ARCHIVUM MATHEMATICUM (BRNO)
Tomus 45 (2009), 47–58
SEVERAL q-SERIES IDENTITIES FROM THE EULER
1
EXPANSIONS OF (a; q)∞ AND (a;q)
∞
Zhizheng Zhang1,2 and Jizhen Yang1
Abstract. In this paper, we first give several operator identities which
extend the results of Chen and Liu, then make use of them to two q-series
1
identities obtained by the Euler expansions of (a; q)∞ and (a;q)
. Several
∞
q-series identities are obtained involving a q-series identity in Ramanujan’s
Lost Notebook.
1. Introduction
Throughout this paper, assume that 0 < q < 1 and adopt the customary notation
in [6] for q-series. Let
(a; q)∞ =
(1)
∞
Y
(1 − aq k ) .
k=0
For any integer n, the q-shifted factorial (a; q)n is given by
(a; q)n =
(2)
(a; q)∞
.
(aq n ; q)∞
The q-binomial coefficient is defined by
n
(q; q)n
.
(3)
=
k
(q; q)k (q; q)n−k
We will also use frequently the following equations:
(4)
−n
) (q a; q)∞
(a; q)∞
n+1
2
(q/a; q)n = (−a)−n q (
or
(5)
n+1
2
(q −n a; q)∞ = (−a)n q −(
) (q/a; q) (a; q) .
n
∞
For convenience, we employ the following notation for multiple q-shifted factorial:
(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n ,
where n is an integer or ∞.
2000 Mathematics Subject Classification: primary 05A30; secondary 33D15, 33D60.
Key words and phrases: exponential operator, operator identity, q-series identity.
Received March 16, 2008. Editor O. Došlý.
48
Z. ZHANG AND J. YANG
The q-differential operator Dq and the q-shifted operator are defined by
1
Dq {f (a)} = (f (a) − f (aq))
a
and
η{f (a)} = f (aq) ,
respectively. These two operators have been introduced in [9, 10, 11] due to Rogers,
who applied them for proving Roger-Ramanujan identities. Built on the Dq and η,
the operator appeared in the work of Roman [12] and will be denoted by
θ = η −1 Dq .
Two operators introduced in the papers [4] and [5] by Chen and Liu are the
exponential operators constructed from Dq and θ:
T (bDq ) =
∞
X
(bDq )n
(q; q)n
n=0
and
E(bθ) =
n
∞
X
(bθ)n q ( 2 )
.
(q; q)n
n=0
Then the following operator identities were obtained.
Theorem 1.1 (Chen and Liu, [4] and [5]). Let T (bDq ) and E(bθ) be respectively
defined as above. Then
n
o
1
1
T (bDq )
(6)
=
,
(at; q)∞
(at, bt; q)∞
o
n
(abst; q)∞
1
(7)
=
,
T (bDq )
(as, at; q)∞
(as, at, bs, bt; q)∞
(8)
E(bθ) (at; q)∞ = (at, bt; q)∞ ,
(9)
(as, at, bs, bt; q)∞
.
E(bθ) (as, at; q)∞ =
(abst/q; q)∞
By these operator identities, in [4, 5, 7, 8, 13], a lot of q-series identities can be derived. In [14], Theorem 1.1 was extended. Applying generalized operator identities to
some terminating basic hypergeometric series and q-integrals, we obtain some summation formulas involving 3 φ2 summation and some q-integrals involving 3 φ2 summation which extend some famous q-integrals, such as the Ismail-Stanton-Viennot
integral, Gasper integral formula.
In this paper, we first give several operator identities which extend the results
of Chen and Liu. Then making use of them to two q-series identities obtained by
1
the Euler expansions of (a; q)∞ and (a;q)
, several q-series identities are obtained
∞
involving a q-series identity in Ramanujan’s Lost Notebook.
Throughout this paper, n and k are two nonnegative integers.
SEVERAL q-SERIES IDENTITIES
49
2. Several operator identities
First of all, by induction we give a result which is to be used later.
Lemma 2.1. We have
(10)
(11)
(12)
(13)
(
0,
θn {ak } =
(−1)n q n (q −k ; q)n ak−n ,
(
0,
if
n
k
Dq {a } =
(q k−n+1 ; q)n ak−n ,
if
if
if
n > k,
n ≤ k,
n > k,
n ≤ k,
θn {a−k } = (−q)n (q k ; q)n a−(k+n) ,
n
Dqn {a−k } = q −( 2 )−kn (q k ; q)n a−(k+n) .
In [13], we obtained the following operator identity.
E(dθ) ak (at, as; q)∞
(14)
k (as, at, ds, dt; q)∞
=a
(adst/q)∞
3 φ2
q −k ,
q/as,
0,
q/at
; q, q
q 2 /adst
.
This identity can be rewritten by
(at, as, dt, ds; q)∞
E(dθ) ak (at, as; q)∞ = ak
(adst/q; q)∞
k
X
k (q/at, q/as; q)j (j+1
×
(−1)j
q 2 )−kj .
2 /adst; q)
(q
j
j
j=0
(15)
Next we give three operator identities which are extensions of Theorem 1.1.
They specialize to Theorem 1.1 by setting k = 0.
Theorem 2.2. We have
(16)
n
T (dDq )
k o
X
ak
(adst; q)∞
k (at, as; q)j d j
k
=a
.
(at, as; q)∞
(at, as, dt, ds; q)∞ j=0 j (adst; q)j a
E(dθ) a−k (at, as; q)∞
(17)
= a−k
n
∞
(at, as, dtq k , dsq k ; q)∞ X (q k , adstq k−1 ; q)n (n+1
d
2 )
q
−
(adstq k−1 ; q)∞
(q, dtq k , dsq k ; q)n
a
n=0
= a−k
∞
d n
(at, as, dt, ds; q)∞ X (q k ; q)n (adst/q; q)n+k (n+1
q 2 ) −
(adst/q; q)∞ n=0 (q; q)n (dt, ds; q)n+k
a
50
Z. ZHANG AND J. YANG
where |d/a| < 1, and
n
T (dDq )
o
a−k
(at, as; q)∞
= a−k
(18)
=
−
∞
X
(adtsq −k ; q)∞
(q k , q 1+k /adst; q)n
(−1)n
qn
−k
−k
(at, as, dtq , dsq ; q)∞ n=0
(q, q 1+k /dt, q 1+k /ds; q)n
∞
X
(adts; q)∞
1 k (k+1
(q k ; q)n (q/adst; q)n+k
q 2 )
(−q)n .
d
(at, as, dt, ds; q)∞ n=0 (q; q)n (q/dt, q/ds; q)n+k
Proof. We note that the rule of Leibniz for Dqn (see [4])
n
X
n
k(k−n) n
(19)
Dq {f (a)g(a)} =
q
Dk {f (a)}Dqn−k {g(aq k )} .
k q
k=0
Therefore, we have
∞
o X
o
n
ak
dn
ak
=
Dqn
(at, as; q)∞
(q; q)n
(at, as; q)∞
n=0
∞
n
o
X
dn X j(j−n) n j k n−j n
1
=
q
Dq {a }Dq
(q; q)n j=0
j
(atq j , asq j ; q)∞
n=0
n
∞
o
X
dn X j(j−n) n (q; q)k k−j n−j n
1
q
=
a Dq
(q; q)n j=0
j (q; q)k−j
(atq j , asq j ; q)∞
n=0
∞
∞ X
o
X
dn j(j−n) n (q; q)k k−j n−j n
1
=
q
a Dq
(q; q)n
(atq j , asq j ; q)∞
j (q; q)k−j
j=0 n=j
n
T (dDq )
=
∞
X
j=0
∞
o
X
dn −nj n n
1
dj (q; q)k
ak−j
q
Dq
(q; q)j (q; q)k−j
(q; q)n
(atq j , asq j ; q)∞
n=0
k j X
∞
o
X
(dq −j Dq )n n
1
k d
=a
j a n=0 (q; q)n
(atq j , asq j ; q)∞
j=0
k
= ak
k j
X
k d
j=0
= ak
a
k j
X
k d
j=0
= ak
j
j
a
· T (dq −j Dq
n
o
1
(atq j , asq j ; q)∞
(adstq j ; q)∞
j
(atq , asq j , dt, ds; q)∞
k X
k (at, as; q)j d j
(adst; q)∞
.
(at, as, dt, ds; q)∞ j=0 j (adst; q)j a
We obtain (16). (17) and (18) can be proved similarly. The proof is completed. Taking s = 0 in Theorem 2.2, we have
SEVERAL q-SERIES IDENTITIES
51
Corollary 2.3.
(20)
n
T (dDq )
k d j
X
1
ak o
k
k
=a
(at; q)j
.
(at; q)∞
(at, dt; q)∞ j=0 j
a
E(dθ) a−k (at; q)∞ = a−k (at, dtq k ; q)∞
×
(21)
∞
d n
X
n+1
1
(q k ; q)n
q( 2 ) −
(q; q)n (dt; q)n+k
a
n=0
where |d/a| < 1, and
n
T (dDq )
(22)
a−k o 1
1 k (k+1
q 2 )
= −
(at; q)∞
adt
(at, dt; q)∞
∞
k
X
(q ; q)n
1
q n
,
×
−
(q; q)n (q/dt; q)n+k
at
n=0
where |q/at| < 1.
Further, setting k = 1 or t = 0 in (21), we get the following results of Chen and
Liu.
Corollary 2.4 (Chen and Liu, [5]).
(23)
∞
X
n+1
E(bθ) a−1 = a−1
(−1)n q ( 2 ) bn a−n ,
n=0
(24)
∞
X
m+1
E(bθ) a−1 (−a; q)∞ = a−1 (−a; q)∞
(−b/a)m q ( 2 ) (−bq m+1 ; q)∞ .
m=0
3. Several q-series identities from expansion of (a; q)∞
From the Euler expansion of (a; q)∞ (see [3]):
n
∞
X
(−1)n q ( 2 ) an
= (a; q)∞ ,
(q; q)n
n=0
the following identity can be verified.
(25)
∞
X
n=0
(−aq n+1 ; q)∞ q n = −a−1 + a−1 (−a; q)∞ .
52
Z. ZHANG AND J. YANG
Theorem 3.1. We have
∞
k+1
X
X (q −(k+1) , −q −n /a, q/at; q)j
(−adt; q)n
qn
qj
1−n /adt; q)
(−aq,
−dq;
q)
(q,
−q
n
j
n=0
j=0
=−a
−1
k
(−adt; q)∞ X (q −k , q/at; q)j
(dt)j
(−aq, −dq; q)∞ j=0
(q; q)j
k
+ a−1
(26)
(1 + a)(1 + d) X (q −k , −q/a, q/at; q)j j
q .
1 + adt/q j=0 (q, −q 2 /adt; q)j
Proof. Multiply both sides in equation (25) by ak+1 (at; q)∞ . Then
(27)
∞
X
ak+1 (−aq n+1 , at; q)∞ q n = −ak (at; q)∞ + ak (−a, at; q)∞ .
n=0
Apply the operator E(dθ) to its both sides with respect to the variable a. Then
∞
X
q n E(dθ) ak+1 (−aq n+1 , at; q)∞ = −E(dθ) ak (at; q)∞
n=0
+ E(dθ) ak (−a, at; q)∞ .
(28)
Applying Theorem 2.2 and Corollary 2.3, we have
(−aq n+1 , at, −dq n+1 , dt; q)∞
E(dθ) ak+1 (−aq n+1 , at; q)∞ = ak+1
(−adtq n ; q)∞
×
k+1
X
j=0
(q −(k+1) , −q −n /a, q/at; q)j j
q ,
(q, −q 1−n /adt; q)j
k
X
(q −k , q/at; q)j
E(dθ) ak (at; q)∞ = ak (at, dt; q)∞
(dt)j ,
(q;
q)
j
j=0
k
(−a, at, −d, dt; q)∞ X (q −k , −q/a, q/at; q)j j
E(dθ) ak (−a, at; q)∞ = ak
q .
(−adt/q; q)∞
(q, −q 2 /adt; q)j
j=0
Substituting these three identities into (27) and then using (4), we obtain (3.1)
after simplifying. This completes the proof.
Taking k = 0 in Theorem 3.1, we have
Corollary 3.2.
∞
∞
X
X
(−adt; q)n
(−adt; q)n
a
q n + d(1 − at/q)
q n+1
(−aq,
−dq;
q)
(−aq;
q)
(−dq;
q)
n
n
n+1
n=0
n=0
(29)
=−
(−adt; q)∞
+ 1 + a.
(−aq, −dq; q)∞
Taking t = 0 in Corollary 3.2 we have
SEVERAL q-SERIES IDENTITIES
Corollary 3.3.
a
∞
X
∞
X
1
1
qn + d
q n+1
(−aq,
−dq;
q)
(−aq;
q)
(−dq;
q)
n
n
n+1
n=0
n=0
=−
(30)
1
+ 1 + a.
(−aq, −dq; q)∞
Taking k = 1 in Theorem 3.1, we have
Corollary 3.4.
a2 q
∞
X
∞
X
(−adt; q)n
(−adt; q)n
q n − ad(1 + q)(at − q)
qn
(−aq,
−dq;
q)
(−aq;
q)
(−dq;
q)
n
n
n+1
n=0
n=0
+ d2 (at − q)(at − q 2 )
(31)
=
∞
X
(−adt; q)n
qn
(−aq;
q)
(−dq;
q)
n
n+2
n=0
(1 + a)(aq + dq + adq 2 − adt)
(−adt; q)∞
− (aq + dq − adt)
.
1 + dq
(−aq, −dq; q)∞
Taking t = 0 in Corollary 3.4, we have
Corollary 3.5.
a2
∞
X
1
1
q n + ad(1 + q)
qn
(−aq,
−dq;
q)
(−aq;
q)
(−dq;
q)
n
n
n+1
n=0
n=0
∞
X
+ d2 q 2
(32)
∞
X
1
qn
(−aq;
q)
(−dq;
q)
n
n+2
n=0
1
(1 + a)(a + d + adq)
− (a + d)
.
1 + dq
(−aq, −dq; q)∞
=
Theorem 3.6. We have
(1 − dtq k )
∞
X
∞
d i
X
(−adtq k ; q)n
(q k , −adtq n+k ; q)i
n
(i+1
2 )
q
q
−
(−aq, −dq k+1 ; q)n i=0 (q, −dq n+k+1 , dtq k ; q)i
a
n=0
= − a−1
∞
d i
X
(−adtq k ; q)∞
(q k+1 ; q)i
(i+1
2 )
q
−
(−aq, −dq k+1 ; q)∞ i=0 (q, dtq k+1 ; q)i
a
−1
+ (1 + a
(33)
)
∞
X
i=0
d i
(q k+1 , −adtq k ; q)i
(i+1
2 )
q
−
,
(q, −dq k+1 , dtq k+1 ; q)i
a
where |d/a| < 1.
Proof. (25) can be rewritten by
(34)
∞
X
n=0
q n a−k (−aq n+1 , at; q)∞ = −a−(k+1) (at; q)∞ + a−(k+1) (−a, at; q)∞
53
54
Z. ZHANG AND J. YANG
Apply the operator E(dθ) to its both sides with respect to the variable a. Then
∞
X
q n E(dθ) a−k (−aq n+1 , at; q)∞ = −E(dθ) a−(k+1) (at; q)∞
n=0
+ E(dθ) a−(k+1) (−a, at; q)∞ .
By Theorem 2.2 and Corollary 2.3, we have
(−aq n+1 , at, −dq n+k+1 , dtq k ; q)∞
E(dθ) a−k (−aq n+1 , at; q)∞ = a−k
(−adtq n+k ; q)∞
i
∞
k
n+k
X (q , −adtq
; q)i
d
(i+1
2 )
,
×
q
−
(q, −dq n+k+1 , dtq k ; q)i
a
i=0
E(dθ) a−(k+1) (at; q)∞ = a−(k+1) (at, dtq k+1 ; q)∞
i
∞
X
(q k+1 ; q)i
d
(i+1
2 )
×
q
,
−
(q, dtq k+1 ; q)i
a
i=0
(−a, at, −dq k+1 , dtq k+1 ; q)∞
E(dθ) a−(k+1) (−a, at; q)∞ = a−(k+1)
(−adtq k ; q)∞
k
d i
X
(q k+1 , −adtq k ; q)i
(i+1
2 )
×
.
q
−
(q, −dq k+1 , dtq k+1 ; q)i
a
i=0
Substituting these three identities into (34) and then using (4), we obtain the
theorem.
Taking k = 0 in Theorem 3.6 we have
Corollary 3.7.
(1 − dt)
∞
d i
i+1
1
(−adt; q)n
(−adt; q)∞ X
q n = − a−1
q( 2 ) −
(−aq, −dq; q)n
(−aq, −dq; q)∞ i=0 (dtq; q)i
a
n=0
∞
X
+ (1 + a−1 )
(35)
∞
d i
X
(−adt; q)i (i+1
q 2) −
,
(−dq, dtq; q)i
a
i=0
where |d/a| < 1.
Taking t = 0 in Corollary 3.7 we obtain the following Ramanujan’s identity.
Corollary 3.8.
∞
X
∞
X
n+1
1
1
q n = − a−1
(−1)n q ( 2 ) (d/a)n
(−aq, −dq; q)n
(−aq, −dq; q)∞ n=0
n=0
(36)
where |d/a| < 1.
−1
+ (1 + a
n+1
∞
X
(−1)n q ( 2 ) (d/a)n
,
)
(−dq; q)n
n=0
SEVERAL q-SERIES IDENTITIES
55
Note. This identity is a formula in Ramanujan’s Lost Notebook, and its proofs
are given by Andrews [1, 2]. In [5], Chen and Liu gave also a simple proof by using
the method of the operator identity.
4. Several q-series identities from expansion of
From the Euler expansion of
1
(a;q)∞
1
(a;q)∞
(see [3]):
∞
X
1
1
an =
,
(q;
q)
(a;
q)∞
n
n=0
the following identity can be verified.
∞
X
1
1
q n = a−1
− a−1 .
n ; q)
(aq
(a;
q)
∞
∞
n=0
(37)
From this identity, by the operator T (dDq we can obtain the following results.
Theorem 4.1. We have
k ∞
k+1
X
X
X k + 1
k (a, at; q)j d j
(a; q)n+j d j
= a−1
q n (d; q)n
(at; q)j
j (adt; q)j a
j
(adt; q)n+j a
n=0
j=0
j=0
(38)
− a−1
j
k (a, d; q)∞ X k
d
(at; q)j
.
(adt; q)∞ j=0 j
a
Proof. By (37), we have
∞
X
ak+1
ak
ak
n
q
=
−
.
(aq n , at; q)∞
(a, at; q)∞
(at; q)∞
n=0
(39)
Apply the operator T (dDq ) to its both sides with respect to the variable a. Then
∞
o
n
o
n
X
ak
ak+1
=
T
(dD
)
q n · T (dDq )
q
(aq n , at; q)∞
(a, at; q)∞
n=0
n ak o
(40)
− T (dDq )
.
(at; q)∞
By Theorem 2.2 and Corollary 2.3, we have
n
o
ak+1
(adtq n ; q)∞
k+1
=
a
T (dDq )
(aq n , at; q)∞
(aq n , at, dq n , dt; q)∞
k+1
X k + 1 (aq n , at; q)j d j
(41)
×
,
j
(adtq n ; q)j a
j=0
(42)
T (dDq )
n
k o
X
(adt; q)∞
ak
k (a, at; q)j d j
= ak
(a, at; q)∞
(a, at, d, dt; q)∞ j=0 j (adt; q)j a
56
Z. ZHANG AND J. YANG
and
T (dDq )
(43)
n
k d j
X
ak o
1
k
.
= ak
(at; q)j
(at; q)∞
(at, dt; q)∞ j=0 j
a
Substituting these three identities into (40) and then using (4), we obtain the
theorem.
Taking k = 0 in Theorem 4.1 we have
Corollary 4.2.
∞
∞
X
X
(a, d; q)n n
(a; q)n+1 n
(a, d; q)∞
(44)
a
(d; q)n
q + d(1 − at)
q =1−
.
(adt;
q)
(adt;
q)
(adt; q)∞
n
n+1
n=0
n=0
Specially, taking t = 0 we have
Corollary 4.3.
∞
∞
X
X
(d; q)n (a; q)n+1 q n = 1 − (a, d; q)∞ .
(a, d; q)n q n + d
(45)
a
n=0
n=0
Taking k = 1 in Theorem 4.1 we have
Corollary 4.4.
∞
∞
X
X
(a, d; q)n n
(a; q)n+1 (d; q)n n
a2
q + ad(1 + q)(1 − at)
q
(adt; q)n
(adt; q)n+1
n=0
n=0
+ d2 (1 − at)(1 − atq)
= a+d
(46)
∞
X
(a; q)n+2 (d; q)n n
q
(adt; q)n+2
n=0
(1 − a)(1 − at) (a, d; q)∞
−
(a + d − adt) .
1 − adt
(adt; q)∞
Specially, taking t = 0 we have
Corollary 4.5.
∞
∞
∞
X
X
X
a2
(a, d; q)n q n + ad(1 + q)
(a; q)n+1 (d; q)n q n + d2
(a; q)n+2 (d; q)n q n
n=0
(47)
n=0
n=0
= (a + d)(1 − (a, d; q)∞ ) − ad .
Theorem 4.6. We have
∞
∞
X
(a, d; q)n X (q k ; q)j (q 1−n /adt; q)k+j
(−q)j
qn
1−n /d; q)
(adt;
q)
(q;
q)
(q/dt,
q
n
j
k+j
n=0
j=0
=−
(48)
+
∞
q k+1 X (q k+1 ; q)j (q/adt; q)k+1+j
(−q)j
d j=0 (q; q)j (q/d, q/dt; q)k+1+j
∞
1 (a, d; q)∞ X (q k+1 ; q)j
(−1)j
(q/at)k+1+j ,
d (adt; q)∞ j=0 (q; q)j (q/dt; q)k+1+j
SEVERAL q-SERIES IDENTITIES
57
where |q/at| < 1.
Proof. We rewrite (37) into the following form:
∞
X
a−(k+1)
a−k
a−(k+1)
n
q
−
.
=
(aq n , at; q)∞
(a, at; q)∞
(at; q)∞
n=0
(49)
Apply the operator T (dDq ) to its both sides with respect to the variable a. Then
∞
X
q n · T (dDq )
n=0
(50)
o
n a−(k+1) o
a−k
=
T
(dD
)
q
(aq n , at; q)∞
(a, at; q)∞
n a−(k+1) o
.
− T (dDq )
(at; q)∞
n
By Theorem 2.2 and Corollary 2.3, we have
n
o
(adtq n ; q)∞
a−k
−k (k+1
2 )
T (dDq )
=
(−d)
q
(aq n , at; q)∞
(aq n , at, dq n , dt; q)∞
∞
X (q k ; q)j (q 1−n /adt; q)k+j
j
(51)
(−q) ,
×
1−n /d; q)
(q;
q)
(q/dt,
q
j
k+j
j=0
(52)
n a−(k+1) o
(adt; q)∞
−(k+1) (k+2
T (dDq )
= (−d)
q 2 )
(a, at; q)∞
(a, at, d, dt; q)∞
∞
X
(q k+1 ; q)j (q/adt; q)k+j+1
j
×
(−q)
(q;
q)
(q/d,
q/dt;
q)
j
k+j+1
j=0
and
(53)
n a−(k+1) o 1 −(k+1) (k+2
1
T (dDq )
= −
q 2 )
(at; q)∞
adt
(at, dt; q)∞
∞
k+1
X
(q
; q)j
1
q j
×
−
.
(q; q)j (q/dt; q)k+j+1
at
j=0
Substituting these three identities into (50) and then using (4), we obtain the
theorem.
Taking k = 0 in Theorem 4.6 we have
Corollary 4.7.
∞
X
∞
qn
n=0
(54)
where |q/at| < 1.
(a, d; q)n
q X (q/adt; q)j+1
=−
(−q)j
(adt; q)n
d j=0 (q/d, q/dt; q)j+1
+
∞
1 (a, d; q)∞ X
(−1)j q j+1
,
d (adt; q)∞ j=0 (q/dt; q)j+1 at
58
Z. ZHANG AND J. YANG
Acknowledgement. This research is supported by the National Natural Science
Foundation of China (Grant No. 10771093).
References
[1] Andrews, G. E., Ramanujan’s “Lost” Notebook I, Partial θ-function, Adv. Math. 41 (1981),
137–172.
[2] Andrews, G. E., Ramanujan: Essays and Surveys, ch. An introduction to Ramanujan’s “Lost”
Notebook, pp. 165–184, 2001.
[3] Andrews, G. E., Askey, R., Roy, R., Special Function, Cambridge University Press, Cambridge,
1999.
[4] Chen, W. Y. C., Liu, Z.-G., Parameter augmentation for basic hypergeometric series, II, J.
Combin. Theory Ser. A 80 (1997), 175–195.
[5] Chen, W. Y. C., Liu, Z.-G., Mathematical essays in honor of Gian-Carlo Rota, ch. Parameter
augmentation for basic hypergeometric series, I, pp. 111–129, Birkhäuser, Basel, 1998.
[6] Gasper, G. G., Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics
and Its Applications, Second edition, vol. 96, Cambridge University Press, 2004.
[7] Liu, Z.-G., A new proof of the Nassrallah-Rahman integral, Acta Math. Sinica 41 (2) (1998),
405–410.
[8] Liu, Z.-G., Some operator identities and q-series transformation formulas, Discrete Math.
265 (2003), 119–139.
[9] Rogers, L. J., On the expansion of some infinite products, Proc. London Math. Soc. 24
(1893), 337–352.
[10] Rogers, L. J., Second Memoir on the expansion of certain infinite products, Proc. London
Math. Soc. 25 (1894), 318–343.
[11] Rogers, L. J., Third Memoir on the expansion of certain infinite products, Proc. London
Math. Soc. 26 (1896), 15–32.
[12] Roman, S., More on the umbral calculus, with emphasis on the q-umbral calculus, J. Math.
Anal. Appl. 107 (1985), 222–254.
[13] Zhang, Z. Z., Liu, M. X., Applications of operator identities to the multiple q-binomial
theorem and q-Gauss summation theorem, Discrete Math. 306 (2006), 1424–1437.
[14] Zhang, Z. Z., Wang, J., Two operator identities and their applications to terminating basic
hypergeometric series and q-integrals, J. Math. Anal. Appl. 312 (2) (2005), 653–665.
1 Center
of Combinatorics and LPMC, Nankai University
Tianjin 300071, P. R. China
2 Department
of Mathematics, Luoyang Normal University
Luoyang 471022, P. R. China
E-mail: zhzhzhang-yang@163.com