R E S E A R C H
Open Access
Bounds for
q
-integrals of
r
+
ψ
r
+
with
applications
Zhefei He
1, Mingjin Wang
1*and Gaowen Xi
2*Correspondence: wang197913@126.com 1Department of Mathematics, Changzhou University, Changzhou, Jiangsu 213164, P.R. China Full list of author information is available at the end of the article
Abstract
In this paper, we establish an inequality for theq-integral of the bilateral basic hypergeometric functionr+1
ψ
r+1. As applications of the inequality, we give somesufficient conditions for the convergence ofq-series.
MSC: Primary 26D15; secondary 33D15
Keywords: inequality;q-integral; the bilateral basic hypergeometric function
r+1
ψ
r+1; convergence1 Introduction and main result
q-series, which are also called basic hypergeometric series, play a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthog-onal polynomials, and physics. The inequality technique is one of the useful tools in the study of special functions. There are many papers about the inequalities and theq-integral; see [–]. In this paper, we derive an inequality for theq-integral of the bilateral basic hypergeometric functionr+ψr+. Some applications of the inequality are also given. The
main result of this paper is the following inequality forq-integrals.
Theorem . Let a,b be any real numbers such that <q<b<a< ,and let ai,bibe
any real numbers such that|ai|>q,|bi|< for i= , , . . . ,r with r≥and|bb· · ·br| ≤
|aa· · ·ar|.Then for any c> ,t> ,such that c>b/a,c+t< ,we have
t
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤ Mt(q,b/a;q)∞
(c+t,b/ac;q)∞, (.)
where
M=max
r
i=
(–|ai|;q)∞
(|bi|;q)∞ , r
i=
(–q/|bi|;q)∞
(q/|ai|;q)∞
.
Before we give the proof of the theorem, we recall some definitions, notation, and well-known results which will be used in this paper. Throughout the whole paper, it is supposed that <q< . Theq-shifted factorials are defined as
(a;q)= , (a;q)n= n–
k=
–aqk, (a;q)∞= ∞
k=
–aqk. (.)
We also adopt the following compact notation for the multipleq-shifted factorial:
(a,a, . . . ,am;q)n= (a;q)n(a;q)n· · ·(am;q)n, (.)
wherenis an integer or∞. We may extend the definition (.) of (a;q)nto
(a;q)α=
(a;q)∞
(aqα;q)
∞, (.)
for any complex numberα. In particular,
(a;q)–n=
(a;q)∞
(aq–n;q)∞ =
(aq–n;q)
n
= (–q/a)
n
(q/a;q)n
q(n). (.)
The following is the well-known Ramanujanψsummation formula [, ],
∞
n=–∞
(a;q)n
(b;q)n
zn=(q,b/a,az,q/az;q)∞
(b,q/a,z,b/az;q)∞ , |b/a|<|z|< . (.)
The bilateral basic hypergeometric seriesrψsis defined by
rψs
a,a, . . . ,ar
b,b, . . . ,bs
;q,z
=
∞
n=–∞
(a,a, . . . ,ar;q)n
(b,b, . . . ,bs;q)n
(–)(s–r)nq(s–r)(n)zn. (.)
Jackson defined theq-integral by []
d
f(t)dqt=d( –q) ∞
n=
f dqnqn (.)
and
d
c
f(t)dqt=
d
f(t)dqt–
c
f(t)dqt. (.)
In [], the author uses Ramanujan’sψ summation formula to give the following
in-equality: Let a, b be any real numbers such that q <b<a< or a<b< , and let
ai, bi be any real numbers such that |ai|>q, |bi|< for i= , , . . . ,r with r≥ and
|bb· · ·br| ≤ |aa· · ·ar|. Then for anyb/a<|z|< , we have
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,z≤M(q,b/a,a|z|,q/a|z|;q)∞
(b,q/a,|z|,b/a|z|;q)∞ , (.)
where
M=max
r
i=
(–|ai|;q)∞
(|bi|;q)∞
,
r
i=
(–q/|bi|;q)∞
(q/|ai|;q)∞
.
2 The proof of theorem
Proof Under the conditions of the theorem ., it is easy to see that
b/a<c+tqn< . (.)
Lettingz=c+tqnin (.) gives
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+tqn
≤M(q,b/a,a(c+tq
n),q/a(c+tqn);q)∞
(b,q/a,c+tqn,b/a(c+tqn);q)∞ , n= , , , . . . . (.)
Since <b<a(c+tqn) <a< , we have
a c+tqn,q/a c+tqn;q
∞< (b,q/a;q)∞ (.)
and
c+tqn,b/a c+tqn;q∞≥(c+t,b/ac;q)∞. (.)
Combining (.), (.), and (.), we get
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+tqn≤M (q,b/a, ;q)∞
(c+t,b/ac;q)∞, n= , , , . . . . (.)
By the definition of theq-integral (.), we get
t
r+ ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
=t( –q)
∞
n=
qnr+ψr+
a,a, . . . ,ar+ b,b, . . . ,br
;q,tqn
. (.)
Consequently,
t
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
=
t( –q)
∞
n=
qnr+ψr+
a,a, . . . ,ar+ b,b, . . . ,br
;q,tqn
≤t( –q)
∞
n=
qnr+ψr+
a,a, . . . ,ar+ b,b, . . . ,br
;q,tqn. (.)
Using (.) one gets
t
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤t( –q)M (q,b/a, ;q)∞
(c+t,b/ac;q)∞ ∞
n=
qn=Mt(q,b/a, ;q)∞
(c+t,b/ac;q)∞. (.)
From (.) and the definition of theq-integral (.), we can easily get the following result.
Corollary . Under the conditions of the theorem,we have
str+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤ M(q,b/a, ;q)∞
(c+t,c+s,b/ac;q)∞
t(c+s;q)∞+s(c+t;q)∞, (.)
where s> and c+s< .
Proof By the definition of theq-integral (.), we get
str+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
=
t
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
–
s
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤
t
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
+
s
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤Mt(q,b/a, ;q)∞
(c+t,b/ac;q)∞ +
Ms(q,b/a, ;q)∞
(c+s,b/ac;q)∞
= M(q,b/a, ;q)∞ (c+t,c+s,b/ac;q)∞
t(c+s;q)∞+s(c+t;q)∞. (.)
Thus, the inequality (.) holds.
3 Some applications of the inequality
In this section, we use the inequality obtained in this paper to give some sufficient con-ditions for the convergence of theq-series. Convergence is an important problem in the study of q-series. There are some results about it. For example, Ito used an inequality technique to give a sufficient condition for the convergence of a specialq-series called the Jackson integral [].
Theorem . Suppose that
() a,b,cbe any positive real numbers such that <q<b<a< ,c>b/a;
() ai,bibe any real numbers such that|ai|>q,|bi|< fori= , , . . . ,rwithr≥and
|bb· · ·br| ≤ |aa· · ·ar|;
() {tn}be any positive number series,such thatc+tn< and
∞
n=tnconverges.
Then the q-series
∞
n=
tn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz (.)
Proof Since∞n=tnconverges, we get
lim
n→∞tn= . (.)
So, there exists an integerNsuch that, whenn>N,
c+tn≤d< . (.)
Whenn>N, lettingt=tnin (.) gives
tnr+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤ Mtn(q,b/a;q)∞
(c+tn,b/ac;q)∞
≤M(q,b/a;q)∞
(d,b/ac;q)∞ tn. (.)
From (.) and the convergence of∞n=tn, it is sufficient to establish that (.) is absolutely
convergent.
Corollary . Let{sn}be any positive number series such that c+sn< and
∞
n=sn
con-verges.Under the conditions of Theorem.,then the q-series
∞
n=
tn
sn r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz (.)
converges absolutely.
Proof By the definition of theq-integral (.), we get
tn
sn r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
=
tn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
–
sn
r+ ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz. (.)
Since both
tn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz (.)
and
sn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz (.)
are absolutely convergent, so (.) is absolutely convergent.
Theorem . Suppose that
() ai,bibe any real numbers such that|ai|>q,|bi|< fori= , , . . . ,rwithr≥and
|bb· · ·br| ≤ |aa· · ·ar|;
() {tn}be any positive number series,such thattn≤dandc+d< ;
() ∞n=enconverges absolutely.
Then the q-series
∞
n= en
tn
r+ ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz (.)
converges absolutely.
Proof Using (.) gives
en
tn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤ Mtn(q,b/a;q)∞
(c+tn,b/ac;q)∞
|en| ≤
Md(q,b/a;q)∞
(c+d,b/ac;q)∞|en|. (.)
Because∞n=enconverges absolutely, (.) is sufficient to establish that (.) is absolutely
convergent.
Corollary . Suppose that
() a,b,c,dbe any positive real numbers such that <q<b<a< ,c>b/a,c+d< ; () ai,bibe any real numbers such that|ai|>q,|bi|< fori= , , . . . ,rwithr≥and
|bb· · ·br| ≤ |aa· · ·ar|;
() {tn},{sn}be any positive number series,such thattn≤d,sn≤d,andc+d< ;
() ∞n=enconverges absolutely.
Then the q-series
∞
n= en
tn
sn r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz (.)
converges absolutely.
Proof By the definition of theq-integral (.), we get
en
tn
sn r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
=en
tn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
–en
sn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
≤en
tn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
dqz
+en
sn
r+ψr+
a,a, . . . ,ar
b,b, . . . ,br
;q,c+z
≤ Mtn(q,b/a;q)∞
(c+tn,b/ac;q)∞
|en|+
Msn(q,b/a;q)∞
(c+sn,b/ac;q)∞
|en|
≤Md(q,b/a;q)∞
(c+d,b/ac;q)∞|en|. (.)
Since∞n=enconverges absolutely, (.) converges absolutely.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors equally have made contributions. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Changzhou University, Changzhou, Jiangsu 213164, P.R. China.2College of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing, 401331, P.R. China.
Acknowledgements
This work was supported by the National Natural Science Foundation (grant No. 11271057) of China.
Received: 15 July 2015 Accepted: 16 December 2015
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