N-THETA FUNCTION IDENTITIES 1. Introduction In

N-THETA FUNCTION IDENTITIES M. S. MAHADEVA NAIKA Abstract. In Chapter 16, of his second notebook, Ramanujan develops the theory of theta-function and recorded several identities without proofs. All these have been proved by C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson. In this paper, we establish several results of N-theta function which are analogous to the results in the Entries in Chapter 16 of Ramanujan’s second notebook.

1. Introduction In Chapter 16, of his second note book [3], Ramanujan develops the theory of theta-function and his theta-function is defined by f (a, b) =

∞ X

a

n(n+1) 2

b

n(n−1) 2

, |ab| < 1.

n=−∞

Following Ramanujan, we define ϕ(q) := f (q, q) =

∞ X

2

qn

(1.1)

n=−∞

3

ψ(q) := f (q, q ) =

∞ X

q

n(n+1) 2

(1.2)

n=0 ∞ X

f (−q) := f (−q, −q 2 ) =

(−1)n q

n(3n−1) 2

.

n=−∞

2000 Mathematics Subject Classification. Primary 33D10, 11F27. Key words and phrases. Theta-function. This paper was typeset using AMS-LATEX. 1

(1.3)

2

M. S. MAHADEVA NAIKA

Following Ramanujan, we define a new N- Theta function by fN (a, b) =

∞ X

a

kN (kN +1) 2

b

kN (kN −1) 2

, |ab| < 1.

(1.4)

k=−∞

If we set a = qe2iz , b = qe−2iz and q = eπiτ , where z is complex and Im(τ ) > 0 in (1.4), then we deduce that ϑN,3 (z, τ ) =

∞ X

2N

N

q k e2ik z ,

(1.5)

k=−∞

where ϑN,3 (z, τ ) is similar to one of the classical theta-functions in its standard notation[4]. Now we obtain special cases of fN (a, b) which are similar to the special cases (1.1)-(1.3) of Ramanujan’s theta-function f (a, b). ϕN (q) := fN (q, q) =

∞ X

qk

2N

(1.6)

k=−∞

ψN (q) =

∞ X

1

1 2N

q 2 (k+ 2 )

(1.7)

k=0

and

∞ X

fN (−q) := fN (−q, −q 2 ) =

(−1)k q

kN (3kN −1) 2

.

(1.8)

k=−∞

Remark: Putting N = 1 in (1.4), then we obtain 1-Theta function which is same as Ramanujan’s f (a, b). 2. Some Results Similar to Ramanujan’s Theta-functions In this section, we establish several results which are similar to the results in the Entries in Chapter 16 of his second notebook[3]. Theorem 2.1. We have fN (a, b) = fN (b, a) if N is odd

(2.1)

Proof. Replacing k by −k in (1.4), we obtained the required result.

N-THETA FUNCTION IDENTITIES

3

Theorem 2.2. We have N

ϕN (q) + ϕN (−q) = 2ϕN (q 4 ).

(2.2)

Proof. Using (1.5), we find that ϕN (q) + ϕN (−q) =

∞ X

q

k2N

+

k=−∞

=

∞ X

∞ X

(−1)k q k

2N

k=−∞ 2N

q k (1 + (−1)k )

k=−∞ ∞ X

=2

2N

q (2k)

k=−∞ N

= 2ϕN (q 4 ). Corollary 2.1. We have ϕ(q) + ϕ(−q) = 2ϕ(q 4 ).

(2.3)

Proof. Putting N = 1 in (2.2), we obtain (2.3). Remark: The identity (2.3) is same as the identity in Entry 25(i) of Chapter 16 of Ramanujan’s second notebook[3]. Theorem 2.3. We have 2N +1

ϕN (q) − ϕN (−q) = 4ψN (q 2

).

(2.4)

Proof. Using (1.5), we find that ϕN (q) − ϕN (−q) =

∞ X

qk

2N

k=−∞

=

∞ X

∞ X

−

k=−∞ 2N

q k (1 − (−1)k )

k=−∞

=2

(−1)k q k

∞ X

2N

q (2k+1)

k=−∞ 2N +1

= 4ψN (q 2

).

2N

4


Corollary 2.2. We have ϕ(q) − ϕ(−q) = 4qψ(q 8 ).

(2.5)

Proof. Putting N = 1 in (2.4), we obtain the require result. Remark: The identity (2.5) is same as the identity in Entry 25(ii) of Chapter 16 of Ramanujan’s second notebook[3]. Theorem 2.4. We have N

2 ϕ2N (q) + ϕ2N (−q) = 2ϕ2N (q 4 ) + 8ψN (q 2

2N +1

).

(2.6)

Proof. Using (1.5), we find that ϕ2N (q)

+

ϕ2N (−q)

=

∞ X

qm

2N +k 2N

(1 + (−1)m+k )

m,k=−∞

=

X

qm

2N +k 2N

(1 + (−1)m+k )

m,k both even

+

X

qm

2N +k 2N

(1 + (−1)m+k )

m,k both odd N

2N +1

2 = 2ϕ2N (q 4 ) + 8ψN (q 2

).

Corollary 2.3. We have ϕ2 (q) + ϕ2 (−q) = 2ϕ2 (q 2 ).

(2.7)

Proof. Putting N = 1 in (2.6), we obtain (2.7). Remark: The identity (2.7) is same as the identity in Entry 25(vi) of Chapter 16 of Ramanujan’s second notebook[3]. Theorem 2.5. We have N

2N +1

ϕ2N (q) − ϕ2N (−q) = 8ϕN (q 4 )ψN (q 2

).

(2.8)

Proof of (2.8) is similar to the proof of (2.6). So we omit the details.


5

Corollary 2.4. We have ϕ2 (q) − ϕ2 (−q) = 8ϕ(q 4 )ψ(q 8 ).

(2.9)

Proof. Putting N = 1 in (2.8), we obtain (2.9). Remark: The identity (2.9) is same as the identity in Entry 25(v) of Chapter 16 of Ramanujan’s second notebook[3]. Theorem 2.6. We have 22N +2N 22N −2N 22N −2N 22N +2N . fN (a, b) + fN (−a, −b) = 2fN a 2 b 2 , a 2 b 2 (2.10) Proof. Using (1.4), we deduce that ∞ X

fN (a, b) + fN (−a, −b) =

a

kN (kN +1) 2

b

kN (kN −1) 2

(1 + (−1)k )

k=−∞

=2

∞ X

a

(2k)N ((2k)N +1) 2

b

(2k)N ((2k)N −1) 2

k=−∞

=2

∞ X

2N

((ab)2 )

k2N −kN 2

(a−1 b)

(22N −2N )kN 2

2N

(a2 )k

N

k=−∞

=2

∞ X

k

((ab)4 )

k2N −kN 2

(a

22N +2N 2

b

22N −2N 2

)k

N

k=−∞

22N +2N 22N −2N 22N −2N 22N +2N . = 2fN a 2 b 2 , a 2 b 2 Corollary 2.5. We have f (a, b) + f (−a, −b) = 2f (a3 b, ab3 ).

(2.11)

Proof. Putting N = 1 in (2.10), we obtain (2.11). Remark: The identity (2.11) is same as the identity in Entry 30(ii) of Chapter 16 of Ramanujan’s second notebook[3].

6


3. Some Results Similar to Classical Theta-functions The theta-function were first systematically studied by Jacobi, Who obtained their properties by purely algebraical methods. The four types of theta functions are as follows: ϑ1 (z, q) = 2

∞ X

1 2

(−1)k q (k+ 2 ) sin(2k + 1)z,

(3.1)

k=0

ϑ2 (z, q) = 2

∞ X

1 2

q (k+ 2 ) cos(2k + 1)z,

(3.2)

k=0

ϑ3 (z, q) = 2

∞ X

2

q k cos2kz

(3.3)

k=−∞

and ∞ X

ϑ4 (z, q) =

2

(−1)k q k cos2kz.

(3.4)

k=−∞

In this section, we introduce generalized classical theta-functions as follows: ϑN,1 (z, q) =

∞ X

2N

q k sin2k N z,

(3.5)

k=0

ϑN,2 (z, q) =

∞ X

1

1 2N

N

q 2 (k+ 2 ) e2i(2k+1) z ,

(3.6)

k=−∞

ϑ0N,2 (z, q)

=

∞ X

1

1 2N

q 2 (k+ 2 ) cos2(2k + 1)N z,

(3.7)

k=0

ϑN,3 (z, q) =

∞ X

2N

N

q k e2ik z ,

(3.8)

k=−∞

ϑ0N,3 (z, q)

=

∞ X

2N

q k cos2k N z,

(3.9)

k=0

and ϑN,4 (z, q) =

∞ X k=−∞

2N

N

(−1)k q k e2ik z .

(3.10)


7

Theorem 3.1. We have

2N

ϑN,3 (z, q) + ϑN,4 (z, q) = 2ϑN,3 (2N z, q 2 ).

(3.11)

Proof. Using (3.8) and (3.10), we find that

ϑN,3 (z, q) + ϑN,4 (z, q) =

∞ X

2N

N

q k e2ik z (1 + (−1)k )

k=−∞

=2

∞ X

2N

2N

(q 2 )k e2ik

N (2N z)

k=−∞ 2N

= 2ϑN,3 (2N z, q 2 ).

Corollary 3.1. We have

2N

ϑN,3 (0, q) + ϑN,4 (0, q) = 2ϑN,3 (0, q 2 ).

(3.12)

Proof. Putting z = 0 in (3.11), we obtain (3.12). Remark: The identity (3.12) is same as (2.2).


ϑN,3 (z, q) − ϑN,4 (z, q) =

  4ϑN,2 (z, q 22N +1 )

if N is even

 4ϑ0 (z, q 22N +1 ) N,2

if N is odd.

(3.13)

8


Proof. Using (3.8) and (3.10), we find that ∞ X

ϑN,3 (z, q) − ϑN,4 (z, q) =

2N

N

q k e2ik z (1 − (−1)k )

k=−∞

=2

∞ X

2N +1

(q 2

1

1 2N

Nz

) 2 (k+ 2 ) e2i(2k+1)

k=−∞

=2

∞ X

(q 2

2N +1

1

1 2N

1

1 2N

Nz

) 2 (k+ 2 ) e2i(2k+1)

k=0

+2

∞ X

2N +1

(q 2

N (2k+1)N z

) 2 (k+ 2 ) e2i(−1)

k=0

=2

∞ X

(q

22N +1

)

1 (k+ 21 )2N 2

h

2i(2k+1)N z

e

+e

2i(−1)N (2k+1)N z

i

k=0

  4 P∞ (q 22N +1 ) 12 (k+ 12 )2N e2i(2k+1)N z k=0 = P  4 ∞ (q 22N +1 ) 21 (k+ 12 )2N cos2(2k + 1)N z k=0   4ϑN,2 (z, q 22N +1 ) if N is even =  4ϑ0 (z, q 22N +1 ) if N is odd. N,2

if N is even if N is odd

Corollary 3.2. We have

2N +1

ϑN,3 (0, q) − ϑN,4 (0, q) = 4ϑN,2 (0, q 2

).

(3.14)

ϑN,3 (z, q) + ϑN,3 (−z, q) = 4ϑ0N,3 (z, q) − 2.

(3.15)

Proof. Putting z = 0 in (3.13), we obtain (3.14). Remark: The identity (3.14) is same as (2.4).



9

Proof. Using (3.8), we find that ϑN,3 (z, q) + ϑN,3 (−z, q) =

∞ X

qk

2N

e2ik

Nz

+ e−2ik

Nz

k=−∞ ∞ X

=2

2N

q k cos2k N z

k=−∞

=2 1+2

∞ X

! q

k2N

cos2k N z

k=1

= 4ϑ0N,3 (z, q) − 2. Theorem 3.4. We have ϑN,3 (z, q) − ϑN,3 (−z, q) = 4iϑ2N,1 (z, q).

(3.16)

Proof of (3.16) is samiliar to the proof of (3.15). So we omit the details. Theorem 3.5. We have 2N

2N +1

2N

2N +1

ϑN,3 (z, q) = ϑN,3 (2N z, q 2 ) + ϑN,2 (z, q 2

ϑN,4 (z, q) = ϑN,3 (2N z, q 2 ) − ϑN,2 (z, q 2 2N

2N +1

ϑ2N,3 (2N z, q 2 ) − ϑ2N,2 (z, q 2

),

(3.17)

),

(3.18)

) = ϑN,3 (z, q)ϑN,4 (z, q),

(3.19)

and ϑ2N,3 (z, q) − ϑ2N,4 (z, q) =

  4ϑN,3 (2N z, q 22N )ϑN,2 (z, q 22N +1 )

if N is even

 4ϑN,3 (2N z, q 22N )ϑ0 (z, q 22N +1 ) N,2

if N is odd. (3.20)

Proofs of the identities (3.17)-(3.20) follows very easily from the definitions (3.8) and (3.10). So we omit the details. Acknowledgement The author wish to thank Prof. Remy Y. Denis for his valuable suggestions.

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References [1] C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Chapter 16 of Ramanujan’s second notebook: Theta-function and q-series, Mem. Amer. Math. Soc., 53, No.315(1985), Amer. Math. Soc., Providence, 1985. [2] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991. [3] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. [4] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th edition, University Press, Cambridge, 1966.

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