Some circular summation formulas for theta functions
© Cai et al.; licensee Springer 2013
Received: 10 December 2012
Accepted: 26 February 2013
Published: 26 March 2013
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.
Keywordscircular summation elliptic functions theta functions theta function identities
Recently, Chan, Liu and Ng  proved Ramanujan’s circular summation formulas and derived identities similar to Ramanujan’s summation formula and connected these identities to Jacobi’s elliptic functions.
Upon a, b, n and k are any positive integer with .
More recently, Liu further obtained the general formulas for theta functions (see ), 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, [4–15]).
In the present paper, we obtain analogues and uniform formulas for theta functions , , and . We now state our result as follows.
Theorem 1 For any positive integer k, n, a and b with , , .
For a, b even, we have(1.8)
For a even, n and b odd, we have(1.9)(1.10)
2 Proof of Theorem 1
Case 1. When , .
In the same manner as in Case 1, we can obtain Case 2 below.
Case 2. When , .
Case 3. When , .
When a and b are even, then kn is also even, we have(2.39)
When a is even, n and b are odd, then kn is also odd, we have(2.44)
In the same manner as in Case 3, we can obtain Case 4 below.
Case 4. When , .
When a and b are even, we have(2.51)(2.52)
When a is even, n and b are odd, we have(2.53)
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.
where and are defined by (2.21) and (2.55), respectively.
Obviously, we find that . □
where , .
Similarly, we have the following.
where are defined by (2.49) and (2.33), respectively.
where , .
Taking , in (1.9), (1.10) and (1.11), we have the following.
where and are defined by (2.49) and (2.55), respectively.
where , , .
where is defined by (1.7).
Substituting n by in the left-hand side of (3.32), we get (3.31). □
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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