An Interchangeable Theorem of -Integral

-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 [1–6]). 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

(1.1)

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

(1.2)

where is an integer or .

The -binomial theorem [2] tells us that

(1.3)

Replace with , and with and then set , we get

(1.4)

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

(1.5)

Thomae [7] defined the -integral on interval by

(1.6)

provided that the series converges.

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

(1.7)

then

(1.8)

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

(1.9)

Proof.

Let

(1.10)

since

(1.11)

is monotonous increasing function with respect to . Hence,

(1.12)

(1.9) is proved.

Lemma 1.2.

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

(1.13)

Proof.

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

(1.14)

and by Lemma 1.1, we have

(1.15)

Consequently,

(1.16)

Thus, (1.13) follows. We complete the 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

(2.1)

Proof.

Using (1.13) and (1.6), we have

(2.2)

Since, the series is convergent, the series

(2.3)

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

(2.4)

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

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

(3.1)

Proof.

By (1.3) and (1.6), we have

(3.2)

From (3.2), (3.1) holds.

Theorem 3.2.

Let be two real numbers, satisfying . Then

(3.3)

Proof.

By (1.6), we have

(3.4)

By (1.3), we have

(3.5)

Using Theorem 2.1, we have

(3.6)

By Lemma 3.1, we have

(3.7)

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

(3.8)

Corollary 3.4.

Let be a real number. Then

(3.9)

Proof.

Taking in (3.8), we have

(3.10)

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

(3.11)

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

Take , (3.9) implies the following result.

Corollary 3.5.

The following equation holds:

(3.12)

Take , (3.9) implies the following result.

Corollary 3.6.

The following equation holds:

(3.13)

Remark 3.7.

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

Corollary 3.8.

Let be a real number, satisfying . Then

(3.14)

Proof.

Taking in (3.8), we have

(3.15)

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

(3.16)

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

Theorem 3.9.

Let be two real numbers, satisfying . Then

(3.17)

Proof.

We recall the Heines transformation formula [7]

(3.18)

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

(3.19)

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

(3.20)

Applying (1.5) to (3.20) yields

(3.21)

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

(3.22)

hence,

(3.23)

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