Open Access

An Interchangeable Theorem of -Integral

Journal of Inequalities and Applications20092009:135693

DOI: 10.1155/2009/135693

Received: 25 August 2008

Accepted: 3 January 2009

Published: 14 January 2009


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 [16]). 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

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


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


When , it is obvious that (1.13) holds; when , for and , we have
and by Lemma 1.1, we have

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


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


By (1.3) and (1.6), we have

From (3.2), (3.1) holds.

Theorem 3.2.

Let be two real numbers, satisfying . Then


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)–(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


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


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


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

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

Authors’ Affiliations

Department of Applied Mathematics, Jiangsu Polytechnic University


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© Hongshun Ruan. 2009

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