- Research Article
- Open Access

# On Convergence of -Series Involving Basic Hypergeometric Series

- Mingjin Wang
^{1}Email author and - Xilai Zhao
^{2}

**2009**:170526

https://doi.org/10.1155/2009/170526

© M.Wang and X. Zhao. 2009

**Received:**18 December 2008**Accepted:**24 March 2009**Published:**31 March 2009

## 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.

## Keywords

- Real Number
- Ratio Test
- Nonnegative Integer
- Hypergeometric Series
- Compact Notation

## 1. Introduction

## 2. Notations and Known Results

where is an integer or .

## 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.

converges absolutely.

Proof.

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

where for .

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

Theorem 3.2.

diverges.

Proof.

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

where for .

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 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.

converges absolutely.

Proof.

one knows that (4.3) converges absolutely.

Theorem 4.2.

diverges.

Proof.

one knows that (4.6) diverges.

## Declarations

### Acknowledgment

The authors would like to express deep appreciation to the referees for the helpful suggestions. In particular, the authors thank the referees for help to improve the presentation of the paper. Mingjin Wang was supported by STF of Jiangsu Polytechnic University.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.