A modular transformation for a generalized theta function with multiple parameters
We obtain a modular transformation for the theta function $$∑_{-∞}^{∞}∑_{-∞}^{∞}q^{a(m^2 + m^n) + cn^2 + λm + μn + ν_{ς}Am + Bn_{Z}Cm + Dn},$$ which enables us to unify and extend several modular transformations known in literature.
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| Дата: | 2009 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3079 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509105987256320 |
|---|---|
| author | Bhargava, S. Mahadeva, Naika M. S. Maheshkumar, M. C. Бхаргава, С. Махадева, Найка М. С. Махешкумар, М. С. |
| author_facet | Bhargava, S. Mahadeva, Naika M. S. Maheshkumar, M. C. Бхаргава, С. Махадева, Найка М. С. Махешкумар, М. С. |
| author_sort | Bhargava, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:57Z |
| description | We obtain a modular transformation for the theta function $$∑_{-∞}^{∞}∑_{-∞}^{∞}q^{a(m^2 + m^n) + cn^2 + λm + μn + ν_{ς}Am + Bn_{Z}Cm + Dn},$$
which enables us to unify and extend several modular transformations known in literature. |
| first_indexed | 2026-03-24T02:35:49Z |
| format | Article |
| fulltext |
UDC 517.5
S. Bhargava, M. S. Mahadeva Naika, M. C. Mahesh Kumar
(Univ. Mysore, Bangalore Univ. India)
A MODULAR TRANSFORMATION FOR A GENERALIZED
THETA FUNCTION WITH MULTIPLE PARAMETERS
MODULQRNE PERETVORENNQ DLQ UZAHAL|NENO}
TETA-FUNKCI} Z BAHAT|MA PARAMETRAMY
We obtain a modular transformation for the theta function
q za m mn cn m n Am Bn Cm Dn( + )+ + + + + +
−∞
∞
−∞
∞
∑∑
2 2 λ µ νζ .
We are thus able to unify and extend several modular transformations in literature.
OderΩano modulqrne peretvorennq dlq teta-funkci]
q za m mn cn m n Am Bn Cm Dn( + )+ + + + + +
−∞
∞
−∞
∞
∑∑
2 2 λ µ νζ ,
wo da[ moΩlyvist\ unifikuvaty ta uzahal\nyty dekil\ka modulqrnyx peretvoren\, vidomyx z
literatury.
1. Definitions. We define, for | q | < 1 and ζ ≠ 0 ≠ z, the function, which we call a
theta function with multiple parameters,
a q z a b c A B C D qQ m n a b c( ) = (, , ; , , ; , , ; , ; , : , ; , , ;ζ λ µ ν λλ µ ν ζ, , ) + +
−∞
∞
−∞
∞
∑∑ Am Bn Cm Dnz , (1.1)
where
Q m n a b c Q m n a b c m n( ) = ( ) + + +, ; , , ; , , , ; , ,λ µ ν λ µ ν�
with
�Q m n a b c am bmn cn( ) = + +, ; , , 2 2 .
Our main concern here, however, will be (1.1) with a = b, that is, with the function
a q z a a c A B C D qa m mn cn( ) = ( + )+ +, , ; , , ; , , ; , ; ,ζ λ µ ν
2 2 λλ µ νζm n Am Bn Cm Dnz+ + + +
−∞
∞
−∞
∞
∑∑ .
(1.2)
In what follows, we freely use many relations that simply follow from the above
notations, for instance,
�
�
Q
Q
m
am bnm = ∂
∂
= +2 ,
�
�
Q
Q
n
bm cnn = ∂
∂
= + 2 ,
� � �Q B A Q D C Q B D A Cm m m( − ) + ( − ) = ( + − − ), , ,ϕ θ ϕ θ ϕ θ ,
where
© S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR, 2009
1040 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION … 1041
� �Q m n Q m n a b cm m( ) = ( ), : , ; , , .
A special case of (1.2), namely,
a q z qm mn n m n( − ) = + + +, , ; , , ; , , ; , ; ,ζ ζ1 1 1 0 0 0 1 1 1 1
2 2
zzn m−
−∞
∞
−∞
∞
∑∑ (1.3)
denoted briefly by a q z( ), ,ζ was introduced by S. Bhargava in [1] and was shown to
have properties which unified and generalized several known properties of the
Hirschhorn – Garvan – Borwein cubic analogues [2] of the classical theta function.
S. Bhargava and S. N. Fathima obtained in [3] a modular transformation a q z( ), ,ξ of
(1.3) which unified several modular transformations established by S. Cooper [4] for
the Hirschhorn – Garvan – Borwein cubic theta functions. Other special cases of (1.2)
with λ = µ = ν = 0 have been studied by various authors including S. Bhargava and
N. Anitha [5], who have recently obtained a triple product for (1.3) and C. Adiga,
M. S. Mahadeva Naika and J. H. Han [6]. Thus, (1.2) unifies and extends several
works in literature [2 – 4, 6 – 8].
The objective of present paper is to obtain a modular transformation for (1.2). It
would be interesting to try and extend our results to (1.1) which we leave as an open
question. It is possible to first treat (1.1) with λ = µ = ν = 0, A = B = 1 = C, D = – 1
and then effect suitable transformations on ζ and z to obtain our main result (4.3).
However, we have preferred to present our main result and all the related lemmas
directly and in detail in order to bring out the motivation and lucidity of the inter play
between the various parameters all through, than would be the case in the abbreviated
alternative approach.
A basic lemma is established in Section 2, which shows that one can express (1.1),
under the condition a | b, in terms of the Jacobian theta function
f qz qz q zn n( ) =−
−∞
∞
∑, 1 2
,
where, in Ramanujan’s notation [7],
f a b a bn n n n( ) = ( + ) ( − )
−∞
∞
∑, : / /1 2 1 2 , | a b | < 1. (1.4)
In Section 3, we obtain modular transformations for the various Jacobian theta
functions which constitute a q z a b c A B C D( ), , ; , , ; , , ; , ; ,ζ λ µ ν ( when a | b ). Our
main result is established in Section 4. The remaining sections are devoted to special
cases.
2. Relation between a q z a b c A B C D(( )), , ; , , ; , , ; , ; ,ζζ λλ µµ νν and the Jacobian
theta function when a |||| b . The following lemma is basic to the remainder of this
work.
Lemma 2.1. Given that a | b , we have
a q z a b c A B C D( ), , ; , , ; , , ; , ; ,ζ λ µ ν =
= q f q z q z f q z qa A C a A C a A C aν λ λ λζ ζ ζ( ) (+ − − − ′+ ′ ′ ′ ′−, , ′′ − ′ − ′ )λ ζ A Cz +
+ q z f q z q z f qC B D a b A C a b A C aν µ λ λζ ζ ζ+ + + + − − − − ′+( ) (, 2 ′′ ′ ′ − ′ − ′ − ′ )λ λζ ζA C A Cz q z, , (2.1)
where
′ = −a a∆ 1 , ∆ = −4 2ac b , ′ = ( − ) −λ µ λ�Q am , 1,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1042 S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR
′ = ( − ) −A Q B A am
� , 1, ′ = ( − ) −C Q D C am
� , 1.
Proof. We define an auxiliary function P , following the method in [8], for
instance,
P q z t q z tam bmn cn m n Am Bn n( ) = + + + + + +, , ˆ; : ˆζ ζλ µ ν2 2 mm
−∞
∞
−∞
∞
∑∑ . (2.2)
We immediately have that the function a of (1.1) satisfies
a q z a b c A B C D P q z zD C( ) = ( ), , ; , , ; , , ; , ; , , , ,ζ λ µ ν ζ . (2.3)
Collecting coefficients of ẑ n2 in (2.2) and setting m̂ m
b
a
n= + , which is valid since
a | b by assumption, we have, on slight manipulations,
P z q t f q t qe
n a n n A n bn a a A a[ ] = (+ ′ + ′ ′ − + −ˆ ,/2 2ν λ λζ ζ λλζ− − )At 1 ,
with f as in (1.4). This immediately gives, on putting back t zC= , ẑ zD= ,
P z z q f q z q zn n a A C a A C
even ˆ ˆ ,[ ] = (
−∞
∞
+ − − −∑ 2 2 ν λ λζ ζ )) ( )′+ ′ ′ ′ ′− ′ − ′ − ′f q z q za A C a A Cλ λζ ζ, .
(2.4)
This establishes a part of (2.1). For the remaining part, we proceed similarly, by
collecting coefficients of ẑ n2 1+ , and we obtain,
P z z q z f qn n C B D a b A
odd ˆ ˆ[ ] = (+ +
−∞
∞
+ + + +∑ 2 1 2 1 ν µ λζ ζ zz q zC a b A C, − − − − )λζ ×
× f q z q za A C A C( )′+ ′ ′ ′ − ′ − ′ − ′2 λ λζ ζ, . (2.5)
We omit the details. Adding (2.4) and (2.5), we get the required result (2.1).
3. Some auxiliary modular transformations. In this section, we obtain modular
transformations for the Jacobian theta functions occurring on the right-hand side of
(2.1). For this, we need the classical theta function transformation [7] (Entry 20)
α βα α α αf e e
nn n( ) =
− + − −2 2 2
4
, exp ff e ein in( )− + − −β β β β2 2
, (3.1)
provided that α β = π and Re ( )α2 > 0. In fact, the following two special cases of
(3.1) are repeatedly used:
f e e
t t
t i t i( ) = −
π
−π + −π −θ θ θ
, exp
1
4
2
− π+ − π−
f e et t
θ θ
, (3.1)′
and
f e e
t
t i
t
t i i( ) = π − −
π
−π + −θ θ θ θ
, exp
2
8 2 2
2
− −
− π+ − π−
f e et t
2 2 2 2θ θ
, . (3.1)′′
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION … 1043
Indeed, putting α = π
t
, β = πt , and n
t
=
π
θ
in (3.1) yields (3.1)′ and putting
α = πt
2
, β = π2
t
and n
t
i
t
= − π +
π2
2 θ in (3.1) gives (3.1)′′.
Lemma 3.1. With q = e t− π2 , ζ = eiϕ , and z = eiθ , we have the modular
transformations
f q z q z q
i A C
a
a A C a A C a( ) ( + )
+ − − −λ λ
λ
ζ ζ λ ϕ θ
, exp
2
4
2
=
=
1
2 8
2
2 2
at
A C
at
f e at
A C
atexp − ( + )
π
− π − + −ϕ θ
ϕ θ ii
a at
A C
at
i
ae
π − π + + + π
λ ϕ θ λ
, 2 2 , (3.2)
f q z q z q
ia A C a A C a( ) ′′+ ′ ′ ′ ′− ′ − ′ − ′
′
′λ λ
λ
ζ ζ λ
, exp
2
4 (( ′ + ′ )
′
A C
a
ϕ θ
2
=
=
1
2 8
2
2
′
− ( ′ + ′ )
π ′
− π
′
− ′ +
a t
A C
a t
f e a t
A
exp
ϕ θ
ϕ ′′
′
− π ′
′
− π
′
+ ′ + ′
′
+ π ′
′
C
a t
i
a a t
A C
a t
i
ae
θ λ ϕ θ λ
2 2 2,
(3.3)
in the notations used in (2.1),
f q z q z q
i ba b A C a b A C
b
a( ) ( ++ + − − − −
( + )
λ λ
λ
ζ ζ λ
, exp
2
4 ))( + )
A C
a
ϕ θ
2
=
=
1
2 8
2
2 2
at
A C
at
f e at
A C
atexp − ( + )
π
− π − + −ϕ θ
ϕ θ ii b
a at
A C
at
i b
ae
π( + ) − π + + + π( + )
λ ϕ θ λ
, 2 2 , (3.4)
and
f q z q z qa A C A C
a
a( )′+ ′ ′ ′ − ′ − ′ − ′
( ′+ ′)
′2 4
2
λ λ
λ
ζ ζ, expp
i a A C
a
( ′ + ′)( ′ + ′ )
′
λ ϕ θ
2
=
=
1
2 8
2
2
′
− ( ′ + ′ )
π ′
−
− π
′
− ′ ′
a t
A C
a t
f e a t
A
exp
ϕ θ
ϕϕ θ λ ϕ θ+ ′
′
− π ′
′
− π
′
+ ′ ′+ ′
′
+ π ′
−
C
a t
i
a a t
A C
a t
i
e2 2 2,
λλ
′
a .
(3.5)
Proof. The proof is straight forward. We need only make repeated use of (3.1) or
its special cases (3.1)′ and (3.1)′′. We omit the details.
Remark 3.1. We have already assumed that a | b. If, further,
b
a
is an odd
integer, then the factors e i b a± π / appearing in the arguments of f on the right-hand
side of (3.4) both equal – 1 and, hence, (3.4) takes the form
f q z q z q
i ba b A C a b A C
b
a( ) ( ++ + − − − −
( + )
λ λ
λ
ζ ζ λ
, exp
2
4 ))( + )
A C
a
ϕ θ
2
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1044 S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR
=
1
2 8
2
2 2
at
A C
at
f e at
A C
atexp − ( + )
π
−
− π − +
ϕ θ
ϕ θ −− π − π + + + π
−
i
a at
A C
at
i
ae
λ ϕ θ λ
, 2 2 . (3.4)′
Lemma 3.2. We have
f q z q z f q z qa A C a A C a A C a( ) (+ − − − ′+ ′ ′ ′ ′− ′λ λ λ λζ ζ ζ, , ζζ− ′ − ′ )A Cz ×
× q
i A C
a
i A C
a
a a
λ λ
λ ϕ θ λ ϕ θ
2 2
4 4
2 2
+ ′
′ ( + ) + ′( ′ + ′ )
′
exp
=
=
1
2 8 8
2 2
t
A C
at
A C
a t∆
exp − ( + )
π
− ( ′ ′ + ′ )
π ′
ϕ θ ϕ θ
×
× f e eat
A C
at
i
a at
A C
at
i
a
− π − + − π − π + + + π
2 2 2 2
ϕ θ λ ϕ θ λ
,
×
× f e ea t
A C
a t
i
a a t
A C− π
′
− ′ + ′
′
− π ′
′
− π
′
+ ′ ′+ ′
2 2 2
ϕ θ λ ϕ
,
θθ λ
2 ′
+ π ′
′
a t
i
a , (3.6)
f q z q z f q z qa b A C a b A C a A C( ) (+ + − − − − ′+ ′ ′ ′λ λ λζ ζ ζ, ,2 −− ′ − ′ − ′ )λ ζ A Cz ×
× q
i b A C
a
i
b
a
a
a
( + ) +( ′+ ′)
′ ( + )( + ) + ( ′
λ λ
λ ϕ θ
2 2
4 4
2
exp
aa A C
a
+ ′)( ′ + ′ )
′
λ ϕ θ
2
=
=
1
2 8 8
2 2
t
A C
at
A C
a t∆
exp − ( + )
π
+ ( ′ + ′ )
π ′
ϕ θ ϕ θ
×
× f e e e e
i b
a at
A C
at
i
a
i b
a at
A− π − π − + − π π − π + +
2 2 2
ϕ θ λ ϕ
,
CC
at
i
a
θ λ
2
+ π
×
× f e ea t
A C
a t
i
a a t
A
− −
− π
′
− ′ + ′
′
− π ′ − π
′
+ ′ ′+ ′
2 2 2
ϕ θ λ ϕ
,
CC
a t
i
a
θ λ
2 ′
+ π ′
′
, (3.7)
and
q
i A C
a
A C
a
a a
λ λ
λ ϕ θ λ ϕ θ
2 2
4 4
2
+ ′
′ ( + ) + ′( ′ + ′ )
′
exp
×
× [ ( ) (+ − − − ′+ ′ ′ ′ ′− ′f q z q z f q z qa A C a A C a A C aλ λ λζ ζ ζ, , λλ ζ− ′ − ′ )A Cz +
+ q z f q z q z f qc B D a b A C a b A C aµ λ λ λζ ζ ζ+ + + − − − − ′+ ′( ) (, 2 ζζ ζλ′ ′ − ′ − ′ − ′ )]A C A Cz q z, =
=
1
2 8 8
2 2
t
A C
at
A C
a t∆
exp − ( + )
π
− ( ′ + ′ )
π ′
(
ϕ θ ϕ θ ααβ α β+ ′ ′) , (3.8)
where
α = f e eat
A C
at
i
a at
A C
at
i
a
− π − + − π − π + + + π
2 2 2 2
ϕ θ λ ϕ θ λ
,
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION … 1045
β = f e ea t
A C
a t
i
a a t
A C− π
′
− ′ + ′
′
− π ′
′
− π
′
+ ′ ′+ ′
2 2 2
ϕ θ λ ϕ
,
θθ λ
2 ′
+ π ′
′
a t
i
a ,
α′ = f e e e e
i b
a at
A C
at
i
a
i b
a at
A− π − π − + − π π − π + +
2 2 2
ϕ θ λ ϕ
,
CC
at
i
a
θ λ
2
+ π
,
β′ = f e ea t
A C
a t
i
a a t
A C− π
′
− ′ + ′
′
− π ′
′
− π
′
+ ′ ′+ ′
2 2 2
ϕ θ λ ϕ
,
θθ λ
2 ′
+ π ′
′
a t
i
a .
Proof. Equations (3.6) follows on multiplying (3.2) and (3.3). Similarly, (3.7)
follows from (3.4) and (3.5). Equation (3.8) follows from (3.6) and (3.7) once we see
b b
a
a a
a
c
2 22
4
2
4
+ + ′ + ′ ′
′
= +λ λ µ
and
b A C
a
A C
B D
( + ) + ′ + ′ = +ϕ θ ϕ θ
2 2
.
However, these are simple consequences of the definitions of ∆, a′, λ′, A′ and C′
given in (2.1).
4. Main result. We first obtain a lemma by employing the following result from
the classical theory of theta functions [7]:
f X Y f Z W f X Y f Z W( ) ( ) + (− − ) (− − ), , , , =
= 2 3 3 3 3 4 4f X Y XY f Z W ZW XZ f
Y
X
X
Y
X Y f
W
Z
( ) ( ) +
, , , ,,
Z
W
Z W4 4
. (4.1)
Lemma 4.1. Assuming
b
a
to be an odd integer, we have, in the notations of
(3.8),
1
2
( + ′ ′)αβ α β = f e eat
A C
at
i
a at
A C
at
i
a
− π − + − π − π + + + π 2 2 2 2ϕ θ λ ϕ θ λ
,
×
× f e ea t
A C
a t
i
a a t
A C− π
′
− ′ + ′
′
− π ′
′
− π
′
+ ′ + ′2 2 2ϕ θ λ ϕ θ
, ′′
+ π ′
′
a t
i
a
2 λ
+
+ e t a a t
A
a
A
a
C
a
C− π +
′
− + ′
′
+ + ′
′
2
2
1 1 1
2
ϕ
aa
i
a a
− π + ′
′
θ λ λ
×
× f e e
A C
at
i
a at
A C
at
i
a
ϕ θ λ ϕ θ λ+ + π − π − + − π
2 4 2
, ×
× f e e
A C
a t
i
a a t
A C
a t
i′ + ′
′
+ π ′
′
− π
′
− ′ + ′
′
− πϕ θ λ ϕ θ2 4 2
,
′′
′
λ
a . (4.2)
Proof. The proof follows from (4.1) on employing for X and Y the arguments in
α defined in (3.8) and for Z and W those of β also defined in (3.8).
We are now finally in a position to establish our main result.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1046 S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR
Theorem 4.1. If q = e t− π2 , ζ = eiϕ , and z = eiθ , we have
a q z a a c A B C D( ), , ; , , ; , , ; , ; ,ζ λ µ ν : =
: = q za m mn cn m n Am Bn Cm Dn( + )+ + + + + +
−∞
∞
−∞
∞
∑∑
2 2 λ µ νζ =
=
1
2
2 2
4 4
t
q
i A C
a
a a
∆
− + ′
′
+
− ( + ) + ′(
λ λ ν λ ϕ θ λ
exp
′′ + ′ )
′
A C
a
ϕ θ
×
× exp −
π
( + ) + ( ′ + ′ )
′
1
8
2 2
t
A C
a
A C
a
ϕ θ ϕ θ
×
× a e e e a a ct i i( − )
− π2
0 0 0 1 1 1 1∆ , , ; , , ; , , ; , ; ,� �ϕ θ (4.3)
with
a e e e a a ct i i( − )
− π2
0 0 0 1 1 1 1∆ , , ; , , ; , , ; , ; ,� �ϕ θ : =
: = � � �q za m mn cn m n m n( + )+ + −
−∞
∞
−∞
∞
∑∑
2 2
ζ , �q e t=
− π2
∆ , �t
t
= 1
∆
, (4.4)
i
A C
at
i
a
A C
a t
i
a
�ϕ ϕ θ λ ϕ θ λ
:= − + − π − ′ + ′
′
− π ′
′4 2
3
4
3
2
, � �ζ ϕ= ei , (4.5)
i
A C
at
i
a
A C
a t
i
a
�θ ϕ θ λ ϕ θ λ
:= + + π − ′ + ′
′
− π ′
′4 2
1
4 2
, �
�
z ei= θ . (4.6)
Proof. From (2.1), (3.8), and (4.2), we need to prove that the right-hand side of
(4.2) equals (4.4). To do this, we make another use of (2.1) with ( q, ζ, z; A, B; C, D ;
λ, µ, ν ) replaced by ( )� � � � � � � � � �q z A B C D, , ; , ; , ; , ,ζ λ µ ν , where �q , �ζ , �z are as in (4.4) –
(4.6) above and
( )� �A B, = ( 1, 1 ), ( )� �C D, = ( 1, – 1 ), ( )� � �λ µ ν, , = ( 0, 0, 0 ). (4.7)
We however retain ( )� � �a b c, , = ( a, b, c ) = ( a, c, c ), so that, from the notations of (2.1),
�∆ ∆= , �′ = ′ =a a
a
∆
, � �′ = ( ) ′−λ Q am 0 0 1, = 0,
�
�
′ =
( − )
A
Q
a
m 1 1,
= 1, �
�
′ =
(− − )
= − −
C
Q
a
a a
a
m 1 1 2,
= – 3. (4.8)
In fact, on using the various definitions in (4.7), (4.8) and in (2.1), we at once have
� � �� � � �
q z ea A C t
a
A C
a± ± ±
− π
± − ′ ′+ ′
′=λ
ϕ θ
ζ
2
∆ tt
i
a
− π ′
′
2 λ
= e
a
t
A C
a t
i
a
− π ′ ′+ ′
′
+ π ′
′
2 2
∆
∓
ϕ θ λ
, (4.9)
� � �� � � � � �
q z ea A C t a
i i′ ± ′ ± ′ ± ′ − π ± ( − )
=λ ϕ θ
ζ
2
3
∆
∆
= e at
A C
at
i
a
− π + + π
2 2
∓
ϕ θ λ
, (4.10)
� � �� � � � � ∓
q z ea b A C a t
a
t± ± ± ±
− π
′
± − π
′
=λζ
2 2
∆
AA C
a t
i
a
′+ ′
′
+ π ′
′
ϕ θ λ2
(as in (4.9)) =
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION … 1047
=
e
e
a t
A C
a t
i
a
A C
a t
− π
′
− ′ ′+ ′
′
− π ′
′
′ ′+ ′
′
+ π
4 2
2
ϕ θ λ
ϕ θ
,
ii
a
′
′
λ
, respectively,
(4.11)
� � �� � � � � �
q z ea A C t a
i i2
4
3′ + ′ ′ ′ − π + ( − )
=λ ϕ θ
ζ ∆
∆
= e at
A C
at
i
a
− π − + − π4 2ϕ θ λ
(as in (4.10)) (4.12)
and, similarly,
� � �
� � �
q z eA C
A C
at
i
a− ′ − ′ − ′
+ + π
=λ
ϕ θ λ
ζ
2
. (4.12)′
Further,
� � � � � � �� � � �
q z e i B DC B D
i C
tµ ζ ϕ θ+ − π
= + ( + )∆ =
= e ac a B
i a
a t
i
− π( + ) + ( − )
( − = = −
∆
∆ ∆
2
4 24
� �
� �ϕ θ
since , DD = )1 =
= e
i
t a
a A C
at
i
a
A C
a t
− π +
− + + π + ′ + ′
′4
1
2 2∆
ϕ θ λ ϕ θ ++ π ′
′
i
a
λ
, (4.13)
on using (4.5) – (4.6). Now using (4.9) – (4.13) in (4.2), (3.8) and (2.1) sequentially
with q, ζ, z; … changed to �q , �ζ , �z ; … we have (4.3) read with (4.4) – (4.6) and in
the notations of (2.1).
The theorem is proved.
In the following sections we specialize our Theorem 4.1 into various known cases.
Before that, we will record a slightly more “balanced” form of (4.3) as follows:
q
i A C
a
A C
a
a a
λ λ ν ϕ θ λ ϕ θ λ
2 2
4 4
2
+ ′
′
− ( + ) + ( ′ + ′ ) ′
′
exp
×
× q za m mn cn m n Am Bn Cm Dn( + )+ + + + + +
−∞
∞
−∞
∞
∑∑
2 2 λ µ νζ =
=
1 1
8
2 2
t t
A C
a
A C
a∆
exp −
π
( + ) + ( ′ + ′ )
′
ϕ θ ϕ θ
×
× � � �q za m mn cn m n m n( + )+ + −
−∞
∞
−∞
∞
∑∑
2 2
ζ . (4.3)′
Or, what is the same,
q
i A C
a
Aa a
1
4 4 4
2 2
2
λ λ ν ϕ θ λ ϕ+ ′
′
− ( + ) + ( ′ + ′
exp
CC
a
θ λ) ′
′
×
× a q z a a c A B C D( ), , ; , , ; , , ; , ; ,ζ λ µ ν =
=
1 1
8
2 2
t t
A C
a
A C
a∆
exp −
π
( + ) + ( ′ + ′ )
′
ϕ θ ϕ θ
×
× a q z a a c( − )� � �, , ; , , ; , , ; , ; ,ζ 0 0 0 1 1 1 1 . (4.3)′′
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1048 S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR
5. The a-functions of [1, 6]. We have the following special case of Theorem 4.1.
Theorem 5.1. If q = e t− π2 , ζ = eiϕ , and z = eiθ , then
q za m mn cn m n m n( + )+ + −
−∞
∞
−∞
∞
∑∑
2 2
ζ =
=
1
4
2 2
2 4
2 2
t a c a
a c c a c
ta c a( − )
− ( + ) + ( − ) +
π ( −
exp
θ θϕ ϕ
))
×
× � � �q za m mn cn m n m n( + )+ + −
−∞
∞
−∞
∞
∑∑
2 2
ζ , (5.1)
where
�q e ta c a=
− π
( − )
2
4 , �ζ θ ϕ= ( − ) − ( + )
( − )
exp
5 2 2
2 4
a c a c
a c a t
,
�z
c a a c
a c a t
= ( + ) − ( − )
( − )
exp
2 2
2 4
θ ϕ
.
Proof. The proof is a direct consequence of Theorem 4.1 on putting λ = µ = ν = 0
and A = B = C = 1, D = – 1, there.
Corollary 5.1. Putting a = 1, we have (5.2) of [6].
Remark 5.1. We note that (5.1) with a = c = 1 is the same as (1.4) of [3], taking
into account that the left-hand side of (5.1) is unchanged with θ and ϕ changed to
– θ and – ϕ and using the simple fact that
q x y
x
y
m mn n m n
m n
2 2 3+ + +
−
−∞
∞
−∞
∞
( )
∑∑ = q y xm mn n m n m n2 2 2 2+ + + −
−∞
∞
−∞
∞
( ) ( )∑∑ .
6. The a′′′′-functions of [1, 6]. We have the following theorem.
Theorem 6.1. If q = e t− π2 , ζ = eiϕ , and z = eiθ , then
q za m mn cn m n( + )+
−∞
∞
−∞
∞
∑∑
2 2
ζ =
1
4 2 4
2 2
t a c a
a c
a c a t( − )
− ( − ) +
π ( − )
exp
θ θϕ ϕ
×
× � � �q za m mn cn m n m n( + )+ + −
−∞
∞
−∞
∞
∑∑
2 2
ζ , (6.1)
where
�q e a c a t=
− π
( − )
2
4 , �ζ ϕ θ= ( − ) −
( − )
exp
2 3
2 4
a c a
a c a t
, �z
c a
a c a t
= −
( − )
exp
2
2 4
ϕ θ
.
Proof. To prove the theorem, it is enough to put λ = µ = ν = 0, A = 1, B = 0,
C = 0, D = 1 in Theorem 4.1.
Corollary 6.1. By rewriting the series on the right-hand side of (6.1) as
� � �q za m mn cn m n( + )+
−∞
∞
−∞
∞
′ ′∑∑
2 2
ζ ,
where
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION … 1049
� � �′ = = −
( − )
ζ ζ ϕ θ
z
c a t
2
4
and �
�
�
′ = = ( − ) −
( − )
z
z
a c a
a c a t
ζ ϕ θ2
4
,
and then putting a = 1, we get main result, namely, Theorem 4.1, of [6].
Remark 6.1. We note that by putting a = c = 1, we have
q zm mn n m n2 2+ +
−∞
∞
−∞
∞
∑∑ ζ =
=
1
3 6
2 2 2 2
t t
qm mn n m nexp − − +
+ + +θ θϕ ϕ ζ� � �zzm n−
−∞
∞
−∞
∞
∑∑ , (6.2)
where
�q e t=
− π2
3 , �ζ θ= −
exp
2t
, �z
t
= −
exp
2
6
ϕ θ
,
which is the same as (1.16) of [3] on noting the simple facts that
q z q zm mn n m n m mn n m n2 2 2 2 1 1+ +
−∞
∞
−∞
∞
+ + − −∑∑ = ( ) ( )ζ ζ
−−∞
∞
−∞
∞
∑∑ =
= q zm mn n m n2 2+ +
−∞
∞
−∞
∞
∑∑ ζ .
Further, (6.2) is also equivalent to (1.15) of [3].
One could also obtain a slightly more general result than (6.1), namely, the
following theorem.
Theorem 6.2. If q = e t− π2 , ζ = eiϕ , and z = eiθ , then
q z
ia m mn cn m n m n( + )+ + + +
−∞
∞
−∞
∞
∑∑
2 2 λ µ νζ exp
((− + ) + ( − )
−
2 2
4
µ λ θ µ λ ϕi
a c
×
× exp 2
2
4
π − −
−
ν µ λ
a c
t =
=
1
4 2 4
2 2
t a c a
a c
a c a t( − )
− ( − ) +
π ( − )
exp
θ θϕ ϕ
( + )+ + −
−∞
∞
−∞
∞
∑∑ � � �q za m mn cn m n m n2 2
ζ , (6.3)
where
�q e a c a t=
− π
( − )
2
4 , �ζ ϕ θ λ µ= ( − ) −
( − )
+ π ( − ) −{ }
exp
2 3
2 4
2 3a c a
a c a t
i c a a
a(( − )
4c a
,
and
�z
c a
a c a t
i c a
a c a
= −
( − )
+ π ( − )
( − )
exp
2
2 4
2
4
ϕ θ λ µ
. (6.4)
7. The b-functions of [1, 6]. The series on the left-hand side of the first formula
in the next theorem reduces to the b-series of [6] on putting a = 1 and λ = µ = ν = 0
and changing ζ to Ω ζ / ω and z to z ω / Ωc with Ω = exp
2
4 1
π
−
i
c
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1050 S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR
Theorem 7.1. If q = e t− π2 , ζ = eiϕ , and z = eiθ , then
exp 2
4
2 2
π − + − +
( − )
ν µ µλ λa a c
a a c
t ×
× exp
i
a c a( − )
( − ) + ( − + π)
{ }
4
2 2 2ϕ θ λ θ ϕ µ ×
× q za m mn cn m n m n m n( + )+ + + + −
−∞
∞
−∞
∞
∑∑
2 2 λ µ νζ ω =
=
1
4
2
4
3
42 2
t a c a
a c a c a
( − )
−
( − ) + + π + π ( − ) + π
exp
θ θϕ ϕ θ ϕ
22
9
2
2 4
( + )
π ( − )
a c
a c a t
×
× � � �q za m mn cn m n m n( + )+ + −
−∞
∞
−∞
∞
∑∑
2 2
ζ , (7.1)
where
�q e a c a t=
− π
( − )
2
4 ,
�ζ ϕ θ π= − ( − ) +
( − )
+ ( + )
( −
exp
2 3
2 4
2 2
3 4
a c a
a c a t
a c
a c a))
+ π ( − ) +
( − )
{ }
t
i c a a
a c a
2 3
4
λ µ
,
�z
c a
a c a t
c a
a c a t
i= −
( − )
+ ( − )
( − )
+ π (
exp
2
2 4
2
3 4
2ϕ θ π cc a
a c a
λ µ− )
( − )
4
. (7.2)
Proof. We need to put in Theorem 4.1 A =1 = D and B = 0 = C and change ϕ
to ϕ π+ 2
3
and θ to θ π+ 4
3
. Routine calculations then give (7.1) read with (7.2).
Remark 7.1. By putting a = c = 1 and λ = µ = ν = 0 in the above, we see that
the series on the left-hand side of (7.1) would be the same as the b-function of (1.27) in
[1] and the series on the right-hand side of (7.1) would be the same as the a-function of
(1.25) there. The equation (7.1) itself will become
q zm mn n m n m n2 2+ + −
−∞
∞
−∞
∞
∑∑ ζ ω =
=
1
3 6 3
2
9
2 2 2
t t t t
qm mexp − − + − −
+θ θϕ ϕ
π
θ π
� nn n m n m nz+ + −
−∞
∞
−∞
∞
∑∑
2 � �ζ (7.3)
with
�q e t=
− π2
3 , �ζ θ π= − −
exp
2
2
3t t
, �z
t
= −
exp
2
6
ϕ θ
. (7.4)
Further, since the series on the left-hand side of (7.3) also equals (on first changing m
to – m and then n to – n and then interchanging m and n )
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
A MODULAR TRANSFORMATION FOR A GENERALIZED THETA FUNCTION … 1051
q zm mn n m n m n2 2 1 1+ + − − −
−∞
∞
−∞
∞
( ) ( )∑∑ ζ ω ,
we have that
q zm mn n m n m n2 2+ + −
−∞
∞
−∞
∞
∑∑ ζ ω =
=
1
3 6 3
2
9
2 2 2
t t t t
qm mnexp − − + + −
+ +θ θϕ ϕ
π
ϕ π
� nn m n m nz
2 � �′ ′+ −
−∞
∞
−∞
∞
∑∑ ζ , (7.3) ′
where
�q e t=
− π2
3 , �′ = −
ζ ϕ π
exp
2
2
3t t
, �′ =
−
z
t
exp
ϕ θ2
6
=
=
1
3 6 3
2 2
1 3 2
t t t
q qm mnexp /− − + +
+ +θ θϕ ϕ
π
ϕ
� � nn m n m n m nz
2+ + + −
−∞
∞
−∞
∞
′′ ′′∑∑ � �ζ , (7.4) ′
�′′ =
ζ ϕ
exp
2t
, �′′ =
−
z
t
exp
2
6
θ ϕ
.
Thus, we have shown that
( ) ( ) ( )− + + −
−∞
∞
−∞
∞
∑∑ e e et m mn n i m i n m n2 2 2π ϕ θ ω =
1
3 6 3
2 2
t t t
exp − − + +
θ θϕ ϕ
π
ϕ
×
× e e et
m mn n m n
t
m n
−
+ + + + + +
2
3
1
3
2
2
2 2
π ϕ θ−− −
−∞
∞
−∞
∞
∑∑
ϕ
6t
m n
,
which is the same as (1.16) of [3].
8. The c -functions of [1, 6]. The series on the left-hand side of the first formula
in the next theorem reduces to c-series of [6] by putting a = 1.
Theorem 8.1. If q = e t− π2 , ζ = eiϕ , and z = eiθ , then
exp
2
4
2 2 3 2
i
a c a
q
a m mn cn
am c a
ϕ
( − )
( + )+ + + +
∆ ∆
nn
a c
m n m nz
+ +
+ −
−∞
∞
−∞
∞
∑∑
2
2∆ ζ =
=
1 2 2
t
q za m mn cn m n m n
∆
� � �( + )+ + −
−∞
∞
−∞
∞
∑∑ ζ ×
× exp − + ( + ) + ( − )
( − )
c a c c a
a c a t
ϕ θ ϕθ
π
2 22 2 2
2 4
, (8.1)
where
�q e t=
− 2π
∆ , ∆ = a c a( − )4 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1052 S. BHARGAVA, M. S. MAHADEVA NAIKA, M. C. MAHESH KUMAR
�ζ ϕ θ π= − ( + ) + ( − )
( − )
+
( − )
exp
2 2 5
2 4
3
4
c a c a
a c a t
i
a c a
,
�z
c a c a
a c a t
i
a c a
= ( − ) + ( + )
( − )
+ π
( − )
exp
2 2
2 4 4
ϕ θ
.
Proof. The proof is straight forward. We need only do some routine calculations
on putting λ =
3a
∆
, µ =
2c a+
∆
, ν =
2
2
a c+
∆
in Theorem 4.1.
Remark 8.1. By putting a = c = 1, we see that the series on the left-hand side of
(8.1) is the same as the c-function of (1.28) in [3] and (8.1) reduces to
exp
2
3
2 2 1
3i
q z
m mn cn m n m n m nϕ ζ
+ + + + + + −
−∞
∞
−
∑
∞∞
∞
∑ =
=
exp − +
+ + + −
−∞
ϕ θ
π ζ
2 23
6
3
2 2t
t
q zm mn n m n m n� � �
∞∞
−∞
∞
∑∑
with
�q e t=
− π2
3 , �ζ θ ϕ= − −
exp
2t
, �z
t
i= − + +
exp
ϕ θ π3
6 3
.
This is the same as (1.18) of [3] with ϕ changed to – ϕ and on realizing that
� � � �
�
�
q z qm mn n m n m n m mn n2 2 2 2+ + + −
−∞
∞
−∞
∞
+ +∑∑ =ζ ζ
ωzz
z
m
m n
( ) −
−∞
∞
−∞
∞
∑∑ ω ζ ω2� � .
Acknowledgement. The first author is thankful to Department of Science and
Technology, Government of India, New Delhi for the financial support under the grant
DST/MS/059/96.
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Received 18.08.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
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| id | umjimathkievua-article-3079 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:49Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2b/819de70cca7da524103156e541a3c72b.pdf |
| spelling | umjimathkievua-article-30792020-03-18T19:44:57Z A modular transformation for a generalized theta function with multiple parameters Модуляре перетворення для yзaгaльненої theta -функції з багатьма параметрами Bhargava, S. Mahadeva, Naika M. S. Maheshkumar, M. C. Бхаргава, С. Махадева, Найка М. С. Махешкумар, М. С. We obtain a modular transformation for the theta function $$∑_{-∞}^{∞}∑_{-∞}^{∞}q^{a(m^2 + m^n) + cn^2 + λm + μn + ν_{ς}Am + Bn_{Z}Cm + Dn},$$ which enables us to unify and extend several modular transformations known in literature. Одержано модулярне перетворення для тета-функції $$∑_{-∞}^{∞}∑_{-∞}^{∞}q^{a(m^2 + m^n) + cn^2 + λm + μn + ν_{ς}Am + Bn_{Z}Cm + Dn},$$ що дає можливість уніфікувати та узагальнити декілька модулярних перетворень, відомих з літератури. Institute of Mathematics, NAS of Ukraine 2009-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3079 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 8 (2009); 1040-1052 Український математичний журнал; Том 61 № 8 (2009); 1040-1052 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3079/2907 https://umj.imath.kiev.ua/index.php/umj/article/view/3079/2908 Copyright (c) 2009 Bhargava S.; Mahadeva Naika M. S.; Maheshkumar M. C. |
| spellingShingle | Bhargava, S. Mahadeva, Naika M. S. Maheshkumar, M. C. Бхаргава, С. Махадева, Найка М. С. Махешкумар, М. С. A modular transformation for a generalized theta function with multiple parameters |
| title | A modular transformation for a generalized theta function with multiple parameters |
| title_alt | Модуляре перетворення для yзaгaльненої theta -функції з багатьма параметрами
|
| title_full | A modular transformation for a generalized theta function with multiple parameters |
| title_fullStr | A modular transformation for a generalized theta function with multiple parameters |
| title_full_unstemmed | A modular transformation for a generalized theta function with multiple parameters |
| title_short | A modular transformation for a generalized theta function with multiple parameters |
| title_sort | modular transformation for a generalized theta function with multiple parameters |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3079 |
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