Abstract
In this paper, we obtain the circular summation formulas of the theta function and show the corresponding alternating summations and inverse relations. Some applications are also considered.
MSC:11F27, 33E05.
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1 Introduction
Throughout this paper we take , , . The classical Jacobi’s four theta functions , , are defined as follows, with the notation of Tannery and Molk (see, e.g., [1–3]):
From the Jacobi theta functions (1.1)-(1.4), via the direct calculation and applying the induction, we have the following properties, respectively:
From the Jacobi theta functions (1.1)-(1.4), via a direct calculation, we also have the following transformation formulas:
From the above equations we easily obtain
From (1.5)-(1.12), we can obtain the following lemmas.
Lemma 1.1 For n any positive integer, we have
Lemma 1.2 For n any positive integer, we have
On p.54 in Ramanujan’s lost notebook (see [[4], p.54, Entry 9.1.1], or [[5], p.337]), Ramanujan recorded the following claim (without proof), which is now well known as Ramanujan’s circular summation. The appellation of ‘circular summation’ is due to Son (see [[5], p.338]).
Theorem 1.3 (Ramanujan’s circular summation)
For each positive integer n and ,
where
Ramanujan’s theta function is defined by
By the definition of Ramanujan’s theta function above and routine calculations, we can rewrite Ramanujan’s circular summation (1.23) as follows (see, for details [[5], p.338]).
Theorem 1.4 (Ramanujan’s circular summation)
Let and . For each positive integer n and ,
where
If we are going to apply the transformation to Ramanujan’s identity, it will be convenient to convert Ramanujan’s theorem into one involving the classical theta function defined by (1.3). Surprisingly Chan et al. [6] prove that Theorem 1.5 below is equivalent to Theorem 1.3.
Theorem 1.5 (Ramanujan’s circular summation)
For any positive integer ,
When ,
Chan et al. [6] also showed that Theorem 1.6 below is equivalent; we have Theorem 1.5 by applying the Jacobi imaginary transformation formula [[3], p.475].
Theorem 1.6 (Ramanujan’s circular summation)
For any positive integer n, there exists a quantity such that
where
Ramanujan’s circular summation is an interesting subject in his notebook. On the subject of Ramanujan’s circular summation and related identities of theta functions and their various extensions, a remarkably large number of investigations have appeared in the literature (see, for example, Andrews, Berndt, Rangachari, Ono, Ahlgren, Chua, Murayama, Son, Chan, Liu, Ng, Chan, Shen, Cai, Zhu, and Xu et al. [5–25]).
Recently, Zeng [23] extended Ramanujan’s circular summation in the following form.
Theorem 1.7 For any nonnegative integers n, k, a, and b with , there exists a quantity such that
Chan and Liu [12] further extended Theorem 1.7 in the following general form.
Theorem 1.8 Suppose are n complex numbers such that ; there exists a quantity such that
In the present paper, motivated by [10, 12], and [6], by applying the theory and method of elliptic functions, we obtain the circular summation formulas of theta functions and show the corresponding alternating summations and inverse relations. We also give some applications and derive some interesting identities of theta functions.
2 Ramanujan’s circular summation formula for theta functions
In the present section, we obtain Ramanujan’s circular summation formula for the theta functions . We now state our main result as follows.
Theorem 2.1 Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
When, we have
-
When, we have
Here
Proof Let be the left-hand side of (2.1) with , . We have
By (1.8) for , we easily obtain
Comparing (2.4) and (2.5), we have
By (1.8), we obtain
-
When in (2.7), we have
(2.8)
We construct the function , by (1.7) for , (2.6) and (2.8), we find that the function is an elliptic function with double periods π and πτ and has only a simple pole at in the period parallelogram. Hence the function is a constant, say , and we have
or, equivalently,
Letting
in (2.9), and then setting
we arrive at (2.1).
Setting
in (1.4), via the direct calculation, we obtain
Setting
in (1.3), we get
Substituting (2.10) and (2.11) into (2.1), we find that
Equating the constants of both sides of (2.12), we get (2.3).
-
When in (2.7), we have
(2.13)
We construct the function , by (1.8) for , (2.6) and (2.13), we find that the function is an elliptic function with double periods π and πτ and has only a simple pole at in the period parallelogram. Hence the function is a constant, say , and we have
or, equivalently,
Letting
in (2.14), and then setting
we arrive at (2.2).
Similarly, we may obtain
Clearly,
The proof is complete. □
In the following we will prove that the following main result of Chan and Liu is a special case of Theorem 2.1.
Corollary 2.2 (Chan and Liu [[12], p.1191, Theorem 4])
Suppose that m, n are any positive integers; are any complex numbers and . We have
where
Proof First, setting in (2.1), we have , noting that p is any integer, hence we say that mn is even. Setting
in (2.1) of Theorem 2.1 and applying the properties (1.12) and (1.17), we have
Next taking in (2.2), we have , noting that p is any integer, hence we say that mn is odd. Setting
in (2.2) of Theorem 2.1 and applying the properties (1.12) and (1.18), this leads to the following:
Obviously, by (2.18) and (2.19), for any positive integers mn, we obtain (2.16).
We easily see that the and are independent of z, therefore, when , we find that
The proof is complete. □
Remark 2.3 Obviously, Theorem 2.1 implies the well-known result (2.16), but we see that from (2.16) we do not obtain Theorem 2.1. However, we can reformulate the results of Chan and Liu as Theorem 2.1.
Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
When , we have
(2.20) -
When , we have
(2.21)
Here
3 The imaginary transformations formulas for Theorem 2.1
In the present section, we derive the corresponding imaginary transformations formulas for Theorem 2.1.
Theorem 3.1 Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
When, we have
(3.1) -
When, we have
(3.2)
Here
Proof In (2.1) making the transformations , and then and for , (2.1) becomes
Applying the imaginary transformation formulas (see, e.g., [1–3])
to (3.7), via the suitable substitutions of the variable z and τ and noting that , and simplifying, we thus obtain (3.1) and (3.3). Applying the series expressions (1.3) and (1.4) in (3.1), via the direct calculation, we obtain (3.4).
In a similar manner, we use the imaginary transformation formula:
and (2.2), and noting that , we can prove (3.2), (3.5), and (3.6). Therefore we complete the proof of Theorem 3.1. □
4 The alternating Ramanujan’s circular summation formula
In this section, we will obtain the corresponding alternating Ramanujan’s circular summation formula from Theorem 2.1.
Theorem 4.1 Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
When, we have
(4.1)(4.2) -
When, we have
(4.3)(4.4)
Here
Proof Let
in Theorem 2.1.
By using (1.11), (1.12), and (1.22) we compute
Substituting (4.6) and (4.7) into (2.1) of Theorem 2.1 and simplifying, we get (4.1) of Theorem 4.1.
Substituting (4.6) and (4.8) into (2.2) of Theorem 2.1 and simplifying, we arrive at (4.3) of Theorem 4.1. This proof is complete. □
Corollary 4.2 Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
When, we have
(4.9)(4.10) -
When, we have
(4.11)(4.12) -
Whenand m, n are odd, we have
(4.13)
Here
Proof If mn is even, setting in , then we have and . We directly deduce (4.9) of Corollary 4.2.
If mn is an even, setting in , we have . We get (4.11) of Corollary 4.2.
If mn is an odd, setting in and , we have . We get (4.13) of Corollary 4.2. The proof is complete. □
Corollary 4.3 Suppose that m, n are any positive integers. We have
where
Proof We set in (4.9), (4.10), and (4.13) of Corollary 4.2. We take in (4.11) and (4.12) of Corollary 4.2. □
Remark 4.4 Taking in (4.15) and (4.17), and noting that
we have
which is just Chan’s result (see [[6], p.634, Theorem 4.2]). Hence (4.15) and (4.17) of Corollary 4.3 are the corresponding extension of Chan’s result.
Remark 4.5 We note that Chan’s result (see [[6], p.632, Theorem 4.2]) has a misprint; we have here corrected this point in (4.21).
Remark 4.6 Setting in (4.18), we have
Setting in (4.18), we have
The above two formulas are the analogs of Boon’s results (see [[8], p.3440]).
Theorem 4.7 Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
Whenand m is even, we have
(4.24) -
Whenand m is odd and n is even, we have
(4.25) -
Whenand m is odd and n is odd, we have
(4.26) -
Whenand m is even, we have
(4.27) -
Whenand m is odd and n is even, we have
(4.28) -
Whenand m is odd and n is odd, we have
(4.29)
Here
Proof Let
in Theorem 2.1.
By using (1.9), (1.12), and (1.22) we have
Substituting (4.31) and (4.32) into (2.1) of Theorem 2.1 and simplifying, we get (4.24), (4.25), and (4.26) of Theorem 4.7.
Substituting (4.31) and (4.33) into (2.2) of Theorem 2.1 and simplifying, we arrive at (4.27), (4.28), and (4.29) of Theorem 4.7. This proof is complete. □
Corollary 4.8 Suppose that m, n are any positive integers; are any complex numbers.
-
Whenand m is even, we have
(4.34) -
Whenand m is odd and n is even, we have
(4.35) -
Whenand m is odd and n is odd, we have
(4.36) -
Whenand m is even, we have
(4.37) -
Whenand m is odd and n is even, we have
(4.38) -
Whenand m, n are odd, we have
(4.39)
Here
Proof If mn is an even, setting in , then we have and . We directly deduce (4.34) and (4.35) of Corollary 4.8 from Theorem 4.7.
If mn is an even, setting in , we have . We get (4.37), (4.38), and (4.36) of Corollary 4.8 from Theorem 4.7.
If mn is an odd, setting in and , we have . We get (4.39) of Corollary 4.8 from Theorem 4.7. The proof is complete. □
Remark 4.9 Equation (4.34) of Corollary 4.8 is just the main result of Zhu (see [[24], p.120, Theorem 1.7]). Of course, we need to make some transformations , , in (4.40); we readily deduce that
which just is the corresponding result of Zhu (see [[24], p.120, (16) of Theorem 1.7]).
Corollary 4.10 Suppose that m, n are any positive integers. We have
where
Proof We set in (4.34), (4.35), and (4.39) of Corollary 4.8. We take in (4.37) and (4.38) of Corollary 4.8. We deduce Corollary 4.10. □
Remark 4.11 Equation (4.42) of Corollary 4.10 is an extension of Boon’s result. On taking in (4.42), and noting that , we have
which is just Boon’s result (see [[8], p.3440, the second identity of (8)]).
Remark 4.12 Equation (4.45) of Corollary 4.10 is also an extension of Boon’s result. On taking in (4.45), and noting that , we have
which is just Boon’s result (see [[8], p.3440, the first identity of (8)]).
Remark 4.13 Equation (4.42) of Corollary 4.10 is an alternating summation of Ramanujan’s circular summation. Taking in (4.42), we have
Remark 4.14 We note that Boon’s result (see [[8], p.3440, the first identity of (8)]) has a misprint, we have here corrected this point in our formula (4.50).
Remark 4.15 If we make the transformations , , , and using the transformation formulas (1.9)-(1.14) for Theorem 3.1, we can also obtain the corresponding alternating circular summation formulas.
5 The inverse formulas for Ramanujan’s circular summation formula
In [15], Liu obtained the following two results.
Lemma 5.1 (see [[15], p.1978, Theorem 1.1])
Suppose that n is a positive integer and is an entire function of z satisfying the functional equations
Then for any positive integer m, there exists a constant independent of z such that
Lemma 5.2 (see [[15], p.1978, Theorem 1.2])
Suppose that n is a positive integer and is an entire function of z satisfying the functional equations
Then for any positive integer m, there exists a constant independent of z such that
Liu also found many new identities of the theta functions from Lemma 5.1 and Lemma 5.2.
Below we give some new inverse relations for theta function by using the results of Liu.
Theorem 5.3 Suppose that m, n are any positive integers, p is any integer; are any complex numbers.
-
When, we have
(5.7)(5.8) -
Whenand n is even, we have
(5.9)(5.10)
Here
Proof Let
We easily obtain by using the first and second identities of (1.8) for , respectively:
-
When in (5.14). From (5.13) and (5.14), we have
(5.15)
which satisfies condition (5.1). By Lemma 5.1 we obtain (5.7) and (5.8) of Theorem 5.3 at once.
Next we compute the constant independent of z. We apply the definitions (1.3) and (1.4).
Setting
in (1.3), we get
Setting
in (1.4), we get
Substituting (5.16) and (5.17) into (5.7), we find that
Comparing the constants of both sides of (5.18) and noting that , we obtain (5.11) of Theorem 5.3.
-
When in (5.14) and n is only even. From (5.13) and (5.14), we have
(5.19)
which satisfies condition (5.4) of Lemma 5.2. From Lemma 5.2 we obtain (5.9) and (5.10) of Theorem 5.3 immediately. In a similar way we can prove (5.12). The proof is complete.
□
Corollary 5.4 Suppose that m, n are any positive integers; are any complex numbers.
-
Whenand n is even, we have
(5.20)(5.21) -
Whenand n is even, we have
(5.22)(5.23)
Here
Proof If n is an even, setting in , then we have . We directly deduce (5.20) (setting ) and (5.21) of Corollary 5.4 from Theorem 5.3.
If n is an even, setting in , we have . We get (5.22) (setting ) and (5.23) of Corollary 5.4 from Theorem 5.3. The proof is complete. □
Setting in (5.20) noting that and , we have
Setting in (5.21), noting that and , , we have
Setting in (5.22) noting that and , we have
Corollary 5.5 Assume m is any positive integer.
-
When n is even, we have
(5.33)(5.34) -
When n is even, we have
(5.35)(5.36)
Here
Proof We set in (5.20) and (5.21) of Corollary 5.4. We take in (5.22) and (5.31) of Corollary 5.4. We obtain Corollary 5.5. □
6 Some applications
The well-known cubic theta function is defined (see [9]) by
We define the multiple theta series as
Obviously, .
We give some applications of Theorem 2.1 below.
Corollary 6.1 Suppose that m, n are any positive integers; are any complex numbers.
-
When, we have
(6.3) -
Whenand mn is even, we have
(6.4)
where
Proof When mn is even, taking in (2.1); when mn is odd, taking in (2.2), we easily get (6.3).
When mn is even, taking in (2.2), we easily obtain (6.4). □
Corollary 6.2 Suppose that m, n are any positive integers, a, b are nonnegative integers and ; y is any complex numbers.
-
When mn is even, we have
(6.7)
Here
Proof When , taking and , in Corollary 6.1, we have (6.6).
When , taking , in Corollary 6.1, we have (6.7). □
Remark 6.3 Let in (6.7) and using the transformation formulas (1.8) and (1.12), we have
which is a new formula; it is also an extension of Boon’s result.
If we take in (6.10) and (6.9), we deduce
which is just Boon’s result (see [[8], p.3440]).
Remark 6.4 We here have corrected a misprint concerning the formulas of and in [[8], p.3440].
Taking in (6.3), we have the following.
Corollary 6.5 Suppose that m, n are any positive integers. We have
Here
Taking in (6.12), we deduce
which is just Boon’s result (see [[8], p.3440, Eq. (7)]), with and the transformation equation (1.7) and (1.12).
Taking in (6.12) and noting that
we have the following.
Corollary 6.6 Suppose that m is any positive integer; we have
The above formula is just Boon’s result [[8], p.3440, Eq. (10)], with and the transformation equation (1.7) and (1.12).
Taking in (6.12) and noting that
we have the following.
Corollary 6.7 Suppose that m is a positive integer.
We set ,
Taking in (6.12), we have
Here
The above formula is just Ramanujan’s circular summation formula (see [[6], p.632, Theorem 3.1]), with and the transformation equation (1.7) and (1.12).
We set in (6.10),
where
We set in (6.10),
We set in (6.10),
Comparing (6.19) and (6.25), we find that
Taking in (6.3) and noting that , we obtain
Hence, we have
where is defined by (6.2).
Taking in the first equation of (6.27), we get
Setting in (6.28), we get
Taking in the second equation of (6.27), we get
Setting in (6.28), we get again
Taking in (6.4) and noting that , we obtain
Hence, we have
Taking in (6.32) and noting that , we get
Setting in (6.33), we get the following new identity for the theta function :
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Acknowledgements
The authors express their sincere gratitude to the referee for many valuable comments and suggestions. The present investigation was supported by Natural Science Foundation Project of Chongqing, China, under Grant CSTC2011JJA00024, 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.
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Zhou, Y., Luo, QM. Circular summation formulas for theta function . Adv Differ Equ 2014, 243 (2014). https://doi.org/10.1186/1687-1847-2014-243
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DOI: https://doi.org/10.1186/1687-1847-2014-243