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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_IEq3_HTML.gif)
-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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ1_HTML.gif)
We also adopt the following compact notation for multiple -shifted factorial:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ2_HTML.gif)
where is an integer or
.
The -binomial theorem [2] tells us that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ3_HTML.gif)
Replace with
, and
with
and then set
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ4_HTML.gif)
Heine [2] introduced the basic hypergeometric series , which is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ5_HTML.gif)
Thomae [7] defined the -integral on interval
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ6_HTML.gif)
provided that the series converges.
Fubini's theorem. Suppose that is absolutely summary, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ7_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ9_HTML.gif)
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ10_HTML.gif)
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_IEq22_HTML.gif)
is monotonous increasing function with respect to . Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ12_HTML.gif)
(1.9) is proved.
Lemma 1.2.
Let ,
be some real numbers, satisfying
with
. Then, for all nonnegative integer
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ13_HTML.gif)
Proof.
When , it is obvious that (1.13) holds; when
, for
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ14_HTML.gif)
and by Lemma 1.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ15_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ17_HTML.gif)
Proof.
Using (1.13) and (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ18_HTML.gif)
Since, the series is convergent, the series
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ19_HTML.gif)
is absolutely convergent. Hence, by the Fubini's theorem, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ21_HTML.gif)
Proof.
By (1.3) and (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ22_HTML.gif)
From (3.2), (3.1) holds.
Theorem 3.2.
Let be two real numbers, satisfying
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ23_HTML.gif)
Proof.
By (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ24_HTML.gif)
By (1.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ25_HTML.gif)
Using Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ26_HTML.gif)
By Lemma 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ28_HTML.gif)
Corollary 3.4.
Let be a real number. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ29_HTML.gif)
Proof.
Taking in (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ30_HTML.gif)
On the other hand, by (1.4) and Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ31_HTML.gif)
which by combining with (3.10), implies (3.9).
Take , (3.9) implies the following result.
Corollary 3.5.
The following equation holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ32_HTML.gif)
Take , (3.9) implies the following result.
Corollary 3.6.
The following equation holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ33_HTML.gif)
Remark 3.7.
Taking , where
is positive integer, (3.9) readily yields many equations.
Corollary 3.8.
Let be a real number, satisfying
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ34_HTML.gif)
Proof.
Taking in (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ35_HTML.gif)
On the other hand, by Theorem 2.1 and set then replace
with
in (1.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ36_HTML.gif)
Combining (3.15) and (3.16), (3.14) follows.
Theorem 3.9.
Let be two real numbers, satisfying
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ37_HTML.gif)
Proof.
We recall the Heines transformation formula [7]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ38_HTML.gif)
In (3.18), replacing by
, respectively, (3.18) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ39_HTML.gif)
Taking the -integral on both sides of (3.19) with respect to variable
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ40_HTML.gif)
Applying (1.5) to (3.20) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ41_HTML.gif)
Applying Theorem 2.1 and Lemma 3.1 to (3.21), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ42_HTML.gif)
hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F135693/MediaObjects/13660_2008_Article_1891_Equ43_HTML.gif)
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.
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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