Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 170526, 9 pages doi:10.1155/2009/170526
r 1
r
1
2
1 Department of Applied Mathematics, Jiangsu Polytechnic University, Changzhou, Jiangsu 213164, China
2 Department of Mechanical and Electrical Engineering, Hebi College of Vocation and Technology,
Hebi, Henan 458030, China
Correspondence should be addressed to Mingjin Wang, wang197913@126.com
Received 18 December 2008; Accepted 24 March 2009
Recommended by Ondrej Dosly
We use inequality technique and the terminating case of the q -binomial formula to give some results on convergence of q -series involving r 1
φ r basic hypergeometric series. As an application of the results, we discuss the convergence for special Thomae q -integral.
Copyright q 2009 M. Wang and X. Zhao. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
q -Series, which are also called basic hypergeometric series, play a very important role in many fields, such as a ffi ne root systems, Lie algebras and groups, number theory, orthogonal polynomials and physics. Convergence of a q -series is an important problem in the study of q -series. There are some results about it in 1 – 3 . For example, Ito used inequality technique to give a su ffi cient condition for convergence of a special q -series called Jackson integral. In this paper, by using inequality technique, we derive the following two theorems on convergence of q -series involving r 1
φ r special Thomae q -integral.
basic hypergeometric series, which can be used for convergence of
We recall some definitions, notations, and known results which will be used in the proofs.
Throughout this paper, it is supposed that 0 < q < 1. The q -shifted factorials are defined as a ; q
0
1 , a ; q n n − 1
1 − aq k , k 0 a ; q
∞
∞ k 0
1 − aq k .
2.1

2 Journal of Inequalities and Applications
We also adopt the following compact notation for multiple q -shifted factorials: a
1
, a
2
, . . . , a m
; q n a
1
; q n a
2
; q n
· · · a m
; q n
, 2.2
where n is an integer or ∞ .
The q -binomial theorem 4 , 5 is
∞ k 0 a ; q q ; q k z k k az ; q z ; q
∞
, | z | < 1 , q < 1 .
∞
When a q
− n , where n denotes a nonnegative integer n k 0 q
− n ; q k z k q ; q k zq
− n
; q n
.
2.3
2.4
Heine introduced the r 1
φ r basic hypergeometric series, which is defined by 4 , 5 r 1
φ r a
1 b
, a
1
2
, b
2
, . . . , a r
, . . . , b r
1
; q, z
∞ n 0 a
1
, a
2
, . . . , a r 1
; q q, b
1
, b
2
, . . . , b r
; q n z n
.
n
2.5
The main purpose of the present paper is to establish the following two theorems on convergence of q -series involving r 1
φ r basic hypergeometric series.
Theorem 3.1. Suppose a i
Let { c n
, b i
, t are any real numbers such that
} be any sequence of numbers. If t > 0 and b i
< 1 with i 1 , 2 , . . . , r .
n lim c n 1 c n p < 1 , 3.1
then the q -series
∞ n 0 c n
· r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, tq n b
1
, b
2
, . . . , b r
3.2
converges absolutely.
Proof. Let b < 1 and f t
1
− at
1 − bt
, 0 ≤ t ≤ 1 ,
It is easy to see that f t is a monotone function with respect to 0 ≤ t ≤ 1.
3.3

Journal of Inequalities and Applications
Consequently, one has
1 − at
1 − bt
≤ max 1 ,
| 1 − a |
1 − b
.
From 3.4
, one knows a i
; q b i
; q k k
1
− a i
1 − b i
·
1 − a i q
1 − b i q
· · ·
1 − a i q k
−
1
1 − b i q k − 1
≤
M i k
, where M i max
{
1 ,
|
1
− a i
|
/ 1
− b i
So, one has
} for i 1 , 2 , . . . , r .
a
1
, a
2
, . . . , a r
; q k b
1
, b
2
, . . . , b r
; q
− 1 k k a
1
, a
2
, . . . , a r
; q k b
1
, b
2
, . . . , b r
; q k
≤ r i 1
M i k
.
It is obvious that q
− n ; q k q ; q
− tq n k k
> 0 , t > 0 , k 1 , 2 , . . . , n.
Multiplying both sides of 3.6
by q
− n ; q k q ; q
− tq n k k gives q
− n , a
1
, a
2
, . . . , a r
; q k q, b
1
, b
2
, . . . , b r
; q k tq n k
≤ q
− n ; q q ; q k k − tq n r i 1
M i k
.
Hence, r 1
φ r a
1
, a
2
, . . . , a r
, q
− n b
1
, b
2
, . . . , b r
; q, tq n n
≤ k 0 n k 0 q
− n , a
1
, a
2
, . . . , a r
; q q, b
1
, b
2
, . . . , b r k
; q k tq n k q
− n , a
1
, a
2
, . . . , a r
; q k q, b
1
, b
2
, . . . , b r
; q k tq n k n
≤ k 0 q
− n ; q q ; q k k − tq n r i 1
M i k
.
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3

4
By using 2.4
one obtains n k 0 q
− n ; q q ; q k k − tq n r i 1
M i k
Journal of Inequalities and Applications r
− t i 1
M i
; q n
.
3.11
Substituting 3.11
into 3.10
, one has r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, tq n b
1
, b
2
, . . . , b r
≤ − t r i 1
M i
; q n
.
Multiplying both sides of 3.12
by | c n
| , one has c n
· r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, tq n b
1
, b
2
, . . . , b r
≤ | c n
| − t r i 1
M i
; q n
.
3.12
3.13
The ratio test shows that the series
∞ n 0 c n
− t r i 1
M i
; q n
3.14
is absolutely convergent. From 3.13
, it is su ffi cient to establish that 3.2
is absolutely convergent.
Theorem 3.2. Suppose a i i 1 , 2 , . . . , r . Let { c n
, b i
, t are any real numbers such that t > 0 and a i
< 1 , b i
< 1 with
} be any sequence of numbers. If n lim c n 1 c n p > 1 , or n lim c n 1 c n
∞ , 3.15
then the q -series
∞ n 0 c n
· r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, − tq n b
1
, b
2
, . . . , b r
3.16
diverges.
Proof. Let a < 1, b < 1 and f t
1 − at
1
− bt
, 0 ≤ t ≤ 1 ,
It is easy to see that f t is a monotone function with respect to 0 ≤ t ≤ 1.
3.17

Journal of Inequalities and Applications
Consequently, one has
1 − at
1
− bt
≥ min 1 ,
1 − a
1
− b
.
From 3.18
, one knows a i
; q k b i
; q k
1 − a
1 − b i i ·
1
− a i q
1 − b i q
· · · where m i min { 1
So, one has
, 1 − a i
/ 1 − b i
} for i 1 , 2 , . . . , r .
1
− a i q k
−
1
1
− b i q k
−
1
≥ m i k
, a
1
, a
2
, . . . , a r
; q k b
1
, b
2
, . . . , b r
; q k
≥ r i 1 m i k
.
It is obvious that q
− n ; q k q ; q
− tq n k k
> 0 , t > 0 , k 1 , 2 , . . . , n.
Multiplying both sides of 3.20
by q
− n ; q k q ; q
− tq n k k gives q
− n , a
1
, a
2
, . . . , a r
; q k q, b
1
, b
2
, . . . , b r
; q
− tq n k k
≥ q
− n ; q q ; q k k − tq n r i 1 m i k
.
Hence, r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, − tq n b
1
, b
2
, . . . , b r n k 0 q
− n , a
1
, a
2
, . . . , a r
; q k q, b
1
, b
2
, . . . , b r
; q
− tq n k k n
≥ k 0 q
− n ; q q ; q k k − tq n r i 1 m i k
.
By using 2.4
one obtains n k 0 q
− n
; q q ; q k k − tq n r i 1 m i k r
− t i 1 m i
; q n
.
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
5

6 Journal of Inequalities and Applications
Substituting 3.25
into 3.24
, one has r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q,
− tq n b
1
, b
2
, . . . , b r r
≥ − t i 1 m i
; q n
.
Multiplying both sides of 3.26
by | c n
| , one has
| c n
| · r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, − tq n b
1
, b
2
, . . . , b r r
≥ | c n
| − t i 1 m i
; q n
.
3.26
3.27
Since n lim
| c n 1
| − t
| c n
| − t r i 1 m i
; q r i 1 m i
; q n 1 n n lim c n 1 c n
.
3.28
By hypothesis n lim c n 1 c n p > 1 , or n lim c n 1 c n
∞ , therefore, in both cases there exists a integer N
0
> 0 such that ∀ n > N
0
| c n 1
| − t
| c n
| − t r i 1 m i
; q r i 1 m i
; q n 1
> 1 .
n
So, one can conclude that
| c n
| − t r i 1 m i
; q n r
>
| c
N
0
| − t i 1 m i
; q
N
0
,
∀ n > N
0
.
Now, from 3.27
and 3.31
| c n
| · r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, − tq n b
1
, b
2
, . . . , b r r
≥ | c n
| − t i 1 m i
; q n r
> | c
N
0
| − t i 1 m i
; q
N
0
> 0 .
3.29
3.30
3.31
3.32
Thereby, 3.16
diverges.
We want to point out that some q -integral can be written as 3.2
or 3.16
. So, the results obtained here can be used to discuss the convergence of q -integrals.

Journal of Inequalities and Applications
In 6 , 7 , Thomae defined the q -integral on the interval 0 , 1 by
1
0 f t d q t 1 − q
∞ n 0 f q n q n
.
4.1
7
The right side of 4.1
corresponds to use a Riemann sum with partition points t n q n , n
0 , 1 , 2 , . . . .
Jackson 8 extended Thomae q -integral via d
0 d f t d q t c f t d q t d 1 − q
∞ f dq n
0 q n , d f t d q t −
0 c
0 f t d q t.
4.2
In this section, we use the theorems derived in this paper to discuss two examples of the convergence for Thomae q -integral. We have the following theorems.
Theorem 4.1. Let a i
, b i
, t be any real numbers such that
α > − 1, then the Thomae q -integral t > 0 and b i
< 1 with i 1 , 2 , . . . , r . If
1 t
α
0
· r 1
φ r a
1
, a
2
, . . . , a r
, t
− 1
; q, t d q t b
1
, b
2
, . . . , b r
4.3
converges absolutely.
Proof. By the definition of Thomae q -integral 4.1
, one has
1 t
α
0
· r 1
φ r
1 − q a
1
, a
2
, . . . , a r
, t
−
1
; q, t d q t b
1
, b
2
, . . . , b r
∞ n 0 q n 1 α r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, q n b
1
, b
2
, . . . , b r
.
Using Theorem 3.1
and noticing, n lim q n 1 1 α q n 1 α q
1 α
< 1 , one knows that 4.3
converges absolutely.
4.4
4.5

8 Journal of Inequalities and Applications
Theorem 4.2. Let a i
, b i
, t be any real numbers such that t > 0 and a i
< 1 , b i
< 1 with i 1 , 2 , . . . , r .
If α > 1, then the Thomae q -integral
1 t
− α
0
· r 1
φ r a
1
, a
2
, . . . , a r
, t
− 1
; q, − t d q t b
1
, b
2
, . . . , b r
4.6
diverges.
Proof. By the definition of Thomae q -integral 4.1
, one has
1 t
− α
0
· r 1
φ r a
1
, a
2
, . . . , a r
, t
− 1
; q, − t d q t b
1
, b
2
, . . . , b r
1 − q
∞ n 0 q 1
−
α n r 1
φ r a
1
, a
2
, . . . , a r
, q
− n
; q, − q n b
1
, b
2
, . . . , b r
.
Using Theorem 3.2
and noticing, n lim q 1
−
α n 1 q 1
−
α n q
1
−
α
> 1 , one knows that 4.6
diverges.
4.7
4.8
The authors would like to express deep appreciation to the referees for the helpful suggestions. In particular, the authors thank the referees for help to improve the presentation of the paper. Mingjin Wang was supported by STF of Jiangsu Polytechnic University.
1 M. Ito, “Convergence and asymptotic behavior of Jackson integrals associated with irreducible reduced root systems,” Journal of Approximation Theory, vol. 124, no. 2, pp. 154–180, 2003.
2 M. Wang, “An inequality for
3, pp. 339–345, 2007.
r 1
φ r and its applications,” Journal of Mathematical Inequalities, vol. 1, no.
3 M. Wang, “Two inequalities for
Article ID 471527, 6 pages, 2008.
r
φ r and applications,” Journal of Inequalities and Applications, vol. 2008,
4 G. E. Andrews, The Theory of Partitions, vol. 2 of Encyclopedia of Mathematics and Its Applications,
Addison-Wesley, Reading, Mass, USA, 1976.
5 G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its
Applications, Cambridge University Press, Cambridge, Mass, USA, 1990.
6 J. Thomae, “Beitr¨age zur Theorie der durch die Heine’sche Reihe darstellbaren Funktionen,” Journal
f ¨ur die reine und angewandte Mathematik, vol. 70, pp. 258–281, 1869.

Journal of Inequalities and Applications 9
7 J. Thomae, “Les s´eries Hein´eennes sup´erieures, ou les s´eries de la forme 1 q
1 − q b n − 1 a 1
· · ·
/ 1 − q 2
1 − q a h
· · · 1 − q
/ 1 − q b h a n − 1
· 1 −
/ 1 − q n q a h
1 /
· 1 − q a h
1 − q b 1
/ 1
· · ·
−
1 q
− b q a
∞ n 1 x n 1 − q
· · · 1 − q a / a n − 1
1 − q
· 1 − q a 1 h n − 1 / 1 −
/ 1 − q b 1 q b h n − 1
/ 1 −
,” Annali di Matematica
·
Pura ed Applicata, vol. 4, pp. 105–138, 1870.
8 F. H. Jackson, “On q -definite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 50, pp.
101–112, 1910.