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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Some circular summation formulas for theta functions

Yi Cai, Si Chen and Qiu-Ming Luo*

Author Affiliations

Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing, 401331, People’s Republic of China

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Boundary Value Problems 2013, 2013:59  doi:10.1186/1687-2770-2013-59


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/59


Received:10 December 2012
Accepted:26 February 2013
Published:26 March 2013

© 2013 Cai et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 identities

1 Introduction

Throughout this paper we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M2">View MathML</a>. The classical Jacobi theta functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M4">View MathML</a>, are defined as follows:

(1.1)

(1.2)

(1.3)

(1.4)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M9">View MathML</a> as follows:

(1.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M11">View MathML</a>

A special case of formula (1.5) yields the following result (see [[1], Theorem 3.1]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M12">View MathML</a>

(1.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M13">View MathML</a>

(1.7)

Upon a, b, n and k are any positive integer with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M14">View MathML</a>.

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, [4-15]).

In the present paper, we obtain analogues and uniform formulas for theta functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M18">View MathML</a>. We now state our result as follows.

Theorem 1For any positive integerk, n, aandbwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M21">View MathML</a>.

Fora, beven, we have

(1.8)

Foraeven, nandbodd, we have

(1.9)

(1.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M25">View MathML</a>

(1.11)

2 Proof of Theorem 1

From Jacobi’s theta functions (1.1)-(1.4), we have the following properties respectively:

(2.1)

(2.2)

(2.3)

(2.4)

From (2.1)-(2.4), by using the induction, we easily obtain

(2.5)

(2.6)

(2.7)

(2.8)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M34">View MathML</a>

(2.9)

Case 1. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M36">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37">View MathML</a> becomes of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M38">View MathML</a>

(2.10)

From (2.10) we easily obtain

(2.11)

(2.12)

Comparing (2.11) and (2.12), when a is even, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M41">View MathML</a>

(2.13)

By (2.5) and (2.7), and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M42">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M43">View MathML</a>

(2.14)

Obviously, when a is even, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M44">View MathML</a>

(2.15)

We construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a>. By (2.13) and (2.15), we find that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a> is an elliptic function with double periods π and πτ and only has a simple pole at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47">View MathML</a> in the period parallelogram. Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a> is a constant, say, this constant is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M49">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M50">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M51">View MathML</a>

(2.16)

By (1.3), (2.10) and (2.16), we have

(2.17)

By (1.1) and (1.3), we obtain

(2.18)

By equating the constant term of both sides of (2.18), we obtain

(2.19)

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M55">View MathML</a>

(2.20)

where

(2.21)

In the same manner as in Case 1, we can obtain Case 2 below.

Case 2. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M36">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37">View MathML</a> becomes of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M60">View MathML</a>

(2.22)

From (2.22) we easily obtain

(2.23)

(2.24)

Comparing (2.23) and (2.24), when a is even, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M63">View MathML</a>

(2.25)

By (2.6) and (2.7), and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M42">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M65">View MathML</a>

(2.26)

Obviously, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M66">View MathML</a>

(2.27)

We construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a>. By (2.25) and (2.27), we find that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a> is an elliptic function with double periods π and πτ and only has a simple pole at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47">View MathML</a> in the period parallelogram. Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a> is a constant, say, this constant is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M71">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M72">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M73">View MathML</a>

(2.28)

By (1.3), (2.22) and (2.28), we have

(2.29)

By (1.2) and (1.3), we obtain

(2.30)

By equating the constant term of both sides of (2.30), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M76">View MathML</a>

(2.31)

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M77">View MathML</a>

(2.32)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M78">View MathML</a>

(2.33)

Case 3. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37">View MathML</a> becomes of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M82">View MathML</a>

(2.34)

From (2.34) we easily obtain

(2.35)

(2.36)

Comparing (2.35) and (2.36), when a is even, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M85">View MathML</a>

(2.37)

By (2.5) and (2.8), and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M42">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M87">View MathML</a>

(2.38)

• When a and b are even, then kn is also even, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M88">View MathML</a>

(2.39)

We construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a>, by (2.37) and (2.39), we find that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a> is an elliptic function with double periods π and πτ and only has a simple pole at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47">View MathML</a> in the period parallelogram. Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M45">View MathML</a> is a constant, say, this constant is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M93">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M94">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M95">View MathML</a>

(2.40)

By (1.3), (2.34) and (2.40), we have

(2.41)

By (1.1) and (1.4), we obtain

(2.42)

By equating the constant term of both sides of (2.42), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M98">View MathML</a>

(2.43)

• When a is even, n and b are odd, then kn is also odd, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M99">View MathML</a>

(2.44)

We construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M100">View MathML</a>. By (2.37) and (2.44), we find that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M100">View MathML</a> is an elliptic function with double periods π and πτ and only has a simple pole at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M47">View MathML</a> in the period parallelogram. Hence the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M100">View MathML</a> is a constant, say, this constant is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M93">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M105">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M106">View MathML</a>

(2.45)

By (1.4), (2.34) and (2.45), we have

(2.46)

By (1.1) and (1.4), we obtain

(2.47)

By equating the constant term of both sides of (2.47), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M109">View MathML</a>

(2.48)

Clearly, in (2.43) and (2.48), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M110">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M111">View MathML</a>

(2.49)

In the same manner as in Case 3, we can obtain Case 4 below.

Case 4. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M37">View MathML</a> becomes of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M115">View MathML</a>

(2.50)

• When a and b are even, we have

(2.51)

(2.52)

• When a is even, n and b are odd, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M118">View MathML</a>

(2.53)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M119">View MathML</a>

(2.54)

Clearly, in (2.52) and (2.54), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M120">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M121">View MathML</a>

(2.55)

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

(3.1)

(3.2)

(3.3)

(3.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M127">View MathML</a>are defined by (2.21) and (2.55), respectively.

Proof Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M36">View MathML</a> in (1.11), we have

(3.5)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80">View MathML</a> in (1.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M135">View MathML</a>

(3.6)

Obviously, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M136">View MathML</a>. □

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M137">View MathML</a> and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M140">View MathML</a> in Corollary 1, we get the following identities for theta functions:

(3.7)

(3.8)

(3.9)

Further, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M144">View MathML</a> in the above identities, we obtain the following additive formulas of the theta function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M17">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M146">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M147">View MathML</a>

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M149">View MathML</a>.

Similarly, we have the following.

Corollary 2For any positive integern, we have

(3.12)

(3.13)

(3.14)

(3.15)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M154">View MathML</a>are defined by (2.49) and (2.33), respectively.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M137">View MathML</a> and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M140">View MathML</a> in Corollary 2, we get the following identities of theta functions:

(3.16)

(3.17)

(3.18)

Further, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M144">View MathML</a> in the above identities, we obtain other additive formulas for the theta function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M17">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M164">View MathML</a>

(3.19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M165">View MathML</a>

(3.20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M149">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M169">View MathML</a> in (1.9), (1.10) and (1.11), we have the following.

Corollary 3Fornodd, we have

(3.21)

(3.22)

(3.23)

(3.24)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M174">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M127">View MathML</a>are defined by (2.49) and (2.55), respectively.

Proof Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M80">View MathML</a> in (1.11), we have

(3.25)

(3.26)

 □

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M137">View MathML</a> and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M185">View MathML</a> in Corollary 3, we get the following identities for theta functions:

(3.27)

and

(3.28)

Further taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M144">View MathML</a> in the above identities, we obtain the following additive formulas for the theta function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M18">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M190">View MathML</a>

(3.29)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M191">View MathML</a>

(3.30)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M149">View MathML</a>.

Corollary 4Whennis odd, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M195">View MathML</a>

(3.31)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M196">View MathML</a>is defined by (1.7).

Proof Replacing z by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M197">View MathML</a> and τ by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M198">View MathML</a> in (1.6), we obtain

(3.32)

Substituting n by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/59/mathml/M200">View MathML</a> in the left-hand 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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