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, Entry 19] 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, Equation (6.13)]. Kongsiriwong and Liu got [2, Equation (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 L’Hospital’s 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 L’Hospital’s 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, Equations (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 [7–9].

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).