Abstract
In this paper, we obtain some circular summation formulas of theta functions using the theory of elliptic functions and show some interesting identities of theta functions and applications.
MSC: 11F27, 33E05, 11F20.
Keywords:
circular summation; elliptic functions; theta functions; theta function identities1 Introduction
Throughout this paper we take , where . The classical Jacobi theta functions , , are defined as follows:
Recently, Chan, Liu and Ng [1] proved Ramanujan’s circular summation formulas and derived identities similar to Ramanujan’s summation formula and connected these identities to Jacobi’s elliptic functions.
Subsequently, Zeng [2] gave a generalized circular summation of the theta function as follows:
where
A special case of formula (1.5) yields the following result (see [[1], Theorem 3.1]):
where
Upon a, b, n and k are any positive integer with .
More recently, Liu further obtained the general formulas for theta functions (see [3]), but from one main result, Theorem 1 of Liu, we do not deduce our results in the present paper. Many people research the circular summation formulas of theta functions and find more interesting formulas (see, for details, [415]).
In the present paper, we obtain analogues and uniform formulas for theta functions , , and . We now state our result as follows.
Theorem 1For any positive integerk, n, aandbwith, , .
• Fora, beven, we have
• Foraeven, nandbodd, we have
where
2 Proof of Theorem 1
From Jacobi’s theta functions (1.1)(1.4), we have the following properties respectively:
From (2.1)(2.4), by using the induction, we easily obtain
Let
The function becomes of the following form:
From (2.10) we easily obtain
Comparing (2.11) and (2.12), when a is even, we get
By (2.5) and (2.7), and noting that , we obtain
Obviously, when a is even, we have
We construct the function . By (2.13) and (2.15), we find that the function is an elliptic function with double periods π and πτ and only has a simple pole at in the period parallelogram. Hence the function is a constant, say, this constant is denoted by , i.e.,
we have
By (1.3), (2.10) and (2.16), we have
By (1.1) and (1.3), we obtain
By equating the constant term of both sides of (2.18), we obtain
Clearly,
where
In the same manner as in Case 1, we can obtain Case 2 below.
The function becomes of the following form:
From (2.22) we easily obtain
Comparing (2.23) and (2.24), when a is even, we get
By (2.6) and (2.7), and noting that , we obtain
Obviously, we have
We construct the function . By (2.25) and (2.27), we find that the function is an elliptic function with double periods π and πτ and only has a simple pole at in the period parallelogram. Hence the function is a constant, say, this constant is denoted by , i.e.,
we have
By (1.3), (2.22) and (2.28), we have
By (1.2) and (1.3), we obtain
By equating the constant term of both sides of (2.30), we obtain
Clearly,
where
The function becomes of the following form:
From (2.34) we easily obtain
Comparing (2.35) and (2.36), when a is even, we have
By (2.5) and (2.8), and noting that , we obtain
• When a and b are even, then kn is also even, we have
We construct the function , by (2.37) and (2.39), we find that the function is an elliptic function with double periods π and πτ and only has a simple pole at in the period parallelogram. Hence the function is a constant, say, this constant is denoted by , i.e.,
we have
By (1.3), (2.34) and (2.40), we have
By (1.1) and (1.4), we obtain
By equating the constant term of both sides of (2.42), we obtain
• When a is even, n and b are odd, then kn is also odd, we have
We construct the function . By (2.37) and (2.44), we find that the function is an elliptic function with double periods π and πτ and only has a simple pole at in the period parallelogram. Hence the function is a constant, say, this constant is denoted by , i.e.,
we have
By (1.4), (2.34) and (2.45), we have
By (1.1) and (1.4), we obtain
By equating the constant term of both sides of (2.47), we obtain
Clearly, in (2.43) and (2.48), we have
where
In the same manner as in Case 3, we can obtain Case 4 below.
The function becomes of the following form:
• When a and b are even, we have
• When a is even, n and b are odd, we have
Clearly, in (2.52) and (2.54), we have
where
Therefore we complete the proof of Theorem 1.
3 Some special cases of Theorem 1
In this section we give some special cases of Theorem 1 and obtain some interesting identities of theta functions.
Corollary 1For any positive integern, we have
whereandare defined by (2.21) and (2.55), respectively.
Proof Taking and , in (1.11), we have
Taking and , in (1.11), we have
Taking and letting , , in Corollary 1, we get the following identities for theta functions:
Further, taking in the above identities, we obtain the following additive formulas of the theta function :
Similarly, we have the following.
Corollary 2For any positive integern, we have
whereare defined by (2.49) and (2.33), respectively.
Taking and letting , , in Corollary 2, we get the following identities of theta functions:
Further, taking in the above identities, we obtain other additive formulas for the theta function as follows:
Taking , in (1.9), (1.10) and (1.11), we have the following.
Corollary 3Fornodd, we have
whereandare defined by (2.49) and (2.55), respectively.
Proof Taking , and , in (1.11), we have
□
Taking and letting , , in Corollary 3, we get the following identities for theta functions:
and
Further taking in the above identities, we obtain the following additive formulas for the theta function as follows:
and
Corollary 4Whennis odd, we have
Proof Replacing z by and τ by in (1.6), we obtain
Substituting n by in the lefthand side of (3.32), we get (3.31). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this paper, and read and approved the final manuscript and.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The present investigation was supported, in part, by Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625, Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07 and National Natural Science Foundation of China under Grant 11226281 and 11271057.
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