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Mathematical Problems in Engineering

VolumeΒ 2011Β (2011), Article IDΒ 462507, 11 pages

http://dx.doi.org/10.1155/2011/462507

## A Sextuple Product Identity with Applications

^{1}Department of Mathematics, East China Normal University, Shanghai 200241, China^{2}Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China

Received 21 February 2011; Revised 20 March 2011; Accepted 24 March 2011

Academic Editor: MingΒ Li

Copyright Β© 2011 Jun-Ming Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We get a new proof of a sextuple product identity depending on the Laurent expansion of an analytic function in an annulus. Many identities, including an identity for , are obtained from this sextuple product identity.

#### 1. Introduction

For convenience, we let throughout the paper. We employ the standard notation Series product has been an interesting topic. The Jacobi triple product is one of the most famous series-product identity. We announce it in the following (see, e.g., [1, page 35, Entryβ19] or [2, Equation (2.1)]):

It is well known that an analytic function has a unique Laurent expansion in an annulus. Bailey [3] used this property to prove the quintuple product identity. By this approach, Cooper [4, 5] and Kongsiriwong and Liu [2] proved many types of the Macdonald identities and some other series-product identities. In this paper, we use this method to deal with a sextuple product identity.

In Section 2, we present the sextuple product identity ((2.1) below) and its proof. Our identity is equivalent to [2, Equation (8.16)] by Kongsiriwong and Liu, which is the simplification of [2, Equationβ(6.13)]. Kongsiriwong and Liu got [2, Equationβ(8.16)] from a more general identity. In this section, we give it a direct proof.

In Section 3, we get many identities from this sextuple product identity.

To simplify notation, we often write for in the following when no confusion occurs.

#### 2. A New Proof of the Sextuple Product Identity

The starting point of our investigation in this section is the identity in the following theorem.

Theorem 2.1. * For any complex number with , one has
*

Before the proof of Theorem 2.1, we need some preparations. The two identities in the following lemma are from [6]. We write them in this version.

Lemma 2.2. * One has
*

*Proof. *For (2.2), see [6, Equation (3.18)]. Equation (2.3) is from [6, Equation (3.21)]. Its proof is similar to that of [6, Equation (3.18)].

The lemma above is used to prove the following two identities.

Lemma 2.3. *One has
*

*Proof. * By (1.2), we have
Adding (2.6) and (2.7), we have
By (2.2), we have (2.4).

Subtracting (2.7) from (2.6), we obtain
Replacing in (2.3) by and, then, applying the resulting identity to the above equation, we get (2.5). This completes the proof.

*Proof of Theorem 2.1. *Set
Then is an analytic function of in the annulus . Put
By (2.10), we can easily verify
Combining (2.11) and (2.12) gives
Equate the coefficients of on both sides to get
Using the above relation, we obtain
Substituting the above four identities into (2.11), we have
By (2.10), we also have
This gives
Then we have
Set to get
By this relation, (2.16) reduces to
Now, it remains to determine , , and .

Putting in (2.21) gives
Set in (2.21) to get
Taking in (2.21) and noting that , we have
Subtracting (2.23) from (2.22) and noting that , we obtain
Add (2.22) and (2.23) to get
Adding (2.24) and (2.26) and, then, using (1.2) in the resulting equation, we obtain
By (2.4), we have
Similarly, subtracting (2.24) from (2.26) and, then using (1.2), we have
Applying (2.5) to this equation gives
which completes the proof.

#### 3. Some Applications

In this section, we deduce many modular identities from Theorem 2.1.

Corollary 3.1. * One has
*

*Proof. *Dividing both sides of (2.1) by , letting , and then using LβHospitalβs rule twice on the right-hand side gives (3.1).

Corollary 3.2. *One has
*

*Proof. *Replace in (2.1) by and, then, by . Using (1.2) in the resulting identity gives (3.2).

Replace in (2.1) by and, then, by . Using (1.2) in the resulting identity gives (3.3).

Replace in (2.1) by and, then, by . Using (1.2) and the fact that in the resulting identity, we obtain
By (1.2), we have
Combining (3.7) and (3.8) gives (3.4).

Replace in (2.1) by and, then, by . Using (1.2) in the resulting identity gives (3.5).

Replace in (2.1) by and, then, by . Using (1.2) in the resulting identity gives (3.6).

Obviously, using the same method above, we can get more identities from (2.1).

Now, we deduce a formula for from (2.1).

Corollary 3.3. *One has
*

*Proof. *Denote the left-hand side of (2.1) by and the right-hand side of (2.1) by . Let be a zero point of . Because (2.1) holds in , is also a zero point of . If , we have
Setting in (3.10) and by LβHospitalβs rule on the right-hand side, we have
Let . Putting and in (3.10) and noting for any integer , we have
Taking and in (3.10), we obtain
Adding the above three identities together gives
Using the fact
in the above identity and, then, replacing by , we get
Replacing in the last two sums on the right-hand side of the above identity by and, then, applying (1.2) to the resulting equation, we get Corollary 3.3.

#### 4. Conclusion

Besides the Jacobi triple product (1.2), well-known series-product identities are known as the quintuple product identity, the Winquist identity, and so forth. The formula (2.1) is also such an identity. Recently, we also obtain some other identities of this kind, including the simplifications of the formulae [2, Equationsβ(6.12) and (6.14)], with a different method. These identities are widely used in number theory, combinatorics, and many other fields. literature on this topic abounds. In (2.1), if we replace by , then the right-hand side of (2.1) turns into fourier series. For recent papers on the applications of fourier analysis, we refer the readers to [7β9].

#### Acknowledgment

This research is supported by the Shanghai Natural Science Foundation (Grant no. 10ZR1409100), the National Science Foundation of China (Grant no. 10771093), and the Natural Science Foundation of Education Department of Henan Province of China (Grant no. 2007110025).

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