On Convergence of Series Involving Basic Hypergeometric Series

Volume 2009, Article ID 170526,9pages doi:10.1155/2009/170526

Research Article

On Convergence of

q

-Series Involving

r

1

φ

r

Basic

Hypergeometric Series

Mingjin Wang

_{and Xilai Zhao}

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,[email protected]

Received 18 December 2008; Accepted 24 March 2009

Recommended by Ondrej Dosly

We use inequality technique and the terminating case of theq-binomial formula to give some results on convergence ofq-series involvingr1φr basic hypergeometric series. As an application

of the results, we discuss the convergence for special Thomaeq-integral.

Copyrightq2009 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.

1. Introduction

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, orthogonal polynomials and physics. Convergence of aq-series is an important problem in the study of

q-series. There are some results about it in1–3. For example, Ito used inequality technique to give a sufficient condition for convergence of a specialq-series called Jackson integral. In this paper, by using inequality technique, we derive the following two theorems on convergence ofq-series involvingr1φr basic hypergeometric series, which can be used for convergence of

special Thomaeq-integral.

2. Notations and Known Results

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. Theq-shifted factorials are defined as

n−1

a1, a2, . . . , am;q_n a1;q_na2;q_n· · ·am;q_n, 2.2

wherenis an integer or∞.

Theq-binomial theorem4,5is

∞

k0

a;q_kzk q;q_k

az;q_∞

z;q_∞ , |z|<1,q<1. 2.3

Whenaq−n_{, wherendenotes a nonnegative integer}

n

k0

q−n_;q kzk

q;q_k zq

−n_;q

n. 2.4

Heine introduced ther1φr basic hypergeometric series, which is defined by4,5

r1φr

a1, a2, . . . , ar1 b1, b2, . . . , br

; q, z

∞

n0

a1, a2, . . . , ar1;qnzn q, b1, b2, . . . , br;qn

. 2.5

3. Main Results

The main purpose of the present paper is to establish the following two theorems on convergence ofq-series involvingr1φr basic hypergeometric series.

Theorem 3.1. Supposeai, bi,tare any real numbers such thatt >0andbi <1withi1,2, . . . , r.

Let{cn}be any sequence of numbers. If

lim

n→ ∞

cn1

cn

p <1, 3.1

then theq-series

∞

n0

cn·r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q, tqn

3.2

converges absolutely.

Proof. Letb <1 and

ft 1−at

1−bt, 0≤t≤1, 3.3

Consequently, one has

1₁−₋at_bt≤max 1,|1−a|

1−b

. 3.4

From3.4, one knows

ai;qk

bi;q_k

1−ai

1−bi

·1−aiq

1−biq

· · ·1−aiqk−1

1−biqk−1

≤Mki, 3.5

whereMimax{1,|1−ai|/1−bi}fori1,2, . . . , r.

So, one has

a1, a2, . . . , ar;qk−1k b1, b2, . . . , br;qk

a1, a2, . . . , ar;qk b1, b2, . . . , br;qk

≤

_r

i1 Mi

k

. 3.6

It is obvious that

q−n_;q k−tqn

k

q;q_k >0, t >0, k1,2, . . . , n. 3.7

Multiplying both sides of3.6by

q−n_;q

k−tqnk

q;q_k 3.8

gives

q−n_{, a1, a2, . . . , a}

r;q_ktqnk q, b1, b2, . . . , br;q_k

≤

q−n_;q k q;q_k

−tqnr i1

Mi

k

. 3.9

Hence,

r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

;q, tqn

n

k0

q−n_{, a1, a2, . . . , a}

r;q_ktqnk q, b1, b2, . . . , br;q_k

≤n

k0

q−n_{, a1, a2, . . . , a}

r;q_ktqnk q, b1, b2, . . . , br;q_k

n

q−n_;q r k

n

k0

q−n_;q k q;q_k

−tqn r

i1 Mi

k

−t r

i1 Mi;q

n

. 3.11

Substituting3.11into3.10, one has

r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q, tqn

≤

−t r

i1 Mi;q

n

. 3.12

Multiplying both sides of3.12by|cn|, one has

cn·r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q, tqn

≤ |cn|

−t r

i1 Mi;q

n

. 3.13

The ratio test shows that the series

∞

n0 cn

−t r

i1 Mi;q

n

3.14

is absolutely convergent. From 3.13, it is sufficient to establish that 3.2 is absolutely convergent.

Theorem 3.2. Suppose ai, bi, tare any real numbers such that t > 0 and ai < 1, bi < 1 with i1,2, . . . , r. Let{cn}be any sequence of numbers. If

lim

n→ ∞

cn1

cn

p >1, or lim

n→ ∞

cn1

cn

∞, 3.15

then theq-series

∞

n0

cn·r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q,−tqn

3.16

diverges.

Proof. Leta <1,b <1 and

ft 1−at

1−bt, 0≤t≤1, 3.17

Consequently, one has

1−at

1−bt ≥min 1,

1−a

1−b

. 3.18

From3.18, one knows

ai;q

k

bi;q

k

1−ai

1−bi ·

1−aiq

1−biq· · ·

1−aiqk−1

1−biqk−1

≥mki, 3.19

wheremimin{1,1−ai/1−bi}fori1,2, . . . , r.

So, one has

a1, a2, . . . , ar;q

k

b1, b2, . . . , br;q

k

≥

_r

i1 mi

k

. 3.20

It is obvious that

q−n_;q

k

−tqnk

q;q_k >0, t >0, k1,2, . . . , n. 3.21

Multiplying both sides of3.20by

q−n_;q

k

−tqnk

q;q_k 3.22

gives

q−n_{, a1, a2, . . . , a} r;q

k

−tqnk

q, b1, b2, . . . , br;q

k

≥

q−n_;q

k

q;q_k

−tqnr i1

mi

k

. 3.23

Hence,

r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q,−tqn

n

k0

q−n_{, a1, a2, . . . , a}

r;q_k−tqnk q, b1, b2, . . . , br;q_k

≥n

k0

q−n_;q k q;q_k

−tqn r

i1 mi

k .

3.24

By using2.4one obtains

n

r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q,−tqn ≥ −t r

i1 mi;q

n

. 3.26

Multiplying both sides of3.26by|cn|, one has

|cn| ·r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q,−tqn

≥ |cn|

−t r

i1 mi;q

n

. 3.27

Since

lim

n→ ∞

|cn1|

−tri1mi;q

n1

|cn|

−tr_i1mi;q

n

lim

n→ ∞

cn1

cn

. 3.28

By hypothesis

lim

n→ ∞

cn1

cn

p >1, or lim

n→ ∞

cn1

cn

∞, 3.29

therefore, in both cases there exists a integerN0>0 such that∀n > N0

|cn1|

−tri1mi;q

n1

|cn|

−tri1mi;q

n

>1. 3.30

So, one can conclude that

|cn|

−t r

i1 mi;q

n >|cN0|

−t r

i1 mi;q

N0

, ∀n > N0. 3.31

Now, from3.27and3.31

|cn| ·r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q,−tqn

≥ |cn|

−t r

i1 mi;q

n

>|cN0|

−t r

i1 mi;q

N0

>0.

3.32

Thereby,3.16diverges.

4. Some Applications

In6,7, Thomae defined theq-integral on the interval0,1by

₁

0

ftdqt

1−q

∞

n0

fqnqn. 4.1

The right side of4.1corresponds to use a Riemann sum with partition pointstn qn, n

0,1,2, . . . .Jackson8extended Thomaeq-integral via

d

0

ftdqtd

1−q

∞

0

fdqn_qn_,

d

c

ftdqt

d

0

ftdqt−

c

0

ftdqt.

4.2

In this section, we use the theorems derived in this paper to discuss two examples of the convergence for Thomaeq-integral. We have the following theorems.

Theorem 4.1. Letai, bi, tbe any real numbers such thatt > 0andbi < 1withi 1,2, . . . , r. If α >−1, then the Thomaeq-integral

1

0

tα·r1φr

a1, a2, . . . , ar, t−1 b1, b2, . . . , br

; q, t

dqt 4.3

converges absolutely.

Proof. By the definition of Thomaeq-integral4.1, one has

1

0

tα·r1φr

a1, a2, . . . , ar, t−1 b1, b2, . . . , br

; q, t

dqt

1−q

∞

n0

qn1αr1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q, qn

.

4.4

UsingTheorem 3.1and noticing,

lim

n→ ∞

qn11α

Ifα >1, then the Thomaeq-integral

₁

0

t−α·r1φr

a1, a2, . . . , ar, t−1 b1, b2, . . . , br

; q,−t

dqt 4.6

diverges.

Proof. By the definition of Thomaeq-integral4.1, one has

₁

0

t−α·r1φr

a1, a2, . . . , ar, t−1 b1, b2, . . . , br

; q,−t

dqt

1−q

∞

n0 q1−αn

r1φr

a1, a2, . . . , ar, q−n b1, b2, . . . , br

; q,−qn

.

4.7

UsingTheorem 3.2and noticing,

lim

n→ ∞

q1−αn1 q1−αn q

1−α_{>1, 4.8}

one knows that4.6diverges.

Acknowledgment

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.

References

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3, pp. 339–345, 2007.

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1−qa1_/1−q2_{· · ·1−q}an−1_/1−qn_·1−qa_/1−qb_·1−qa1_/1−qb1_{· · ·1−q}an−1_/1− qbn−1_{· · ·1−q}ah

/1−qbh

·1−qah₁

/1−qbh₁

· · ·1−qah_n−1

/1−qbh_n−1

,”Annali di Matematica Pura ed Applicata, vol. 4, pp. 105–138, 1870.