Volume 2008, Article ID 471527,6pages doi:10.1155/2008/471527
Research Article
Two Inequalities for
r
φ
r
and Applications
Mingjin Wang1, 2
1Department of Mathematics, East China Normal University, Shanghai 200062, China 2Department of Information Science, Jiangsu Polytechnic University, Jiangsu Province 213164,
Changzhou, China
Correspondence should be addressed to Mingjin Wang,[email protected]
Received 26 August 2007; Accepted 13 November 2007
Recommended by Nikolaos S. Papageorgiou
We use theq-binomial formula to establish two inequalities for the basic hypergeometric seriesrφr.
As applications of the inequalities, we discuss the convergence ofq-series.
Copyrightq2008 Mingjin Wang. 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.
1. Introduction and main results
q-series, which is 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 polyno-mials, physics, and so on. Inequality technique is one of the useful tools in the study of special functions. There are many papers about it1–6. In1, the authors gave some inequalities for hypergeometric functions. In this paper, we derive two inequalities for the basic hypergeomet-ric seriesrφr, which can be used to study the convergence ofq-series.
The main results of this paper are the following two inequalities.
Theorem 1.1. Supposeai,bi, andzare any real numbers such that|bi|<1withi1,2, . . . , r. Then
rφr
a1, a2, . . . , ar
b1, b2, . . . , br;q, z
≤
− |z|;q∞
r
i1
−ai;q
∞
bi;q
∞
. 1.1
Theorem 1.2. Supposeai,bi, andzare any real numbers such thatz <0 and|ai| < 1,|bi| < 1with
i1,2, . . . , r. Then
rφr
a1, a2, . . . , ar
b1, b2, . . . , br;q, z
≥z;q∞
r
i1
ai;q
∞
−bi;q
∞
Before the proof of the theorems, we recall some definitions, notations, and known results which will be used in this paper. Throughout the whole paper, it is supposed that 0< q <1. Theq-shifted factorials are defined as
a;q01, a;qn
n−1
k0
1−aqk, a;q∞
∞
k0
1−aqk. 1.3
We also adopt the following compact notation for multipleq-shifted factorial:
a1, a2, . . . , am;q
n
a1;q
n
a2;q
n· · ·
am;q
n, 1.4
wherenis an integer or∞.
Theq-binomial theorem7
∞
k0
a;qkzk
q;qk
az;q∞
z;q∞ , |z|<1. 1.5
Replacingawith 1/a, andzwithazand then settinga0, we get
∞
k0
−1kq
k
2
zk
q;qk z;q∞. 1.6
Heine introduced therφsbasic hypergeometric series, which is defined by7
rφs
a1, a2, . . . , ar
b1, b2, . . . , bs;q, z
∞
k0
a1, a2, . . . , ar;q
k
q, b1, b2, . . . , bs;q
k
−1kq
k 2
1s−r
zk. 1.7
2. The proof ofTheorem 1.1
In this section, we use theq-binomial formula1.6to proveTheorem 1.1.
Proof. Since
a;qnn−1
i0
1−aqi≤n−1 i0
1|a|qi− |a|;qn≤− |a|;q∞,
b;qnn−1
i0
1−bqi≥n−1 i0
1− |b|qi|b|;qn≥|b|;q∞>0,
2.1
we have
a;qn b;qn
≤
− |a|;q∞
|b|;q∞ . 2.2
Hence,
a1, a2, . . . , ar;q
n
b1, b2, . . . , br;q
n
≤
r
i1
−ai;q
∞
bi;q
∞
Multiplying both sides of2.3by
q
n 2
| −z|n
q;qn 2.4
gives
a1, a2, . . . , ar;q
n
q, b1, b2, . . . , br;q
n
−1nq
n 2
zn≤ q
n 2
|z|n
q;qn ·
r
i1
−ai;q
∞
bi;q
∞
. 2.5
Consequently,
rφr
a1, a2, . . . , ar
q, b1, b2, . . . , br;q, z
∞
n0
a1, a2, . . . , ar;q
n
q, b1, b2, . . . , br;q
n
−1nq
n 2
zn≤
∞
n0
a1, a2, . . . , ar;q
n
q, b1, b2, . . . , br;q
n
−1nq
n 2
zn
∞
n0
q
n 2
|z|n
q;qn ·
a1, a2, . . . , ar;q
n
b1, b2, . . . , br;q
n ≤∞
n0 q
n 2
|z|n
q;qn ·
r
i1
−ai;q
∞
bi;q
∞
.
2.6
Using theq-binomial theorem1.6obtains
∞
n0 q
n 2
|z|n
q;qn
− |z|;q∞. 2.7
Substituting2.7into2.6gets1.1. Thus, we complete the proof.
3. The proof ofTheorem 1.2
In this section, we use again theq-binomial formula1.6in order to proveTheorem 1.2.
Proof. Since
a;qn
n−1
i0
1−aqi≥
n−1
i0
1− |a|qi|a|;qn≥|a|;q∞>0,
0<b;qn n−1
i0
1−bqi≤
n−1
i0
1|b|qi− |b|;qn≤− |b|;q∞,
3.1
we have
a;qn b;qn ≥
|a|;q∞
Hence,
a1, a2, . . . , ar;q
n
b1, b2, . . . , br;q
n
≥r
i1
ai;q
∞
−bi;q
∞
. 3.3
Multiplying both sides of3.3by
−1nq
n 2
zn
q;qn >0 3.4
gives
a1, a2, . . . , ar;q
n
q, b1, b2, . . . , br;q
n
−1nq
n 2
zn≥ −1
nqn2zn
q;qn ·
r
i1
ai;q
∞
−bi;q
∞
. 3.5
Consequently, we have
rφr
a1, a2, . . . , ar
q, b1, b2, . . . , br;q, z
∞
n0
a1, a2, . . . , ar;q
n
q, b1, b2, . . . , br;q
n
−1nq
n 2
zn
≥∞
n0
−1nq
n 2
zn
q;qn ·
r
i1
ai;q
∞
−bi;q
∞
.
3.6
Using theq-binomial theorem1.6obtains
∞
n0
−1nq
n 2
zn
q;qn z;q∞. 3.7
Substituting3.7into3.6gets1.2. Thus, we complete the proof.
4. Some applications of the inequalities
Convergence is an important problem in the study of q-series. There are some results about it. For example, Ito used inequality technique to give a sufficient condition for convergence of a special q-series called Jackson integral 8. In this section, we use the inequalities obtained in this paper to give some sufficient conditions for convergence of a q-series and sufficient conditions for divergence of aq-series.
Theorem 4.1. Supposeai,bi, andzare any real numbers such that|bi|<1withi1,2, . . . , r. Let{cn} and{dn}be any number series. If
lim
n→∞
cn1
cn
p <1, dn1≤dn, n1,2, . . . , 4.1 then theq-series
∞
n0 cn rφr
a1, a2, . . . , ar
b1, b2, . . . , br;q, dn
4.2
Proof. Lettingzdnin1.1and then multiplying both sides of1.1by|cn|give
cn rφr
a1, a2, . . . , ar
b1, b2, . . . , br;q, dn
≤cn−dn;q
∞
r
i1
−ai;q
∞
bi;q
∞
. 4.3
From|dn1| ≤ |dn|, we know
−dn1;q
∞
−dn;q
∞
≤1. 4.4
The ratio test shows that the series
∞
n0
cn−dn;q
∞
r
i1
−ai;q
∞
bi;q
∞
4.5
is convergent. From4.3, it is sufficient to establish that4.2is absolutely convergent.
Theorem 4.2. Supposeai,bi, andzare any real numbers such that|ai|<1,|bi|<1withi1,2, . . . , r. Let{cn}and{dn}be any number series. If
lim
n→∞
cn1
cn
p >1, dn1≤dn <0, n0,1,2, . . . , 4.6
then theq-series
∞
n0 cn rφr
a1, a2, . . . , ar
b1, b2, . . . , br;q, dn
4.7
diverges.
Proof. Lettingzdnin1.2and then multiplying both sides of1.2by|cn|give
cnrφra1, a2, . . . , ar
b1, b2, . . . , br;q, dn
≥cndn;q
∞
r
i1
ai;q
∞
−bi;q
∞
. 4.8
Fromdn1≤dn, we know
dn1;q
∞
dn;q
∞
≥1. 4.9
Since
lim
n→∞inf
cn1dn1;q
∞
cndn;q
∞
≥ lim
n→∞
cn1
there exists an integerN0such that, whenn > N0,
cnrφra1, a2, . . . , ar
b1, b2, . . . , br;q, dn
≥cndn;q
∞
r
i1
ai;q
∞
−bi;q
∞
>|cN0|
dN0;q
∞
r
i1
ai;q
∞
−bi;q
∞
>0. 4.11
So,4.7diverges.
Acknowledgment
This work was supported by innovation program of Shanghai Education Commission.
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