As applications of the inequality, we discuss the convergence of someq-series involving3φ2

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AN INEQUALITY ABOUT₃φ₂ AND ITS APPLICATIONS

MINGJIN WANG AND HONGSHUN RUAN DEPARTMENT OFINFORMATIONSCIENCE

JIANGSUPOLYTECHNICUNIVERSITY

CHANGZHOUCITY213164 JIANGSUPROVINCE, P.R. CHINA.

wmj@jpu.edu.cn rhs@em.jpu.edu.cn

Received 17 August, 2007; accepted 18 May, 2008 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we use the terminating case of theq-binomial formula, theq-Chu- Vandermonde formula and the Grüss inequality to drive an inequality about3φ2. As applications of the inequality, we discuss the convergence of someq-series involving3φ2.

Key words and phrases: Basic hypergeometric function3φ2;q-binomial theorem;q-Chu-Vandermonde formula; Grüss inequality.

2000 Mathematics Subject Classification. Primary 26D15; Secondary 33D15.

1. STATEMENT OFMAINRESULTS

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, etc. Inequalities techniques provide useful tools in the study of spe- cial functions (see [1, 6, 7, 8, 9, 10]). For example, Ito used inequalities techniques to give a sufficient condition for convergence of a specialq-series called the Jackson integral [6]. In this paper, we derive the following new inequality aboutq-series involving₃φ₂.

Theorem 1.1. Leta1, a2, b1, b2be some real numbers such thatbi <1fori= 1,2. Then for all positive integersn, we have:

(1.1)

³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ b1, b2

;q,−a₁a₂qⁿ

− (a₁, a₂;q)_n (−1, b1, b2;q)n

≤λµ(−1;q)_n,

where

λ= max{1, Mⁿ}, M= max

|a1|,|a₁−b₁| 1−b₁

, µ= max{1, Nⁿ}, N= max

|a2|,|a₂−b₂| 1−b₂

. Applications of this inequality are also given.

The author would like to express deep appreciation to the referee for the helpful suggestions. In particular, the author thanks the referee for helping to improve the presentation of the paper.

271-07

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2. NOTATIONS ANDKNOWN RESULTS

We recall some definitions, notations and known results which will be used in the proof.

Throughout this paper, it is supposed that0< q <1. Theq-shifted factorials are defined as (2.1) (a;q)₀ = 1, (a;q)_n =

n−1

Y

k=0

(1−aq^k), (a;q)∞=

∞

Y

k=0

(1−aq^k).

We also adopt the following compact notation for multipleq-shifted factorials:

(2.2) (a₁, a₂, . . . , a_m;q)_n = (a₁;q)_n(a₂;q)_n· · ·(a_m;q)_n, wherenis an integer or∞.

Theq-binomial theorem (see [2, 3, 4]) is given by (2.3)

∞

X

k=0

(a;q)_kz^k

(q;q)_k = (az;q)∞

(z;q)∞

, |z|<1.

Whena =q⁻ⁿ, wherendenotes a nonnegative integer, we have (2.4)

n

X

k=0

(q⁻ⁿ;q)_kz^k

(q;q)_k = (zq⁻ⁿ;q)_n.

Heine introduced the_r+1φ_r basic hypergeometric series, which is defined by

(2.5) _r+1φ_r

a₁, a₂, . . . , a_r+1 b₁, b₂, . . . , b_r ;q, x

=

∞

X

n=0

(a₁, a₂, . . . , a_r+1;q)_nxⁿ (q, b₁, b₂, . . . , b_r;q)_n . Theq-Chu-Vandermonde sums (see [2, 3, 4]) are

(2.6) 2φ₁

a, q⁻ⁿ c ;q, q

= aⁿ(c/a;q)_n (c;q)_n and, reversing the order of summation, we have

(2.7) ₂φ₁

a, q⁻ⁿ

c ;q, cqⁿ/a

= (c/a;q)_n (c;q)_n . At the end of this section, we recall the Grüss inequality (see [5]):

(2.8)

1 b−a

Z b

a

f(x)g(x)dx− 1

b−a Z b

a

f(x)dx 1

b−a Z b

a

g(x)dx

≤ (M −m)(N −n)

4 ,

provided thatf, g : [a, b]→ Rare integrable on[a, b]andm≤ f(x) ≤M, n ≤g(x)≤ N for allx∈[a, b], wherem,M,n,N are given constants.

By simple calculus, one can prove that the discrete version of the Grüss inequality can be stated as:

if a ≤ λ_i ≤ A and b ≤ µ_i ≤ B, i = 1,2, . . . , n, then for all sequences (p_i)0≤i≤n of nonnegative real numbers satisfyingPn

i=1p_i = 1, we have (2.9)

n

X

i=1

λ_iµ_ip_i−

n

X

i=1

λ_ip_i

!

·

n

X

i=1

µ_ip_i

!

≤ (A−a)(B−b)

4 ,

wherea,A,b,B are some given real constants.

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3. PROOF OF THETHEOREM

In this section, we use the terminating case of theq-binomial formula, theq-Chu-Vandermonde formula and the Grüss inequality to prove (1.1). For this purpose, we need the following lemma.

Lemma 3.1. Letaandbbe two real numbers such thatb <1, and let0≤t ≤1. Then, (3.1)

a−bt 1−bt

≤max

|a|,|a−b|

1−b

.

Proof. Let

f(t) = a−bt

1−bt, 0≤t≤1, then

f⁰(t) = b(a−1)

(1−bt)², 0≤t≤1.

Sof(t)is a monotonic function with respect to0 ≤ t ≤ 1. Sincef(0) = aand f(1) = ^a−b_1−b,

(3.1) holds.

Now, we are in a position to prove the inequality (1.1).

Proof. Put

(3.2) p_k = (q⁻ⁿ;q)_k(−qⁿ)^k

(q;q)_k(−1;q)_n , k = 0,1, . . . , n.

It is obvious thatpk ≥0.

On the other hand, using (2.4), we obtain

n

X

k=0

p_k = 1

(−1;q)_n

n

X

k=0

(q⁻ⁿ;q)_k(−qⁿ)^k (q;q)_k = 1.

Let

(3.3) λ_k = (−a₁)^k(b₁/a₁;q)_k

(b₁;q)_k , and

(3.4) µ_k = (−a2)^k(b2/a2;q)k

(b₂;q)_k , wherek = 0,1, . . . , n.

According to the definitions ofM, N, λandµ, it is easy to see that M^k ≤λ and N^k≤µ, 0≤k ≤n.

Using the lemma, one can get for all0≤k ≤n,

(3.5) |λ_k|=

a₁−b₁

1−b₁ · a₁−b₁q

1−b₁q · · · a₁−b₁q^k−1 1−b₁q^k−1

≤M^k ≤λ

and

(3.6) |µ_k|=

a2−b2

1−b₂ · a2−b2q

1−b₂q · · · a2−b2q^k−1 1−b₂q^k−1

≤N^k ≤µ.

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Substitution of (3.2), (3.3), (3.4), (3.5) and (3.6) into (2.9), gives (3.7)

n

X

k=0

(q⁻ⁿ;q)_k(−qⁿ)^k

(q;q)_k(−1;q)_n · (−a₁)^k(b₁/a₁;q)_k

(b₁;q)_k ·(−a₂)^k(b₂/a₂;q)_k (b₂;q)_k

−

n

X

k=0

(q;q)_k(−1;q)_n · (−a₁)^k(b₁/a₁;q)_k (b₁;q)_k

×

n

X

k=0

(q;q)_k(−1;q)_n · (−a₂)^k(b₂/a₂;q)_k (b₂;q)_k

≤λµ.

Using (2.5) and (2.7), we get

n

X

k=0

(q;q)_k(−1;q)_n ·(−a₁)^k(b₁/a₁;q)_k

(b₁;q)_k · (−a₂)^k(b₂/a₂;q)_k (b₂;q)_k (3.8)

= 1

(−1;q)_n

n

X

k=0

(q⁻ⁿ, b₁/a₁, b₂/a₂;q)_k

(q, b₁, b₂;q)_k (−a1a2qⁿ)^k

= 1

(−1;q)_n³φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

b₁, b₂ ;q,−a₁a₂qⁿ

,

(3.9)

n

X

k=0

(q;q)_k(−1;q)_n ·(−a₁)^k(b₁/a₁;q)_k

(b₁;q)_k = (a₁;q)_n (−1, b₁;q)_n and

(3.10)

n

X

k=0

(q;q)_k(−1;q)_n · (−a₂)^k(b₂/a₂;q)_k

(b₂;q)_k = (a₂;q)_n (−1, b₂;q)_n.

Substituting (3.8), (3.9) and (3.10) into (3.7), we obtain (1.1).

Takinga₂ = 1in (1.1), we get the following corollary.

Corollary 3.2. We have (3.11)

2φ1

b₁/a₁, q⁻ⁿ

b₁ ;q,−a1qⁿ

≤λ(−1;q)n.

4. SOME APPLICATIONS OF THEINEQUALITY

Convergence ofq-series is an important problem which is at times very difficult. As applications of the inequality derived in this paper, we obtain some results about the convergence of theq-series involving₃φ₂. In this section, we mainly discuss the convergence of the following q-series:

(4.1)

∞

X

n=0

c_n3φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

.

Theorem 4.1. Suppose |a_i| ≤ 1 and b_i < ^aⁱ₂⁺¹ for i = 1,2. Let {c_n} be a real sequence satisfying

n→∞lim

c_n+1 c_n

=p <1.

Then the series (4.1) is absolutely convergent.

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Proof. It is obvious thatb_i <1fori= 1,2. Combining the following inequality (4.2)

³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

−

(a₁, a₂;q)_n (−1, b₁, b₂;q)_n

≤ ³

φ₂

b1/a1, b2/a2, q⁻ⁿ

− (a1, a2;q)n

(−1, b₁, b₂;q)_n with (2.1), shows that

(4.3) ³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ b1, b2

;q,−a₁a₂qⁿ

≤

(a₁, a₂;q)_n (−1, b1, b2;q)n

+λµ(−1;q)_n. Since

|a_i| ≤1, b_i < a_i+ 1

2 , i= 1,2, which is equivalent to

|a_i| ≤1, |a_i−b_i| 1−bi

<1, i= 1,2, then

(4.4) λ =µ= 1.

Substituting (4.4) into (4.3), we obtain (4.5)

³

φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

≤

(a₁, a₂;q)_n (−1, b₁, b₂;q)_n

+ (−1;q)_n.

Multiplication of the two sides of (4.5) by|c_n|gives (4.6)

c_n3φ₂

b₁/a₁, b₂/a₂, q⁻ⁿ

≤

c_n(a₁, a₂;q)_n (−1, b₁, b₂;q)_n

+|c_n(−1;q)_n|. The ratio test shows that both

∞

X

n=0

c_n(a₁, a₂;q)_n

(−1, b₁, b₂;q)_n and

∞

X

n=0

c_n(−1;q)_n

are absolutely convergent. From (4.6), we get that (4.1) is absolutely convergent.

Theorem 4.2. Suppose|a₁| >1ora₁ <2b₁−1,b₁ <1, |a₂| ≤ 1andb₂ ≤ ^a²₂⁺¹. Let{c_n}be a real sequence satisfying

n→∞lim

c_n+1 cn

=p < 1 M, whereM = maxn

|a₁|,^|a_1−b¹^−b¹^|

1

o

. Then the series (4.1) is absolutely convergent.

Proof. First we point out thata₁ <2b₁−1implies

|a₁−b₁| 1−b₁ >1.

So, under the conditions of the theorem, we know

λ=Mⁿ and µ= 1.

Multiplying both sides of (4.3) by|cn|, one gets (4.7)

c_n3φ₂

b1/a1, b2/a2, q⁻ⁿ

≤

cn(a1, a2;q)n

(−1, b₁, b₂;q)_n

+|c_nMⁿ(−1;q)_n|.

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The ratio test shows that both

∞

X

n=0

c_n(a₁, a₂;q)_n

(−1, b₁, b₂;q)_n and

∞

X

n=0

cnMⁿ(−1;q)n

are absolutely convergent. From (4.7), we get that (4.1) is absolutely convergent.

Similarly, we have

Theorem 4.3. Suppose |a_i| > 1 or a_i < 2b_i −1, b_i < 1 with i = 1,2. Let {c_n} be a real sequence satisfying

n→∞lim

c_n+1 cn

=p < 1 M N, where M = maxn

|a₁|,^|a_1−b¹^−b¹^|

1

o

and N = maxn

|a₂|,^|a_1−b²^−b²^|

2

o

. Then the series (4.1) is abso- lutely convergent.

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