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# An Interchangeable Theorem of -Integral

*Journal of Inequalities and Applications*
**volumeÂ 2009**, ArticleÂ number:Â 135693 (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

-series, which are also called basic hypergeometric series, plays a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials, and physics. Inequality technique is one of the useful tools in the study of special functions. There are many papers about it (see [1â€“6]). First, we recall some definitions, notations, and known results which will be used in this paper. Throughout this paper, it is supposed that . The -shifted factorials are defined as

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

where is an integer or .

The -binomial theorem [2] tells us that

Replace with , and with and then set , we get

Heine [2] introduced the basic hypergeometric series , which is defined by

Thomae [7] defined the -integral on interval by

provided that the series converges.

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

then

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

Lemma 1.1.

Let be a given real number, satisfying . Then, for , one has

Proof.

Let

since

is monotonous increasing function with respect to . Hence,

(1.9) is proved.

Lemma 1.2.

Let , be some real numbers, satisfying with . Then, for all nonnegative integer , one has

Proof.

When , it is obvious that (1.13) holds; when , for and , we have

and by Lemma 1.1, we have

Consequently,

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.

Let , be some real numbers, satisfying with . Suppose real function is -integrable absolutely with and series is convergent. Then

Proof.

Using (1.13) and (1.6), we have

Since, the series is convergent, the series

is absolutely convergent. Hence, by the Fubini's theorem, we have

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.

Let be a real number, satisfying . Then, for all nonnegative integer , one has

Proof.

By (1.3) and (1.6), we have

From (3.2), (3.1) holds.

Theorem 3.2.

Let be two real numbers, satisfying . Then

Proof.

By (1.6), we have

By (1.3), we have

Using Theorem 2.1, we have

By Lemma 3.1, we have

Combining (3.4)â€“(3.7), (3.3) holds.

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

Corollary 3.3.

Let be some real numbers, satisfying . Then

Corollary 3.4.

Let be a real number. Then

Proof.

Taking in (3.8), we have

On the other hand, by (1.4) and Theorem 2.1, we have

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

Take , (3.9) implies the following result.

Corollary 3.5.

The following equation holds:

Take , (3.9) implies the following result.

Corollary 3.6.

The following equation holds:

Remark 3.7.

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

Corollary 3.8.

Let be a real number, satisfying . Then

Proof.

Taking in (3.8), we have

On the other hand, by Theorem 2.1 and set then replace with in (1.3), we have

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

Theorem 3.9.

Let be two real numbers, satisfying . Then

Proof.

We recall the Heines transformation formula [7]

In (3.18), replacing by , respectively, (3.18) yields

Taking the -integral on both sides of (3.19) with respect to variable , we have

Applying (1.5) to (3.20) yields

Applying Theorem 2.1 and Lemma 3.1 to (3.21), we have

hence,

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

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Ruan, H. An Interchangeable Theorem of -Integral.
*J Inequal Appl* **2009**, 135693 (2009). https://doi.org/10.1155/2009/135693

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DOI: https://doi.org/10.1155/2009/135693