On Convergence of -Series Involving Basic Hypergeometric Series

Abstract

We use inequality technique and the terminating case of the -binomial formula to give some results on convergence of -series involving basic hypergeometric series. As an application of the results, we discuss the convergence for special Thomae -integral.

1. Introduction

-Series, which are also called basic hypergeometric series, play a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials and physics. Convergence of a -series is an important problem in the study of -series. There are some results about it in [1–3]. For example, Ito used inequality technique to give a sufficient condition for convergence of a special -series called Jackson integral. In this paper, by using inequality technique, we derive the following two theorems on convergence of -series involving basic hypergeometric series, which can be used for convergence of special Thomae -integral.

2. Notations and Known Results

We recall some definitions, notations, and known results which will be used in the proofs. Throughout this paper, it is supposed that . The -shifted factorials are defined as

(2.1)

We also adopt the following compact notation for multiple -shifted factorials:

(2.2)

where is an integer or .

The -binomial theorem [4, 5] is

(2.3)

When , where denotes a nonnegative integer

(2.4)

Heine introduced the basic hypergeometric series, which is defined by [4, 5]

(2.5)

3. Main Results

The main purpose of the present paper is to establish the following two theorems on convergence of -series involving basic hypergeometric series.

Theorem 3.1.

Suppose , are any real numbers such that and with . Let be any sequence of numbers. If

(3.1)

then the -series

(3.2)

converges absolutely.

Proof.

Let and

(3.3)

It is easy to see that is a monotone function with respect to .

Consequently, one has

(3.4)

From (3.4), one knows

(3.5)

where for .

So, one has

(3.6)

It is obvious that

(3.7)

Multiplying both sides of (3.6) by

(3.8)

gives

(3.9)

Hence,

(3.10)

By using (2.4) one obtains

(3.11)

Substituting (3.11) into (3.10), one has

(3.12)

Multiplying both sides of (3.12) by , one has

(3.13)

The ratio test shows that the series

(3.14)

is absolutely convergent. From (3.13), it is sufficient to establish that (3.2) is absolutely convergent.

Theorem 3.2.

Suppose are any real numbers such that and with . Let be any sequence of numbers. If

(3.15)

then the -series

(3.16)

diverges.

Proof.

Let , and

(3.17)

It is easy to see that is a monotone function with respect to .

Consequently, one has

(3.18)

From (3.18), one knows

(3.19)

where for .

So, one has

(3.20)

It is obvious that

(3.21)

Multiplying both sides of (3.20) by

(3.22)

gives

(3.23)

Hence,

(3.24)

By using (2.4) one obtains

(3.25)

Substituting (3.25) into (3.24), one has

(3.26)

Multiplying both sides of (3.26) by , one has

(3.27)

Since

(3.28)

By hypothesis

(3.29)

therefore, in both cases there exists a integer such that

(3.30)

So, one can conclude that

(3.31)

Now, from (3.27) and (3.31)

(3.32)

Thereby, (3.16) diverges.

We want to point out that some -integral can be written as (3.2) or (3.16). So, the results obtained here can be used to discuss the convergence of -integrals.

4. Some Applications

In [6, 7], Thomae defined the -integral on the interval by

(4.1)

The right side of (4.1) corresponds to use a Riemann sum with partition points Jackson [8] extended Thomae -integral via

(4.2)

In this section, we use the theorems derived in this paper to discuss two examples of the convergence for Thomae -integral. We have the following theorems.

Theorem 4.1.

Let be any real numbers such that and with . If , then the Thomae -integral

(4.3)

converges absolutely.

Proof.

By the definition of Thomae -integral (4.1), one has

(4.4)

Using Theorem 3.1 and noticing,

(4.5)

one knows that (4.3) converges absolutely.

Theorem 4.2.

Let ,  be any real numbers such that and with . If , then the Thomae -integral

(4.6)

diverges.

Proof.

By the definition of Thomae -integral (4.1), one has

(4.7)

Using Theorem 3.2 and noticing,

(4.8)

one knows that (4.6) diverges.