∞ q S1 n = L im . (3.2) Φ . α →∞ 3 2 2 (q ) = ∑ 2 2 −q ; q n=0 −q n T0
∞ q (q ) = ∑ n=0
( n + 1 )( n + 2 ) − q 2 ; q 2
n
− q; q 2
n + 1 α , − q 2 , q 2 ; q 2 ; − q 4 / α = L im Φ 1 + q α → ∞ 3 2 −q3
q2
(iv)
(v)
T1
U0
∞ q (q ) = ∑ n=0
n(n + 1)
.
(3.3)
− q 2 ; q 2
n − q; q 2 n + 1 α , − q 2 , q 2 ; q 2 ; − q 4 / α 1 = L im Φ (1 + q ) α → ∞ 3 1 − q 3
n2 − q; q 2 ∞ q n = L im Φ (q ) = ∑ α →∞ 3 2 2 4 n=0 −q ; q n +1
.
α , − q , q 2 ; q 2 ; − q α . 2 2 iq , − iq
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
(3.4)
(3.5)
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011)
561
Making use of (1.3) one can represent U0 (q) in continued fraction by replacing q by q2 and then taking b = α , c = − q, d = i, e = −iq 2 in it, q2 (1 − α U0 1 αq im ( q ) = αL→ ∞ 1 +
(1 − q 2 ) 1 − iqα
1+ i 1 + iq 2
(
2
)(
(
)(1 + q ) / 1 + iq 2 1 − iq 2
(1 + iq ) i / 1 + iq 2
)
)(1 − iq 2 )(1 − iq 4 )
1+ q3
α
(
1 − iq 2
)(
1−α q2
(
)(
1+ i 1 + iq 4
4 1 q (1 + q ) / 1 + q = 1 − (1 + i ) 1 + iq 2
(
)(
1 + q 3 / 1 + iq 4
)
)(
1 − iq 4
)(
... .
)
1 − iq 6
) (1 − q 2 ) i (1 + iq ) / (1 + q 4 )(1 + iq 4 ) 1−
(1 + q 4 ) q 5 (1 − iq 2 )(1 + q 3 ) / (1 + q 8 )(1 − iq 6 ) (1 + i ) (1 − iq 4 ) ... . 8 1 + q ) ( i (1 − q 2 ) (1 + iq ) / (1 − iq 4 ) q (1 + q ) 1 = 1 − (1 + i ) 1 − iq 2 − 1− ) ( (
)(1 − i q 2 ) / (1 − i q 6 ) (1 + i ) (1 − i q 4 ) ... .
q5 1 + q3
(vi)
U1
2 n +1 − q; q 2 ∞ q n (q ) = ∑ 2 4 n=0 −q ;q n +1
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
(3.6)
562
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011) 2 2 q3 α , − q , q ; q ; − α = L im Φ . 3 2 2 α → ∞ 1+ q iq 3 , − iq 3
q
(3.7)
Replacing q by q2 and then taking b = α , c = − q, d = iq and e = −iq3 in (1.3) we get. q3 (1 − α )(1 + q ) / 1 + iq 3 1 − iq 3 U1 q 1 α L im (q ) = 2 α → ∞ 1 + 1 + iq 1+ q − 1 + iq 3 3 iq 1 + iq 2 / 1 + iq 3 1 − iq 3 1 − iq 5 iq 1 − q 2 1 − α
)
(
)(
(
)(
(
)(
)(
)
)
1+ q5
α
(
1 − iq 3
)(
1−α q2
)(
1 + iq − ... . 1 + iq 5
( q = 1+ q2
)(
1 + q 3 / 1 + iq 5
(
)
)(
1 − iq 5
) (
)
)(
1 − iq 6
)(
)(
3 3 iq 1 − q 2 1 + iq 2 / 1 − iq 3 1 q (1 + q ) / 1 − iq 1 − 1− (1 + iq ) − q 7 1 + q 3 1 − iq 3 / 1 − iq 5 1 − iq 6
(
)(
)(
(1 + i q ) ... .
)(
) (1 − iq 5 )
)
(3.8)
Thus we find that U0 (q) and U1(q) can be expersed in continued fraction.
(vii)
2 2 ∞ q − q ; q n 1 + Vo (q ) = 2 ∑ q; q 2 n=0 n 2n2 −q 2 ; q 4 ∞ q n = 2 ∑ 2 q; q n−0 2 n + 1 α , − q , q 2 ; q 2 ; − q α = 2 L im Φ α → ∞ 3 1 q
.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
(3.9)
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011)
563
2
( n + 1) q
− q; q 2 n 2 q; q n +1
∞ V (q ) = ∑ 1 n=0
2 n 2 + 2 n +1 − q 4 ; q 4 ∞ q ∞ = ∑ q; q 2 n−0 2 n + 2
q 1 − q
=
im Φ αL→ ∞ 3 1
q3 2 2 α , − q , q ; q ; − . α q
(3.10)
4. RELATIONS AMONG MOCK THETA FUNCTIONS In this section, an attempt has been mode to establish relations among the mock theta functions of order six and eight we shall use the following identity,
n m n n n −1 v ∑ δm ∑ αr = ∑ αv ∑ δm − ∑ αr +1 ∑ δm. m =0 r =0 r =0 m =0 r =0 m =0
(i)
Choosing α = r
(− )
r
2 q r q ; q 2
r and δ m =
− q; q 2 2 r
(4.1)
( m + 1) (− ) q
2
m
q; q 2 m
[ − q ; q ]2 m +1
in (4.1) we find, n (− ) ∑ m =0
=Φ
m
q
m +1
2
q; q 2 m
[ − q ; q ]2 m +1
n −1 ( − )
n (q ) Ψ n (q ) − ∑
r=0
r +1
Φ m (q )
q
r + 1
2
q; q2 r + 1 Ψ r
[ − q ; q ]2 r + 2
(q ),
(4.2)
r r2 q ; q 2 n (− ) q r is partial mock theta function of order six. As n → where Φ ( q ) ∑ n [ − q ; q ]2 r r =0 ∞, (4.2) yields Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
564
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011)
m +1
2
m q; q 2 ∞ (− ) q m Φ (q ) Ψ (q ) = ∑ Φ m (q ) [ − q ; q ]2 m +1 m =0
2 r r +1 − q q ; q 2 ∞ ( ) r +1 Ψ − ∑ r ( q ). q q − ; [ ] r =0 2r +2
(4.3)
(4.3) gives the relations between two mock theta functions of order six. Again, choosing 2
r ( − ) q r q ; q 2 r αr = [ − q ; q ]2 r
and
δm =
qm
2 − q ; q 2
m in (2.4.1) we get, −q2; q2 m
m 2 − q; q 2 n q m Φ n ( q ) So , n ( q ) = ∑ Φ m (q ) 2 2 m =0 − q ; q m 2 r ( r + 1) q ; q 2 n −1 ( − ) q r +1 S o , q . − ∑ ) m ( [ − q ; q ]2 r + 2 r =0
(4.4)
As n → ∞, (4.4) yields; m 2 − q; q 2 ∞ q m Φ (q ) S (q ) = ∑ Φ m (q ) 0 2 2 m =0 − q ; q m 2 r ( r + 1) − q q ; q 2 ∞ ( ) r +1 So , q − ∑ m ( ) − q ; q [ ]2 r + 2 r =0
(4.5)
(4.5) gives a relation between a mock theta function of order six and a mock theta function of order eight. Also in (4.5) So, n (q) is partial mock theta function of order eight. Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
Shikha Srivastava, et al., J. Comp. & Math. Sci. Vol.2 (3), 557-567 (2011)
(iii)
Choosing
αr =
2 q r + 2 r − q ; q 2
r and
−q2 ; q2 r
δm =
qm
565
2 − q ; q 2
m −q2; q2 m
in (4.1) we get, So,n (q) S1,n
2 q m − q ; q 2 n m (q) = s q ∑ 1, m ( ) m =0 − q 2 ; q 2 m
As n → ∞, we get; S0(q) S1(q)
( r + 1) 2 + 2 r + 2 − q ; q 2 n −1 q r + 1 S q + ∑ 0 ,r ( ) 2 2 −q ; q r=0 r + 1
m 2 − q; q 2 ∞ q m = ∑ S (q ) 1, m 2 2 m =0 − q ; q m
(4.6)
(4.7)
(4.7) gives the relation between two mock theta functions of order eight. Similarly one can establish relations between any two theta functions of order six and eight.
REFERENCES 1. Andrews, G. E. and Hickerson, D., Ramanujan Lost Notebook VII: The Sixth order mock theta functions, Adv. Math. 89, p. 60-105, (1991).
2. Gorden, B. and Mcintosh, R.J., Some eight order mock theta functions, J. London Math. Soc., 321-335, (2000). 3. Singh S.N., Basic hypergeometric series and continued fractions, The Mathematics Student, Vol. 56, No. 1-4, p. 91-96, (1998).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)