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An expectation formula with applications
J. Math. Anal. Appl. 379 (2011) 461–468
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
An expectation formula with applications Mingjin Wang Department of Applied Mathematics, Changzhou University, Changzhou, Jiangsu, 213164, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 September 2010 Available online 20 January 2011 Submitted by Goong Chen
In this paper, we derive an expectation formula of a random variable having distribution W (x; q). As applications of the expectation formula, we give a transformation formula and an expansion of Sears’ 3 φ2 transformation formula. © 2011 Elsevier Inc. All rights reserved.
Keywords: Andrews–Askey integral Probability distribution W (x, q) Al-Salam and Verma q-integral Lebesgue’s dominated convergence theorem Basic hypergeometric function 3 φ2
1. Introduction and main result The probabilistic method is a useful tool in the study of basic hypergeometric functions. There are some works available in the literature. For example, K.W.J. Kadell [11] gave a probabilistic proof of Ramanujan’s 1 ψ1 sum based on the order statistics. J. Fulman [7] presented a probabilistic proof of Rogers–Ramanujan identity using Markov chain on the nonnegative integers. R. Chapman [6] extended J. Fulman’s methods to prove the Andrews–Gordon identity. In particular, the present author [19] established the following new probability distribution W (x; q):
P ξ = xn qk =
(−x)n (xn−1 qk+1 , xn qk+1 ; q)∞ qk , (q, q/x, x; q)∞
(1.1)
where x < 0; 0 < q < 1; n = 0, 1; k = 0, 1, 2, . . . , and gave some applications of this distribution in q-series. In this paper, we derive an expectation formula of W (x; q) and provide some probabilistic derivations of q-identities which include an extension of Sears’ 3 φ2 transformation formula. We recall some definitions, notation and known results in [4,8] which will be used in this paper. Throughout the whole 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 − aqk ,
(a; q)∞ =
k =0
∞
1 − aqk .
(1.2)
k =0
We also adopt the following compact notation for multiple q-shifted factorials:
(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n , E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.01.044
©
2011 Elsevier Inc. All rights reserved.
(1.3)
462
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
where n is an integer or ∞. We may extend the definition (1.2) of (a; q)n to
(a; q)α =
(a; q)∞ , (aqα ; q)∞
α . In particular,
for any complex number
(a; q)−n =
(1.4)
1 (a; q)∞ (−q/a)n (n) = = q 2 . (aq−n ; q)∞ (aq−n ; q)n (q/a; q)n
(1.5)
Heine introduced the r +1 φr basic hypergeometric series, which is defined by
r +1 φr
a 1 , a 2 , . . . , a r +1 b 1 , b 2 , . . . , br
; q, x =
∞ (a1 , a2 , . . . , ar +1 ; q)n xn n =0
(q, b1 , b2 , . . . , br ; q)n
.
(1.6)
The q-binomial theorem ∞ (a; q)n xn
(q; q)n
n =0
=
(ax; q)∞ , (x; q)∞
|x| < 1.
The q-Gauss summation formula
2 φ1
a, b c
; q,
c ab
=
(c /a, c /b; q)∞ , (c , c /ab; q)∞
(1.7)
c < 1. ab
(1.8)
Sears’ 3 φ2 transformation formula
(b2 /a3 , b1 b2 /a1 a2 ; q)∞ a1 , a2 , a3 b1 b2 b1 /a1 , b1 /a2 , a3 b2 = , φ ; q , φ ; q , 3 2 3 2 b1 , b2 a1 a2 a3 (b2 , b1 b2 /a1 a2 a3 ; q)∞ b1 , b1 b2 /a1 a2 a3 b2 b1 b2 < 1. < 1, a a a a 1 2 3
(1.9)
3
The bilateral basic hypergeometric series r ψs is defined by
r ψs
∞ n (a1 , a2 , . . . , ar ; q)n ; q, z = (−1)(s−r )n q(s−r )(2) zn . b1 , b2 , . . . , b s (b1 , b2 , . . . , b s ; q)n n=−∞ a 1 , a 2 , . . . , ar
(1.10)
The following is the well-known Ramanujan’s 1 ψ1 summation formula ∞ (a; q)n n (q, b/a, az, q/az; q)∞ z = , (b; q)n (b, q/a, z, b/az; q)∞ n=−∞
|b/a| < | z| < 1.
(1.11)
F.H. Jackson defined the q-integral by [10]
d f (t ) dq t = d(1 − q)
∞
f dqn qn ,
(1.12)
n =0
0
and
d
d f (t ) dq t =
c
c f (t ) dq t −
0
f (t ) dq t .
(1.13)
0
He also defined an integral on (0, ∞) by
∞
∞
f (t ) dq t = (1 − q) 0
f qn qn .
(1.14)
n=−∞
On the interval (−∞, ∞) the bilateral q-integral is defined by
∞ f (t ) dq t = (1 − q) −∞
∞
n
f q
n=−∞
+ f −qn qn .
(1.15)
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
463
The q-integral is important in the theory and applications of basic hypergeometric series. For example, the present author gives some applications of the q-integral [14–18,20]. These papers can help the reader to understand the main idea of the present paper. The following is the Andrews–Askey integral [2], which can be derived from Ramanujan’s 1 ψ1 summation:
d c
(qt /c , qt /d; q)∞ d(1 − q)(q, dq/c , c /d, abcd; q)∞ dq t = , (at , bt ; q)∞ (ac , ad, bc , bd; q)∞
(1.16)
provided that no zero factors occur in the denominator of the integrals. Al-Salam and Verma gave an extension of the Andrews–Askey integral, which is called the Al-Salam and Verma [1] q-integral
d c
(qt /c , qt /d, et ; q)∞ d(1 − q)(q, dq/c , c /d, e /a, e /b, e / f ; q)∞ dq t = , (at , bt , f t ; q)∞ (ac , ad, bc , bd, f c , f d; q)∞
(1.17)
provided that no zero factors occur in the denominator of the integrals, where e = abcdf . The main result of this paper is the following theorem: Theorem 1. Let ξ denote random variable having distribution W (x; q), −1 < x < 0, then we have
E
(cxξ ; q)∞ (aξ, bξ, c ξ ; q)∞
=
(acx, bcx, x; q)∞ ab, c , cx ; q, x , 3 φ2 (a, ax, b, bx, c , cx; q)∞ acx, bcx
(1.18)
provided that max{|a|, |b|, |c |} < 1, where E( X ) denotes expected value of the random variable X . 2. The proof of Theorem 1 In this section, we use the Al-Salam and Verma [1] q-integral to give the proof of Theorem 1. We also need to use the following well-known theorems: Analytic continuation theorem: If f and g are analytic at z0 and agree at infinitely many points which include z0 as an accumulation point, then f = g. Lebesgue’s dominated convergence theorem: Suppose that { X n , n 1} is a sequence of random variables, that X n → X pointwise almost everywhere as n → ∞, and that | X n | Y for all n, where random variable Y is integrable. Then X is integrable, and
lim E X n = E X .
(2.1)
n→∞
The analytic continuation theorem is useful in the study of basic hypergeometric functions. Two well-known examples are, Ismail’s proof of the Ramanujan’s 1 ψ1 sum [9] and the Askey–Ismail proof of Bailey’s sum of the very well poised 6 ψ6 [5]. Now we give the proof. Proof. Considering the following sequence of random variables (on a probability space):
ηn =
n (abcxξ ; q)∞ (ab, c ξ ; q)k xk , (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k k =0
where n = 0, 1, 2, . . . . Since,
n (abcxξ ; q) (ab, c ξ ; q)k xk ∞ |ηn | = (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k k =0 n (abcxqk ξ ; q) (ab; q)k xk ∞ = · (q; q)k (aξ, bξ, cqk ξ ; q)∞ k =0 n (−|abcx|; q)∞ (ab; q)k xk (q; q) (|a|, |b|, |c |; q)∞ k k =0
and the series
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M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
∞ (ab; q)k xk
(q; q)k
k =0
converges absolutely. Using Lebesgue’s dominated convergence theorem obtains:
lim Eηn = E
n→∞
lim
n→∞
ηn .
(2.2)
Using the q-Gauss theorem (1.8), we have
lim
n→∞
ηn =
∞ (abcxξ ; q)∞ (ab, c ξ ; q)k xk (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k k =0
=
(abcxξ ; q)∞ (cxξ, abx; q)∞ · (aξ, bξ, c ξ ; q)∞ (abcxξ, x; q)∞
=
(abx; q)∞ (cxξ ; q)∞ · . (x; q)∞ (aξ, bξ, c ξ ; q)∞
(2.3)
Hence, we get the right-hand side of (2.2):
E
lim
n→∞
ηn =
(abx; q)∞ (cxξ ; q)∞ . E (x; q)∞ (aξ, bξ, c ξ ; q)∞
(2.4)
Now we begin to calculate the left-hand side of (2.2). Observe that
n (abcxξ ; q)∞ (ab, c ξ ; q)k xk Eηn = E (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k
=
k =0
n (abcxqk ξ ; q)∞ (ab; q)k xk E . (q; q)k (aξ, bξ, cqk ξ ; q)∞
(2.5)
k =0
Employing the Al-Salam and Verma q-integral (1.17) gives
E
(abcxqk ξ ; q)∞ (aξ, bξ, cqk ξ ; q)∞
1 ∞ (−x)n (xn−1 qm+1 , xn qm+1 , abcxn+1 qk+m ; q)∞ qm (q, q/x, x, axn qm , bxn qm , cqk+m xn ; q)∞ n =0 m =0 ∞ 1 (qm+1 /x, qm+1 , abcxqk+m ; q)∞ qm = (1 − q) (1 − q)(q, q/x, x; q)∞ (aqm , bqm , cqk+m ; q)∞ m =0 ∞ (qm+1 , xqm+1 , abcx2 qk+m ; q)∞ qm − x(1 − q) (axqm , bxqm , cqk+m x; q)∞
=
m =0
=
1
1
(1 − q)(q, q/x, x; q)∞
x
(qt /x, qt , abcxqk t ; q)∞ dq t (at , bt , cqk t ; q)∞
(abx, acxqk , bcxqk ; q)∞ = . (a, ax, b, bx, cqk , cxqk ; q)∞
(2.6)
Substituting (2.6) into (2.5) yields
Eηn =
n n (abcxqk ξ ; q)∞ (abx, acx, bcx; q)∞ (ab, c , cx; q)k xk (ab; q)k xk E . = (q; q)k (a, ax, b, bx, c , cx; q)∞ (q, acx, bcx; q)k (aξ, bξ, cqk ξ ; q)∞ k =0
(2.7)
k =0
Hence, we get the left-hand side of (2.2)
lim Eηn =
n→∞
(abx, acx, bcx; q)∞ ab, c , cx . φ ; q , x 3 2 (a, ax, b, bx, c , cx; q)∞ acx, bcx
Substituting (2.4) and (2.8) into (2.2) yields (1.18).
2
(2.8)
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
465
3. Some applications The expectation formula (1.18) can be used to derive identities about the basic hypergeometric functions. In this section, we show some applications of Theorem 1. The following transformation formula can be derived directly from Theorem 1 and q-binomial theorem. Theorem 2. Let |a| < 1, |b| < 1, |x| < 1 and | z| < 1, then ∞ (a, ax; q)n zn n =0
(q, abx; q)n
3 φ2
(az, azx; q)∞ b, bx ; q , x φ ; q , x = . 2 1 ( z, abzx; q)∞ abxqk , abz abx
a2 zqk , b, bx
(3.1)
Proof. Rewrite q-binomial theorem (1.7) as ∞ n =0
zn
1
1
1
· = · . (q; q)n (aqn , az; q)∞ ( z; q)∞ (a; q)∞
(3.2)
Letting a = aξ in (3.2) gives ∞ n =0
zn
1
1
1
· = · , (q; q)n (aqn ξ, azξ ; q)∞ ( z; q)∞ (aξ ; q)∞
(3.3)
where ξ denotes random variable having distribution W (x; q) and −1 < x < 0. Multiplying both sides of (3.3) by
(bxξ ; q)∞ (bξ ; q)∞ and then applying the expectation operator E on both sides of (3.3), it follows that
E
∞ n =0
(bxξ ; q)∞ · (q; q)n (aqn ξ, azξ, bξ ; q)∞ zn
=
1
( z; q)∞
E
(bxξ ; q)∞ . (aξ, bξ ; q)∞
(3.4)
Since,
n z (bxξ ; q)∞ | z|n (−|b|; q)∞ · · (q; q) (aqn ξ, azξ, bξ ; q) (|a|, |az|, |b|; q) n ∞ ∞ (q; q)n and the series ∞ | z|n (q; q)n
m =0
converges. Using Lebesgue’s dominated convergence theorem yields
E
∞ n =0
(bxξ ; q)∞ · n (q; q)n (aq ξ, azξ, bξ ; q)∞ zn
=
∞ n =0
zn
(q; q)n
E
(bxξ ; q)∞ . (aqn ξ, azξ, bξ ; q)∞
(3.5)
Hence, ∞ n =0
zn
(q; q)n
E
(bxξ ; q)∞ (aqn ξ, azξ, bξ ; q)∞
=
1
( z; q)∞
E
(bxξ ; q)∞ . (aξ, bξ ; q)∞
(3.6)
Employing (1.18) gives
(bxξ ; q)∞ E (aqn ξ, azξ, bξ ; q)∞
2 n (abxqn , abzx, x; q)∞ a zq , b, bx = ; q, x 3 φ2 (aqn , axqn , az, azx, b, bx; q)∞ abxqn , abz
(3.7)
and
E
(bxξ ; q)∞ (aξ, bξ ; q)∞
=
(abx, x; q)∞ b, bx . φ ; q , x 2 1 (a, ax, b, bx; q)∞ abx
(3.8)
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M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
Substituting (3.7) and (3.8) into (3.6) yields ∞ (a, ax; q)n zn
(q, abx; q)n
n =0
3 φ2
(az, azx; q)∞ b, bx ; q , x φ ; q , x = , 2 1 ( z, abzx; q)∞ abxqn , abz abx
a2 zqn , b, bx
(3.9)
where −1 < x < 0. By analytic continuation, we may replace the assumption −1 < x < 0 by |x| < 1. Thus, we get (3.1).
2
From the theorem, we can easily get the summation formula involving 3 φ2 : Corollary 1. Let |a| < |b| < 1 and | z| < 1, then ∞ (a, a2 /b; q)n zn
(q, a2 ; q)n
n =0
3 φ2
a, b, a2 zqn abz, a2 qn
; q,
a
b
=
(az, a2 z/b, a, a2 /b; q)∞ . ( z, a2 z, a2 , a/b; q)∞
=
(az, a2 z/b; q)∞ a, b a φ ; q , 2 1 b a2 ( z, a2 z; q)∞
=
(az, a2 z/b, a, a2 /b; q)∞ , ( z, a2 z, a2 , a/b; q)∞
(3.10)
Proof. Letting x = a/b in (3.1) gives ∞ (a, a2 /b; q)n zn
(q, a2 ; q)n
n =0
3 φ2
a, b, a2 zqn abz, a2 qn
; q,
a b
(3.11)
2
as desired.
Sears’ 3 φ2 transformation formula has been used by Andrews [3,12] in proving many of Ramanujan’s identities for partial theta functions. Using the expectation formula (1.18), we can easily derive the following expansion of Sears’ 3 φ2 transformation formula. Theorem 3. Let |a| < |c | < |b| < 1, |d| < 1, |x| < 1 and | z| < 1, then ∞ (b, a, ax; q)n zn
(q, c , adx; q)n
n =0
=
3 φ2
a2 bzqn /c , d, dx adxqn , abdzx/c
; q, x
2 n ∞ (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n a bzq , d, dx . φ ; q , x 3 2 ( z, c ; q)∞ (q, bz, abdzx/c ; q)n abdzxqn /c , adx
(3.12)
n =0
Proof. By the following Rogers’ iteration of Heine’s transformation formula [4,8]
2 φ1
(c /b, bz; q)∞ b, abz/c c , ; q, z = ; q, 2 φ1 ( z, c ; q)∞ b c bz
a, b
(3.13)
which can be rewritten as ∞ (b; q)n zn n =0
·
1
(q, c ; q)n (aqn , abz/c ; q)∞
=
∞ (c /b, bz; q)∞ (b; q)n (c /b)n 1 · . ( z, c ; q)∞ (q, bz; q)n (a, abzqn /c ; q)∞
(3.14)
n =0
Letting a = aξ in (3.14) and multiplying both sides of (3.14) by
(dxξ ; q)∞ (dξ ; q)∞ yields, ∞ (b; q)n zn n =0
(q, c ; q)n
·
∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n (dxξ ; q)∞ = · , (aqn ξ, abzξ/c , dξ ; q)∞ ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞
(3.15)
n =0
where ξ denotes random variable having distribution W (x; q) and −1 < x < 0. Applying the expectation operator E on both sides of (3.15), it follows that
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
E
∞ (b; q)n zn
467
(dxξ ; q)∞ · (q, c ; q)n (aqn ξ, abzξ/c , dξ ; q)∞ n =0 ∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n =E · . ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞
(3.16)
n =0
Since,
(b; q)n zn (|a|, b; q)n zn (−|d|; q)∞ (dxξ ; q)∞ · (q, c ; q) (aqn ξ, abzξ/c , dξ ; q) (|a|, |abz/c |, |d|; q) (q, c ; q) n ∞ ∞ n and the series
∞ (|a|, b; q)n zn (q, c ; q) n
m =0
is convergent. Using Lebesgue dominated convergence theorem, we obtain
E
∞ (b; q)n zn n =0
(dxξ ; q)∞ · (q, c ; q)n (aqn ξ, abzξ/c , dξ ; q)∞
=
∞ (b; q)n zn n =0
(dxξ ; q)∞ E . (q, c ; q)n (aqn ξ, abzξ/c , dξ ; q)∞
(3.17)
Employing (1.18) gives
(dxξ ; q)∞ E (aqn ξ, abzξ/c , dξ ; q)∞
2 n (adxqn , abdzx/c , x; q)∞ a bzq /c , d, dx ; q, x . = 3 φ2 (aqn , axqn , abz/c , abzx/c , d, dx; q)∞ adxqn , abdzx/c
(3.18)
Substituting (3.18) into (3.17), then we get the left-hand side of (3.16)
E
∞ (b; q)n zn n =0
(dxξ ; q)∞ · n (q, c ; q)n (aq ξ, abzξ/c , dξ ; q)∞
2 n ∞ (b, a, ax; q)n zn a bzq /c , d, dx (adx, abdzx/c , x; q)∞ ; q, x . = 3 φ2 (a, ax, abz/c , abzx/c , d, dx; q)∞ (q, c , adx; q)n adxqn , abdzx/c
(3.19)
n =0
Similarly, we can obtain the right-hand side of (3.16)
∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n E · ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞
n =0
=
∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n E ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞ n =0
=
∞ (adx, abdzx/c , x; q)∞ (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n (a, ax, abz/c , abzx/c , d, dx; q)∞ ( z, c ; q)∞ (q, bz, abdzx/c ; q)n n =0 2 n a bzq , d, dx × 3 φ2 ; q, x . abdzxqn /c , adx
(3.20)
Substituting (3.19) and (3.20) into (3.17), we have ∞ (b, a, ax; q)n zn n =0
(q, c , adx; q)n
3 φ2
a2 bzqn /c , d, dx adxqn , abdzx/c
; q, x
2 n ∞ a bzq , d, dx (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n = ; q, x , 3 φ2 abdzxqn /c , adx ( z, c ; q)∞ (q, bz, abdzx/c ; q)n
(3.21)
n =0
where −1 < x < 0. By analytic continuation, we may replace the assumption −1 < x < 0 by |x| < 1. Thus, we get (3.12).
2
468
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
We point out that the two terms 3 φ2 transformation formula of Sears is a special case of (3.12). The two terms 3 φ2 transformation formula of Sears [13] can be stated as Corollary 2. We have
3 φ2
a1 , a2 , a3 b1 , b2
; q,
b1 b2 a1 a2 a3
=
(b2 /a3 , b1 b2 /a1 a2 ; q)∞ b1 /a1 , b1 /a2 , a3 b2 . ; q, 3 φ2 (b2 , b1 b2 /a1 a2 a3 ; q)∞ b1 , b1 b2 /a1 a2 a3
(3.22)
Proof. Letting d = 0 in (3.12) and using q-binomial theorem (1.7) gives ∞ (b, a, ax; q)n zn (a2 bzxqn /c ; q)∞ · (q, c ; q)n (x; q)∞ n =0
=
∞ (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n (a2 bzxqn ; q)∞ · , ( z, c ; q)∞ (q, bz; q)n (x; q)∞
(3.23)
n =0
which can be rewritten as
3 φ2
(c /b, bz; q)∞ b, abz/c , abzx/c c ; q , z φ ; q , = . 3 2 ( z, c ; q)∞ b c , a2 bzx/c bz, a2 bzx/c b, a, ax
(3.24)
After replacing (a, b, ax, c , z) by (a1 , a3 , a2 , b2 , b1 b2 /a1 a2 a3 ) in (3.23), we obtain the two terms 3 φ2 transformation formula of Sears (3.22). 2 Acknowledgments The author would like to express deep appreciation to the anonymous referee for the helpful suggestions. In particular, the author thanks the referee for help to improve the presentation of the paper.
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W.A. Al-Salam, A. Verma, Some remarks on q-beta integral, Proc. Amer. Math. Soc. 85 (1982) 360–362. G.E. Andrews, R. Askey, Another q-extension of the beta function, Proc. Amer. Math. Soc. 81 (1981) 97–100. G.E. Andrews, Ramanujan’s “lost” notebook. 1. Partial θ -functions, Adv. Math. 41 (1981) 137–172. G.E. Andrews, q-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, CBMS Reg. Conf. Ser. Math., vol. 66, Amer. Math. Soc., Providence, RI, 1986. R. Askey, M.E.H. Ismail, The very well poised 6 ψ6 , Proc. Amer. Math. Soc. 77 (1979) 218–222. R. Chapman, A probabilistic proof of the Andrews–Gordon identities, Discrete Math. 290 (2005) 79–84. J. Fulman, A probabilistic proof of the Rogers–Ramanujan identities, Bull. Lond. Math. Soc. 33 (2001) 397–407. G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, MA, 1990. M.E.H. Ismail, A simple proof of Ramanujan’s 1 ψ1 sum, Proc. Amer. Math. Soc. 63 (1977) 185–186. F.H. Jackson, On q-definite integrals, Q. J. Pure Appl. Math. 50 (1910) 101–112. K.W.J. Kadell, A probabilistic proof of Ramanujan’s 1 ψ1 sum, SIAM J. Math. Anal. 18 (1987) 1539–1548. Z.G. Liu, Some operator identities and q-series transformation formulas, Discrete Math. 265 (2003) 119–139. D.B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951) 158–180. M. Wang, A remark on Andrews–Askey integral, J. Math. Anal. Appl. 341 (2) (2008) 1487–1494. M. Wang, q-Integral representation of the Al-Salam–Carlitz polynomials, Appl. Math. Lett. 22 (2009) 943–945. M. Wang, Generalizations of Milne’s U (n + 1) q-binomial theorems, Comput. Math. Appl. 58 (2009) 80–87. M. Wang, A recurring q-integral formula, Appl. Math. Lett. 23 (2010) 256–260. M. Wang, An extension of the q-beta integral with applications, J. Math. Anal. Appl. 365 (2010) 653–658. M. Wang, A new probability distribution with applications, Pacific J. Math. 247 (1) (2010) 241–255. M. Wang, On another extension of q-Pfaff–Saalschutz formula, Czechoslovak Math. J. 60 (4) (2010) 1131–1137.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
An expectation formula with applications Mingjin Wang Department of Applied Mathematics, Changzhou University, Changzhou, Jiangsu, 213164, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 September 2010 Available online 20 January 2011 Submitted by Goong Chen
In this paper, we derive an expectation formula of a random variable having distribution W (x; q). As applications of the expectation formula, we give a transformation formula and an expansion of Sears’ 3 φ2 transformation formula. © 2011 Elsevier Inc. All rights reserved.
Keywords: Andrews–Askey integral Probability distribution W (x, q) Al-Salam and Verma q-integral Lebesgue’s dominated convergence theorem Basic hypergeometric function 3 φ2
1. Introduction and main result The probabilistic method is a useful tool in the study of basic hypergeometric functions. There are some works available in the literature. For example, K.W.J. Kadell [11] gave a probabilistic proof of Ramanujan’s 1 ψ1 sum based on the order statistics. J. Fulman [7] presented a probabilistic proof of Rogers–Ramanujan identity using Markov chain on the nonnegative integers. R. Chapman [6] extended J. Fulman’s methods to prove the Andrews–Gordon identity. In particular, the present author [19] established the following new probability distribution W (x; q):
P ξ = xn qk =
(−x)n (xn−1 qk+1 , xn qk+1 ; q)∞ qk , (q, q/x, x; q)∞
(1.1)
where x < 0; 0 < q < 1; n = 0, 1; k = 0, 1, 2, . . . , and gave some applications of this distribution in q-series. In this paper, we derive an expectation formula of W (x; q) and provide some probabilistic derivations of q-identities which include an extension of Sears’ 3 φ2 transformation formula. We recall some definitions, notation and known results in [4,8] which will be used in this paper. Throughout the whole 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 − aqk ,
(a; q)∞ =
k =0
∞
1 − aqk .
(1.2)
k =0
We also adopt the following compact notation for multiple q-shifted factorials:
(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n , E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2011.01.044
©
2011 Elsevier Inc. All rights reserved.
(1.3)
462
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
where n is an integer or ∞. We may extend the definition (1.2) of (a; q)n to
(a; q)α =
(a; q)∞ , (aqα ; q)∞
α . In particular,
for any complex number
(a; q)−n =
(1.4)
1 (a; q)∞ (−q/a)n (n) = = q 2 . (aq−n ; q)∞ (aq−n ; q)n (q/a; q)n
(1.5)
Heine introduced the r +1 φr basic hypergeometric series, which is defined by
r +1 φr
a 1 , a 2 , . . . , a r +1 b 1 , b 2 , . . . , br
; q, x =
∞ (a1 , a2 , . . . , ar +1 ; q)n xn n =0
(q, b1 , b2 , . . . , br ; q)n
.
(1.6)
The q-binomial theorem ∞ (a; q)n xn
(q; q)n
n =0
=
(ax; q)∞ , (x; q)∞
|x| < 1.
The q-Gauss summation formula
2 φ1
a, b c
; q,
c ab
=
(c /a, c /b; q)∞ , (c , c /ab; q)∞
(1.7)
c < 1. ab
(1.8)
Sears’ 3 φ2 transformation formula
(b2 /a3 , b1 b2 /a1 a2 ; q)∞ a1 , a2 , a3 b1 b2 b1 /a1 , b1 /a2 , a3 b2 = , φ ; q , φ ; q , 3 2 3 2 b1 , b2 a1 a2 a3 (b2 , b1 b2 /a1 a2 a3 ; q)∞ b1 , b1 b2 /a1 a2 a3 b2 b1 b2 < 1. < 1, a a a a 1 2 3
(1.9)
3
The bilateral basic hypergeometric series r ψs is defined by
r ψs
∞ n (a1 , a2 , . . . , ar ; q)n ; q, z = (−1)(s−r )n q(s−r )(2) zn . b1 , b2 , . . . , b s (b1 , b2 , . . . , b s ; q)n n=−∞ a 1 , a 2 , . . . , ar
(1.10)
The following is the well-known Ramanujan’s 1 ψ1 summation formula ∞ (a; q)n n (q, b/a, az, q/az; q)∞ z = , (b; q)n (b, q/a, z, b/az; q)∞ n=−∞
|b/a| < | z| < 1.
(1.11)
F.H. Jackson defined the q-integral by [10]
d f (t ) dq t = d(1 − q)
∞
f dqn qn ,
(1.12)
n =0
0
and
d
d f (t ) dq t =
c
c f (t ) dq t −
0
f (t ) dq t .
(1.13)
0
He also defined an integral on (0, ∞) by
∞
∞
f (t ) dq t = (1 − q) 0
f qn qn .
(1.14)
n=−∞
On the interval (−∞, ∞) the bilateral q-integral is defined by
∞ f (t ) dq t = (1 − q) −∞
∞
n
f q
n=−∞
+ f −qn qn .
(1.15)
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
463
The q-integral is important in the theory and applications of basic hypergeometric series. For example, the present author gives some applications of the q-integral [14–18,20]. These papers can help the reader to understand the main idea of the present paper. The following is the Andrews–Askey integral [2], which can be derived from Ramanujan’s 1 ψ1 summation:
d c
(qt /c , qt /d; q)∞ d(1 − q)(q, dq/c , c /d, abcd; q)∞ dq t = , (at , bt ; q)∞ (ac , ad, bc , bd; q)∞
(1.16)
provided that no zero factors occur in the denominator of the integrals. Al-Salam and Verma gave an extension of the Andrews–Askey integral, which is called the Al-Salam and Verma [1] q-integral
d c
(qt /c , qt /d, et ; q)∞ d(1 − q)(q, dq/c , c /d, e /a, e /b, e / f ; q)∞ dq t = , (at , bt , f t ; q)∞ (ac , ad, bc , bd, f c , f d; q)∞
(1.17)
provided that no zero factors occur in the denominator of the integrals, where e = abcdf . The main result of this paper is the following theorem: Theorem 1. Let ξ denote random variable having distribution W (x; q), −1 < x < 0, then we have
E
(cxξ ; q)∞ (aξ, bξ, c ξ ; q)∞
=
(acx, bcx, x; q)∞ ab, c , cx ; q, x , 3 φ2 (a, ax, b, bx, c , cx; q)∞ acx, bcx
(1.18)
provided that max{|a|, |b|, |c |} < 1, where E( X ) denotes expected value of the random variable X . 2. The proof of Theorem 1 In this section, we use the Al-Salam and Verma [1] q-integral to give the proof of Theorem 1. We also need to use the following well-known theorems: Analytic continuation theorem: If f and g are analytic at z0 and agree at infinitely many points which include z0 as an accumulation point, then f = g. Lebesgue’s dominated convergence theorem: Suppose that { X n , n 1} is a sequence of random variables, that X n → X pointwise almost everywhere as n → ∞, and that | X n | Y for all n, where random variable Y is integrable. Then X is integrable, and
lim E X n = E X .
(2.1)
n→∞
The analytic continuation theorem is useful in the study of basic hypergeometric functions. Two well-known examples are, Ismail’s proof of the Ramanujan’s 1 ψ1 sum [9] and the Askey–Ismail proof of Bailey’s sum of the very well poised 6 ψ6 [5]. Now we give the proof. Proof. Considering the following sequence of random variables (on a probability space):
ηn =
n (abcxξ ; q)∞ (ab, c ξ ; q)k xk , (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k k =0
where n = 0, 1, 2, . . . . Since,
n (abcxξ ; q) (ab, c ξ ; q)k xk ∞ |ηn | = (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k k =0 n (abcxqk ξ ; q) (ab; q)k xk ∞ = · (q; q)k (aξ, bξ, cqk ξ ; q)∞ k =0 n (−|abcx|; q)∞ (ab; q)k xk (q; q) (|a|, |b|, |c |; q)∞ k k =0
and the series
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M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
∞ (ab; q)k xk
(q; q)k
k =0
converges absolutely. Using Lebesgue’s dominated convergence theorem obtains:
lim Eηn = E
n→∞
lim
n→∞
ηn .
(2.2)
Using the q-Gauss theorem (1.8), we have
lim
n→∞
ηn =
∞ (abcxξ ; q)∞ (ab, c ξ ; q)k xk (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k k =0
=
(abcxξ ; q)∞ (cxξ, abx; q)∞ · (aξ, bξ, c ξ ; q)∞ (abcxξ, x; q)∞
=
(abx; q)∞ (cxξ ; q)∞ · . (x; q)∞ (aξ, bξ, c ξ ; q)∞
(2.3)
Hence, we get the right-hand side of (2.2):
E
lim
n→∞
ηn =
(abx; q)∞ (cxξ ; q)∞ . E (x; q)∞ (aξ, bξ, c ξ ; q)∞
(2.4)
Now we begin to calculate the left-hand side of (2.2). Observe that
n (abcxξ ; q)∞ (ab, c ξ ; q)k xk Eηn = E (aξ, bξ, c ξ ; q)∞ (q, abcxξ ; q)k
=
k =0
n (abcxqk ξ ; q)∞ (ab; q)k xk E . (q; q)k (aξ, bξ, cqk ξ ; q)∞
(2.5)
k =0
Employing the Al-Salam and Verma q-integral (1.17) gives
E
(abcxqk ξ ; q)∞ (aξ, bξ, cqk ξ ; q)∞
1 ∞ (−x)n (xn−1 qm+1 , xn qm+1 , abcxn+1 qk+m ; q)∞ qm (q, q/x, x, axn qm , bxn qm , cqk+m xn ; q)∞ n =0 m =0 ∞ 1 (qm+1 /x, qm+1 , abcxqk+m ; q)∞ qm = (1 − q) (1 − q)(q, q/x, x; q)∞ (aqm , bqm , cqk+m ; q)∞ m =0 ∞ (qm+1 , xqm+1 , abcx2 qk+m ; q)∞ qm − x(1 − q) (axqm , bxqm , cqk+m x; q)∞
=
m =0
=
1
1
(1 − q)(q, q/x, x; q)∞
x
(qt /x, qt , abcxqk t ; q)∞ dq t (at , bt , cqk t ; q)∞
(abx, acxqk , bcxqk ; q)∞ = . (a, ax, b, bx, cqk , cxqk ; q)∞
(2.6)
Substituting (2.6) into (2.5) yields
Eηn =
n n (abcxqk ξ ; q)∞ (abx, acx, bcx; q)∞ (ab, c , cx; q)k xk (ab; q)k xk E . = (q; q)k (a, ax, b, bx, c , cx; q)∞ (q, acx, bcx; q)k (aξ, bξ, cqk ξ ; q)∞ k =0
(2.7)
k =0
Hence, we get the left-hand side of (2.2)
lim Eηn =
n→∞
(abx, acx, bcx; q)∞ ab, c , cx . φ ; q , x 3 2 (a, ax, b, bx, c , cx; q)∞ acx, bcx
Substituting (2.4) and (2.8) into (2.2) yields (1.18).
2
(2.8)
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
465
3. Some applications The expectation formula (1.18) can be used to derive identities about the basic hypergeometric functions. In this section, we show some applications of Theorem 1. The following transformation formula can be derived directly from Theorem 1 and q-binomial theorem. Theorem 2. Let |a| < 1, |b| < 1, |x| < 1 and | z| < 1, then ∞ (a, ax; q)n zn n =0
(q, abx; q)n
3 φ2
(az, azx; q)∞ b, bx ; q , x φ ; q , x = . 2 1 ( z, abzx; q)∞ abxqk , abz abx
a2 zqk , b, bx
(3.1)
Proof. Rewrite q-binomial theorem (1.7) as ∞ n =0
zn
1
1
1
· = · . (q; q)n (aqn , az; q)∞ ( z; q)∞ (a; q)∞
(3.2)
Letting a = aξ in (3.2) gives ∞ n =0
zn
1
1
1
· = · , (q; q)n (aqn ξ, azξ ; q)∞ ( z; q)∞ (aξ ; q)∞
(3.3)
where ξ denotes random variable having distribution W (x; q) and −1 < x < 0. Multiplying both sides of (3.3) by
(bxξ ; q)∞ (bξ ; q)∞ and then applying the expectation operator E on both sides of (3.3), it follows that
E
∞ n =0
(bxξ ; q)∞ · (q; q)n (aqn ξ, azξ, bξ ; q)∞ zn
=
1
( z; q)∞
E
(bxξ ; q)∞ . (aξ, bξ ; q)∞
(3.4)
Since,
n z (bxξ ; q)∞ | z|n (−|b|; q)∞ · · (q; q) (aqn ξ, azξ, bξ ; q) (|a|, |az|, |b|; q) n ∞ ∞ (q; q)n and the series ∞ | z|n (q; q)n
m =0
converges. Using Lebesgue’s dominated convergence theorem yields
E
∞ n =0
(bxξ ; q)∞ · n (q; q)n (aq ξ, azξ, bξ ; q)∞ zn
=
∞ n =0
zn
(q; q)n
E
(bxξ ; q)∞ . (aqn ξ, azξ, bξ ; q)∞
(3.5)
Hence, ∞ n =0
zn
(q; q)n
E
(bxξ ; q)∞ (aqn ξ, azξ, bξ ; q)∞
=
1
( z; q)∞
E
(bxξ ; q)∞ . (aξ, bξ ; q)∞
(3.6)
Employing (1.18) gives
(bxξ ; q)∞ E (aqn ξ, azξ, bξ ; q)∞
2 n (abxqn , abzx, x; q)∞ a zq , b, bx = ; q, x 3 φ2 (aqn , axqn , az, azx, b, bx; q)∞ abxqn , abz
(3.7)
and
E
(bxξ ; q)∞ (aξ, bξ ; q)∞
=
(abx, x; q)∞ b, bx . φ ; q , x 2 1 (a, ax, b, bx; q)∞ abx
(3.8)
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M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
Substituting (3.7) and (3.8) into (3.6) yields ∞ (a, ax; q)n zn
(q, abx; q)n
n =0
3 φ2
(az, azx; q)∞ b, bx ; q , x φ ; q , x = , 2 1 ( z, abzx; q)∞ abxqn , abz abx
a2 zqn , b, bx
(3.9)
where −1 < x < 0. By analytic continuation, we may replace the assumption −1 < x < 0 by |x| < 1. Thus, we get (3.1).
2
From the theorem, we can easily get the summation formula involving 3 φ2 : Corollary 1. Let |a| < |b| < 1 and | z| < 1, then ∞ (a, a2 /b; q)n zn
(q, a2 ; q)n
n =0
3 φ2
a, b, a2 zqn abz, a2 qn
; q,
a
b
=
(az, a2 z/b, a, a2 /b; q)∞ . ( z, a2 z, a2 , a/b; q)∞
=
(az, a2 z/b; q)∞ a, b a φ ; q , 2 1 b a2 ( z, a2 z; q)∞
=
(az, a2 z/b, a, a2 /b; q)∞ , ( z, a2 z, a2 , a/b; q)∞
(3.10)
Proof. Letting x = a/b in (3.1) gives ∞ (a, a2 /b; q)n zn
(q, a2 ; q)n
n =0
3 φ2
a, b, a2 zqn abz, a2 qn
; q,
a b
(3.11)
2
as desired.
Sears’ 3 φ2 transformation formula has been used by Andrews [3,12] in proving many of Ramanujan’s identities for partial theta functions. Using the expectation formula (1.18), we can easily derive the following expansion of Sears’ 3 φ2 transformation formula. Theorem 3. Let |a| < |c | < |b| < 1, |d| < 1, |x| < 1 and | z| < 1, then ∞ (b, a, ax; q)n zn
(q, c , adx; q)n
n =0
=
3 φ2
a2 bzqn /c , d, dx adxqn , abdzx/c
; q, x
2 n ∞ (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n a bzq , d, dx . φ ; q , x 3 2 ( z, c ; q)∞ (q, bz, abdzx/c ; q)n abdzxqn /c , adx
(3.12)
n =0
Proof. By the following Rogers’ iteration of Heine’s transformation formula [4,8]
2 φ1
(c /b, bz; q)∞ b, abz/c c , ; q, z = ; q, 2 φ1 ( z, c ; q)∞ b c bz
a, b
(3.13)
which can be rewritten as ∞ (b; q)n zn n =0
·
1
(q, c ; q)n (aqn , abz/c ; q)∞
=
∞ (c /b, bz; q)∞ (b; q)n (c /b)n 1 · . ( z, c ; q)∞ (q, bz; q)n (a, abzqn /c ; q)∞
(3.14)
n =0
Letting a = aξ in (3.14) and multiplying both sides of (3.14) by
(dxξ ; q)∞ (dξ ; q)∞ yields, ∞ (b; q)n zn n =0
(q, c ; q)n
·
∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n (dxξ ; q)∞ = · , (aqn ξ, abzξ/c , dξ ; q)∞ ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞
(3.15)
n =0
where ξ denotes random variable having distribution W (x; q) and −1 < x < 0. Applying the expectation operator E on both sides of (3.15), it follows that
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
E
∞ (b; q)n zn
467
(dxξ ; q)∞ · (q, c ; q)n (aqn ξ, abzξ/c , dξ ; q)∞ n =0 ∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n =E · . ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞
(3.16)
n =0
Since,
(b; q)n zn (|a|, b; q)n zn (−|d|; q)∞ (dxξ ; q)∞ · (q, c ; q) (aqn ξ, abzξ/c , dξ ; q) (|a|, |abz/c |, |d|; q) (q, c ; q) n ∞ ∞ n and the series
∞ (|a|, b; q)n zn (q, c ; q) n
m =0
is convergent. Using Lebesgue dominated convergence theorem, we obtain
E
∞ (b; q)n zn n =0
(dxξ ; q)∞ · (q, c ; q)n (aqn ξ, abzξ/c , dξ ; q)∞
=
∞ (b; q)n zn n =0
(dxξ ; q)∞ E . (q, c ; q)n (aqn ξ, abzξ/c , dξ ; q)∞
(3.17)
Employing (1.18) gives
(dxξ ; q)∞ E (aqn ξ, abzξ/c , dξ ; q)∞
2 n (adxqn , abdzx/c , x; q)∞ a bzq /c , d, dx ; q, x . = 3 φ2 (aqn , axqn , abz/c , abzx/c , d, dx; q)∞ adxqn , abdzx/c
(3.18)
Substituting (3.18) into (3.17), then we get the left-hand side of (3.16)
E
∞ (b; q)n zn n =0
(dxξ ; q)∞ · n (q, c ; q)n (aq ξ, abzξ/c , dξ ; q)∞
2 n ∞ (b, a, ax; q)n zn a bzq /c , d, dx (adx, abdzx/c , x; q)∞ ; q, x . = 3 φ2 (a, ax, abz/c , abzx/c , d, dx; q)∞ (q, c , adx; q)n adxqn , abdzx/c
(3.19)
n =0
Similarly, we can obtain the right-hand side of (3.16)
∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n E · ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞
n =0
=
∞ (dxξ ; q)∞ (c /b, bz; q)∞ (b; q)n (c /b)n E ( z, c ; q)∞ (q, bz; q)n (aξ, abzqn ξ/c , dξ ; q)∞ n =0
=
∞ (adx, abdzx/c , x; q)∞ (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n (a, ax, abz/c , abzx/c , d, dx; q)∞ ( z, c ; q)∞ (q, bz, abdzx/c ; q)n n =0 2 n a bzq , d, dx × 3 φ2 ; q, x . abdzxqn /c , adx
(3.20)
Substituting (3.19) and (3.20) into (3.17), we have ∞ (b, a, ax; q)n zn n =0
(q, c , adx; q)n
3 φ2
a2 bzqn /c , d, dx adxqn , abdzx/c
; q, x
2 n ∞ a bzq , d, dx (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n = ; q, x , 3 φ2 abdzxqn /c , adx ( z, c ; q)∞ (q, bz, abdzx/c ; q)n
(3.21)
n =0
where −1 < x < 0. By analytic continuation, we may replace the assumption −1 < x < 0 by |x| < 1. Thus, we get (3.12).
2
468
M. Wang / J. Math. Anal. Appl. 379 (2011) 461–468
We point out that the two terms 3 φ2 transformation formula of Sears is a special case of (3.12). The two terms 3 φ2 transformation formula of Sears [13] can be stated as Corollary 2. We have
3 φ2
a1 , a2 , a3 b1 , b2
; q,
b1 b2 a1 a2 a3
=
(b2 /a3 , b1 b2 /a1 a2 ; q)∞ b1 /a1 , b1 /a2 , a3 b2 . ; q, 3 φ2 (b2 , b1 b2 /a1 a2 a3 ; q)∞ b1 , b1 b2 /a1 a2 a3
(3.22)
Proof. Letting d = 0 in (3.12) and using q-binomial theorem (1.7) gives ∞ (b, a, ax; q)n zn (a2 bzxqn /c ; q)∞ · (q, c ; q)n (x; q)∞ n =0
=
∞ (c /b, bz; q)∞ (b, abz/c , abzx/c ; q)n (c /b)n (a2 bzxqn ; q)∞ · , ( z, c ; q)∞ (q, bz; q)n (x; q)∞
(3.23)
n =0
which can be rewritten as
3 φ2
(c /b, bz; q)∞ b, abz/c , abzx/c c ; q , z φ ; q , = . 3 2 ( z, c ; q)∞ b c , a2 bzx/c bz, a2 bzx/c b, a, ax
(3.24)
After replacing (a, b, ax, c , z) by (a1 , a3 , a2 , b2 , b1 b2 /a1 a2 a3 ) in (3.23), we obtain the two terms 3 φ2 transformation formula of Sears (3.22). 2 Acknowledgments The author would like to express deep appreciation to the anonymous referee for the helpful suggestions. In particular, the author thanks the referee for help to improve the presentation of the paper.
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