A sextuple product identity with applications.(Research Article)(Report)

1. Introduction

For convenience, we let [absolute value of q] < 1 throughout the paper. We employ the standard notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)

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)]):

[(q,z, q/z;q).sub.[infinity]] = [[infinity].summation over (n=-[infinity]][(-1).sup.n][q.sup.(1/2)n(n-1)][z.sup.n], z [not equal to] 0. (1.2)

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 [[summation].sub.n] for [[summation].sup.[infinity].sub.n=-[infinity]] 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 z with z [not equal to] 0, one has

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