- Research Article
- Open Access

# An Interchangeable Theorem of -Integral

- Hongshun Ruan
^{1}Email author

**2009**:135693

https://doi.org/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.

## Keywords

- Real Number
- Root System
- Special Function
- Number Theory
- Orthogonal Polynomial

## 1. Introduction and Some Lemmas

where is an integer or .

provided that the series converges.

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

## References

- Anderson GD, Barnard RW, Richards KC, Vamanamurthy MK, Vuorinen M:
**Inequalities for zero-balanced hypergeometric functions.***Transactions of the American Mathematical Society*1995,**347**(5):1713–1723. 10.2307/2154966MathSciNetView ArticleMATHGoogle Scholar - Gasper G, Rahman M:
*Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications*.*Volume 35*. Cambridge University Press, Cambridge, UK; 1990:xx+287.MATHGoogle Scholar - Ito M:
**Convergence and asymptotic behavior of Jackson integrals associated with irreducible reduced root systems.***Journal of Approximation Theory*2003,**124**(2):154–180. 10.1016/j.jat.2003.08.006MathSciNetView ArticleMATHGoogle Scholar - Wang M:
**An inequality for**and its applications.*Journal of Mathematical Inequalities*2007,**1**(3):339–345.MathSciNetView ArticleMATHGoogle Scholar - Wang M:
**Two inequalities for**and applications.*Journal of Inequalities and Applications*2008,**2008:**-6.Google Scholar - Wang M, Ruan H:
**An inequality about**and its applications.*Journal of Inequalities in Pure and Applied Mathematics*2008,**9**(2, article 48):-6.Google Scholar - Rogers LJ:
**On a three-fold symmetry in the elements of Heine's series.***Proceedings of the London Mathematical Society*1893,**24:**171–179.MATHGoogle Scholar

## Copyright

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