# An Interchangeable Theorem of -Integral

- Hongshun Ruan
^{1}Email author

**2009**:135693

**DOI: **10.1155/2009/135693

© Hongshun Ruan. 2009

**Received: **25 August 2008

**Accepted: **3 January 2009

**Published: **14 January 2009

## Abstract

We give a sufficient condition for the interchangeability of the order of sum and -integral by using inequality technique. As the application of the theorem, some interesting results on the hypergeometric series are obtained.

## 1. Introduction and Some Lemmas

where is an integer or .

provided that the series converges.

*Fubini's theorem*. Suppose that is absolutely summary, that is

In order to prove the main result, we need to introduce two lemmas.

Lemma 1.1.

Proof.

(1.9) is proved.

Lemma 1.2.

Proof.

Thus, (1.13) follows. We complete the proof.

## 2. Main Result and Its Proof

We know that, whether the order of sum and -integral is interchangeable is an important problem in the study of -series. We obtain following result on the interchangeability.

Theorem 2.1.

Proof.

From (2.4) and (1.6), (2.1) holds. The proof is completed.

## 3. Applications

As the application of Theorem 2.1, in this section, we obtain some results. First, we give following lemma.

Lemma 3.1.

Proof.

From (3.2), (3.1) holds.

Theorem 3.2.

Proof.

Combining (3.4)–(3.7), (3.3) holds.

In (3.5), replacing by , we obtain the following result.

Corollary 3.3.

Corollary 3.4.

Proof.

which by combining with (3.10), implies (3.9).

Take , (3.9) implies the following result.

Corollary 3.5.

Take , (3.9) implies the following result.

Corollary 3.6.

Remark 3.7.

Taking , where is positive integer, (3.9) readily yields many equations.

Corollary 3.8.

Proof.

Combining (3.15) and (3.16), (3.14) follows.

Theorem 3.9.

Proof.

From (3.23) and (1.5), (3.17) follows.

## Authors’ Affiliations

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## Copyright

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