- Research Article
- Open access
- Published:
An Interchangeable Theorem of -Integral
Journal of Inequalities and Applications volume 2009, Article number: 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 [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
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)–(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.
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. 10.2307/2154966
Gasper G, Rahman M: Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications. Volume 35. Cambridge University Press, Cambridge, UK; 1990:xx+287.
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.2003.08.006
Wang M: An inequality for and its applications. Journal of Mathematical Inequalities 2007,1(3):339–345.
Wang M: Two inequalities for and applications. Journal of Inequalities and Applications 2008, 2008:-6.
Wang M, Ruan H: An inequality about and its applications. Journal of Inequalities in Pure and Applied Mathematics 2008,9(2, article 48):-6.
Rogers LJ: On a three-fold symmetry in the elements of Heine's series. Proceedings of the London Mathematical Society 1893, 24: 171–179.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ruan, H. An Interchangeable Theorem of -Integral. J Inequal Appl 2009, 135693 (2009). https://doi.org/10.1155/2009/135693
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/135693