q-Difference equation and the Cauchy operator identities

J. Math. Anal. Appl. 359 (2009) 265–274

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

q-Difference equation and the Cauchy operator identities Da-qian Lu East China Normal University, Department of Mathematics, Shanghai 200241, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 22 December 2008 Available online 27 May 2009 Submitted by B. Bongiorno

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers–Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu–Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins–Miller transformation. © 2009 Elsevier Inc. All rights reserved.

Keywords: q-Series Basic hypergeometric series q-Differential operator The Cauchy operator Multiple basic hypergeometric series

1. Introduction In an attempt to find efficient q-shift operators to deal with basic hypergeometric series identities in the framework of the q-umbral calculus [1,2,10], Chen and Liu [7,8] introduced two q-exponential operators, Fang [9] introduced a new q-exponential operator, Chen and Gu [6] introduced a Cauchy operator for deriving identities from their special cases. In this paper, motivated by their work, we study some applications of the Cauchy operator for basic hypergeometric series. Following [5] we will define the q-shifted factorial by

(a; q)0 = 1,

(a; q)n =

n −1





1 − aqk ,

(a; q)∞ =

k =0

∞  



1 − aqk ,

k =0

where a is a complex variable. And for convenience, we always assume 0 < q < 1 throughout the paper. For a complex number α , we define

  (a; q)α = (a; q)∞ / aqα ; q ∞ .

(1.1)

We also adopt the following compact notation

(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n . . . (am ; q)n ,

n = 0, 1, 2, . . . , ∞.

In this paper, we will frequently use the following property





−n

aq1−n /c ; q ∞ = (−a/c )n q( 2 ) (c /a; q)n (aq/c ; q)∞ ,

E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.05.048

©

2009 Elsevier Inc. All rights reserved.

n = 0, 1, 2, . . . , ∞.

(1.2)

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D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

The q-binomial coefficient and the q-binomial theorem are given by

  n k

=

(q; q)n (q; q)k (q; q)n−k

∞  (a; q)n

and

n =0

(q; q)n

xn =

(ax; q)∞ , (x; q)∞

|x| < 1,

(1.3)

respectively. Recall that the q-difference operator is defined by





f (a) − f (aq)

D q f (a) =

(1.4)

a

and the Leibniz rule for D q is referred to the following identity





D nq f (a) g (a) =

n 

qk(k−n)

k =0

 

    n D kq f (a) D nq −k g aqk . k

(1.5)

The following relations are easily verified. Proposition 1.1. Let k be a nonnegative integer. Then we have





1

D kq

tk

=

, (at ; q)∞ (at ; q)∞   n  D kq (at ; q)∞ = (−t )k q(2) atqk ; q ∞ ,

 (avqk ; q)∞ (av ; q)∞ D kq . = t k ( v /t ; q)k (at ; q)∞ (at ; q)∞ We recall that Chen and Gu [6] introduced the Cauchy operator

T(a, b; D q ) =

∞  (a; q)n n =0

(q; q)n

(bD q )n ,

(1.6)

as the basis of parameter augmentation which serves as a method for proving extensions of the Askey–Wilson integral, the Askey–Roy integral and so on. Liu [12] established two general q-exponential operator identities by solving two simple q-difference equations. Zhu [15] established the following q-exponential operator identity by solving a simple q-difference equation. Proposition 1.2. Let f (a, b, c ) be a three variables analytic function in a neighborhood of (a, b, c ) = (0, 0, 0) ∈ C 3 , satisfying the q-difference equation

(c − b) f (a, b, c ) = abf (a, bq, cq) − bf (a, b, cq) + (c − ab) f (a, bq, c ).

(1.7)

Then we have





f (a, b, c ) = T(a, b; D q ) f (a, 0, c ) .

(1.8)

Proof. We write (1.7) in the form









c f (a, b, c ) − f (a, bq, c ) = b f (a, b, c ) − f (a, b, cq) − af (a, bq, c ) + af (a, bq, cq) .

(1.9)

Now we begin to solve this q-difference equation. From the theory of several complex variables (see, for example, [14]), we may assume that

f (a, b, c ) =

∞ 

A n (a, c )bn

(1.10)

n =0

and then substitute the above equation into (1.9) to obtain

c

∞  



1 − qn A n (a, c )bn =

n =0

∞   n =0

Equating coefficients of b , we readily find that, for each integer n  1, n



A n (a, c ) − A n (a, cq) − aqn A n (a, c ) + aqn A n (a, cq) bn+1 .

D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

A n (a, c ) =

1 − aqn−1 1 − qn



267



D q,c A n−1 (a, c ) .

By iteration, we easily deduce that

A n (a, c ) =

(a; q)n n  D q,c A 0 (a, c ) . (q; q)n

(1.11)

It remains to calculate A 0 (a, c ). Putting b = 0 in (1.10), we immediately deduce that A 0 (a, c ) = f (a, 0, c ). Substituting (1.11) back into (1.10), we find that

f (a, b, c ) =

∞  (a; q)n (bD q )n 

(q; q)n

n =0







f (a, 0, c ) = T(a, b; D q ) f (a, 0, c ) ,

which completes the proof of proposition.

2

If we take a = 0 and then substitute c with a in Proposition 1.2, it reduces to Theorem 1 of [12]. Proposition 1.2 tell us that if a analytic function f (a, b, c ) in three variables a, b and c satisfies q-difference equation (1.7), then we can recover f (a, b, c ) from its special case f (a, 0, c ). To get f (a, b, c ) we should use the Cauchy operator T(a, b; D q ) to act on f (a, 0, c ). In Section 2, we verify four operator identities. In Section 3, we use the operator identities to obtain a generating function for Rogers–Szegö polynomials for hn (x, y |q). And it can be stated in the equivalent forms in terms of the continuous big q-Hermite polynomial. In Section 4, applying the technique of parameter augmentation to two multiple generalizations of q-Chu–Vandermonde summation theorem given by Milne, we obtain two multiple generalizations of the Kalnins–Miller transformation which extend the results of Zhang [16]. 2. Cauchy operator identities In fact, Proposition 1.2 contain the following two operator identities as special cases. Theorem 2.1. We have

T(a, b; D q )



1

(ct ; q)∞

=

provided |bt | < 1.

T(a, b; D q )

1

(abt ; q)∞ , (bt , ct ; q)∞



(cs, ct ; q)∞

=



(abt ; q)∞ a, ct ; q, bs , 2 φ1 abt (bt , cs, ct ; q)∞

provided max{|bs|, |bt |} < 1. Proof. We first prove (2.1). Using the identity, (x; q)∞ = (1 − x)(xq; q)∞ , by direct calculation, we find that

f (a, b, c ) :=

(abt ; q)∞ (bt , ct ; q)∞

satisfies the functional equation

(c − b) f (a, b, c ) = abf (a, bq, cq) − bf (a, b, cq) + (c − ab) f (a, bq, c ). And the identity (1.8) becomes

 (abt ; q)∞ 1 = T(a, b; D q ) (bt , ct ; q)∞ (ct ; q)∞ which is (2.1). Similarly we can verify that

f (a, b, c ) :=



(abt ; q)∞ a, ct ; q, bs 2 φ1 abt (bt , cs, ct ; q)∞

satisfies the functional equation

(c − b) f (a, b, c ) = abf (a, bq, cq) − bf (a, b, cq) + (c − ab) f (a, bq, c ). And the identity (1.8) becomes

(2.1)

(2.2)

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D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

T(a, b; D q ) which is (2.2).



1



(abt ; q)∞ a, ct ; q, bs = 2 φ1 abt (bt , cs, ct ; q)∞

(cs, ct ; q)∞

2

We can verify the following operator identity by using (2.1) directly. Theorem 2.2. We have



(cv ; q)∞ (cv ; q)∞ a, v /t = , T(a, b; D q ) φ ; q , bt 2 1 cv (ct ; q)∞ (ct ; q)∞

(2.3)

provided |bt | < 1. Proof. Recall the operator identity in (2.1), namely

T(a, b; D q )



1

(ct ; q)∞

=

(abt ; q)∞ . (bt , ct ; q)∞

(2.4)

We now introduce the following linear transform



L t n = ( v /t ; q)n t n ,

n = 0, 1, 2, . . . , ∞.

By the q-binomial theorem, we find that

L

1



(ct ; q)∞

=

∞  n =0

=

(q; q)n

∞  n =0

=



cn

cn

(q; q)n

L tn

( v /t ; q)n t n

(cv ; q)∞ . (ct ; q)∞

Employing the same type argument as the above, we have

L

(abt ; q)∞ (bt , ct ; q)∞

 =

∞ (cv ; q)∞  (a, v /t ; q)n (bt )n . (ct ; q)∞ (q, cv ; q)n

(2.5)

n =0

Applying the operator L to both sides of (2.4) and then use the above two equations, we conclude that



(cv ; q)∞ (cv ; q)∞ a, v /t T(a, b; D q ) ; q, bt , = 2 φ1 cv (ct ; q)∞ (ct ; q)∞ which is (2.3). Thus we complete the proof of theorem.

2

By using (2.2), we can verify the following operator identity. Theorem 2.3.



(abt , cv ; q)∞ (cv ; q)∞ a, ct , v /s = T(a, b; D q ) ; q, bs , 3 φ2 abt , cv (cs, ct ; q)∞ (bt , ct , cs; q)∞

(2.6)

provided max{|bs|, |bt |} < 1. Proof. Recall the operator identity in (2.2), namely

T(a, b; D q )

1



(cs, ct ; q)∞

=



(abt ; q)∞ a, ct ; q, bs . 2 φ1 abt (bt , cs, ct ; q)∞

=

∞ n (abt ; q)∞   cn−k bk (a, ct ; q)k n s . (bt , ct ; q)∞ (q; q)n−k (q, abt ; q)k

It can be rewritten as

T(a, b; D q )

1

(cs, ct ; q)∞



n =0 k =0

(2.7)

D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

269

We now introduce the following linear transform



L sn = ( v /s; q)n sn ,

n = 0, 1, 2, . . . , ∞.

By the q-binomial theorem, we find that

L



1

∞ 

=

(cs; q)∞

n =0

cn

(q; q)n

∞ 

=

n =0

cn

(q; q)n



L sn

( v /s; q)n sn

(cv ; q)∞ . (cs; q)∞

=

Applying the operator L to both sides of (2.7) and then use the above equation, we have

 ∞ n (cv ; q)∞ (abt ; q)∞   cn−k bk (a, ct ; q)k T(a, b; D q ) ( v /s; q)n sn = (cs, ct ; q)∞ (bt , ct ; q)∞ (q; q)n−k (q, abt ; q)k n =0 k =0

n ∞  (a, ct , v /s; q)k (bs)k  ( vqk /s; q)n−k (cs)n−k = (q, abt ; q)k (q; q)n−k n =0

k =0

=

∞ ∞  (a, ct , v /s; q)k (bs)k  ( vqk /s; q)n (cs)n

(q, abt ; q)k

k =0

=

∞  k =0

=

n =0

(q; q)n

(a, ct , v /s; q)k (bs)k (cvqk ; q)∞ (q, abt ; q)k (cs; q)∞

∞ (abt , cv ; q)∞  (a, ct , v /s; q)k (bs)k , (bt , ct , cs; q)∞ (q, abt , cv ; q)k k =0

which is (2.6). Thus we complete the proof of theorem.

2

3. The bivariate Rogers–Szegö The bivariate Rogers–Szegö polynomials are introduced by Chen, Fu and Zhang [5], as defined by

hn (x, y |q) =

n    n

k

k =0

P k (x, y ).

(3.1)

Setting y = 0, the polynomials hn (x, y |q) reduce to the classical Rogers–Szegö polynomials hn (x| y ) defined by

hn (x| y ) =

n    n k =0

k

xk .

(3.2)

The continuous big q-Hermite polynomials [11] are defined by

H n (x, a|q) =

n    n  k =0

k



ae i θ ; q k e i (n−2k)θ ,

x = cos θ.

We observe that the bivariate Rogers–Szegö polynomials hn (x, y |q) are equivalent to the continuous big q-Hermite polynomials owing to the following relation

 



H n (x, a|q) = e inθ hn e −2i θ , ae −i θ q ,

x = cos θ.

(3.3)

The polynomials hn (x, y |q) have the generating function ∞  n =0

hn (x, y |q)

tn

(q; q)n

=

( yt ; q)∞ , (t , xt ; q)∞

|t | < 1, |xt | < 1.

(3.4)

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D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

A direct calculation shows that





D kq an =

an−k (q; q)n /(q; q)n−k ,

0  k  n;

0,

k > n.

(3.5)

From the identity (3.5), we can easily establish the following lemma. Lemma 3.1. We have n    n  n T(a, b; D q ) c = (a; q)k bk cn−k .

(3.6)

k

k =0

From (3.1) and (3.6), we can easily obtain



hn (x, y |q) = lim T( y /x, x; D q ) cn .

(3.7)

c →1

Carlitz [4] studied generating functions for Rogers–Szegö polynomials systematically and gave a formula ∞ 

hm+n (a|q)hn (b|q)

n =0

zn

(q; q)n

=

−m

(az; q)m (abz2 ; q)∞ q q , bz , φ q , ; 2 1 q1−m /(az) z (abz2 ; q)m ( z, az, bz, abz; q)∞

(3.8)

where m ∈ N and max{| z|, |az|, |bz|, |abz|} < 1. Cao [3] used the q-exponential operator to prove (3.8). In this section, we will use the Cauchy operator to derive (3.8) for hn (x, y |q). Theorem 3.1. We have ∞ 

hm+n (x, y |q)hn (u , v |q)

n =0

zn

(q; q)n

=

 m   m i

i =0

ai (b/a; q)i

i

(buzqi , v zqi ; q)∞ bq /a, uzqi , v , φ ; q , az 3 2 buzqi , v zqi (auz, uzqi , zqi ; q)∞

where max{|az|, |auz|} < 1. Proof. By Lemma 3.1, the left side of (3.9) can be written as ∞  n =0

 lim T(b/a, a; D q ) cm+n hn (u , v |q)

c →1

zn



∞ 

(cz)n = lim T(b/a, a; D q ) c hn (u , v |q) (q; q)n c →1 (q; q)n n =0

 (cv z; q)∞ . = lim T(b/a, a; D q ) cm c →1 (cz, cuz; q)∞ m

In view of (1.6) and (1.5), the above sum equals

lim

c →1

∞  (b/a; q)n n =0

= lim

c →1

(q; q)n

n

a

D nq

∞  (b/a; q)n n =0

(q; q)n

(cv z; q)∞ c (cz, cuz; q)∞   n 



m

qi (i −n)

an

i =0





 (cqi v z; q)∞ n D qi cm D nq −i . i (cqi z, cqi uz; q)∞

In view of (3.5), the above sum equals

 (q; q)m m−i n−i (cqi v z; q)∞ c Dq c →1 (q; q)n (q; q)m−i (cqi z, cqi uz; q)∞ n =0 i =0

 n ∞  (cqi v z; q)∞ (q; q)m ai cm−i (b/a; q)i  (bqi /a)n−i i (i −n) n−i n−i = lim q a Dq c →1 (q; q)i (q; q)m−i (q; q)n−i (cqi z, cqi uz; q)∞ n =0 i =0 

 m  ∞   (cqi v z; q)∞ (bqi /a; q)n −in n n m m −i i = lim c a (b/a; q)i q a Dq i c →1 (q; q)n (cqi z, cqi uz; q)∞ n =0 i =0 

 m  ∞   (cqi v z; q)∞ (bqi /a; q)n  −i n m m −i i = lim c a (b/a; q)i aq D q . i c →1 (q; q)n (cqi z, cqi uz; q)∞ lim

∞  (b/a; q)n

i =0

an

n 

qi (i −n)

  n i

n =0



(3.9)

D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

271

In view of (1.6), the above sum equals ∞    m

lim

c →1

i =0

= lim

c →1

=

i



cm−i ai (b/a; q)i T bqi /a, aq−i ; D q

∞    m

i

i =0

∞    m

i

i =0

c





(cqi v z; q)∞ (cqi z, cqi uz; q)∞



i

(buzqi , cv zqi ; q)∞ bq /a, cuzqi , v a (b/a; q)i ; q, az 3 φ2 buzqi , cv zqi (auz, cuzqi , czqi ; q)∞

m −i i

ai (b/a; q)i

i

(buzqi , v zqi ; q)∞ bq /a, uzqi , v , φ ; q , az 3 2 buzqi , v zqi (auz, uzqi , zqi ; q)∞

where max{|az|, |auz|} < 1. This complete the proof of theorem.

2

Remark 3.1. Setting b = 0, v = 0 and u = b, (3.9) reduce to (3.8). From the above theorem and (1.3), we get the following equivalent formula for H n (x, a|q). Corollary 3.1. We have ∞ 

H m+n (x, a|q) H n (u , b|q)

n =0

zn

(q; q)n

 m   m

(bze i (θ −β) q j , bze i (θ +2β) q j ; q)∞ j (aze i (θ −β) , ze i (θ −β) q j , ze i (θ +β) q j ; q)∞ j =0

bq j /a, ze i (θ −β) q j , be i β i (θ +β) × 3 φ2 ; q, aze , bze i (θ −β) q j , bze i (θ +2β) q j

= e imθ

a j (b/a; q) j

where x = cos θ , u = cos β and max{|aze i (θ−β) |, |aze i (θ+β) q j |} < 1. 4. The U (n + 1) generations of the Kalnins–Miller transformation Proposition 4.1 (The U (n + 1) generations of the q-Chu–Vandermonde summation theorem). (See [13, Theorem 5.10].) Let b, c and x1 , . . . , xn be indeterminate, let N i be nonnegative integers for i = 1, 2, . . . , n with n  1. Suppose that none of the denominators in the following identity vanishes. Then

 b

N 1 +···+ N n

n  ( xxi c /b; q) N i n

i =1

( xxni c ; q) N i







=

0 y i  N i i =1,2,...,n

×

xn

xr y r − y s q xs 1 − xxr s

1−

1r
n   xi i =1





   n  xr − N s (x q ; q) yr s

r , s =1



−1  c; q

(q xxrs ; q) yr

(b; q) y 1 +···+ yn q

y 1 +2 y 2 +···+nyn

.

(4.1)

yi

2

Proof. See [13].

Theorem 4.1 (The U (n + 1) generalization of the fourth Kalnins–Miller transformation). Let b, c, x, y and x1 , . . . , xn be indeterminate, let N i be nonnegative integers for i = 1, 2, . . . , n with n  1. Suppose that none of the denominators in the following identity vanishes, and that max{|dx|, |dy |, |dyq y 1 +···+ yn |, |dxq y 1 +···+ yn |} < 1. Then

 0 y i  N i i =1,2,...,n

=







xr y r − y s q xs 1 − xxr s

1−

1r
   n  n  xr − N s (x q ; q) yr  xi s

r , s =1

(q xxrs ; q) yr

i =1

xn

−1  cx; q yi



(bx, dx; q) y 1 +···+ yn a, bxq y 1 +···+ yn y 1 +2 y 2 +···+nyn × ; q, dy q 2 φ1 adxq y 1 +···+ yn (adx; q) y 1 +···+ yn  N +···+ Nn  n  1 −  ( xxi c y ; q) N i (dx, ady ; q)∞ x 1 n

(dy , adx; q)∞

y

( xi cx; q) N i 0 y  N i =1 xn i i i =1,2,...,n

1r
xr y r − y s q xs 1 − xxr s



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D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

 n  n  xr − N s  (x q ; q) yr  xi

×

−1 

s

(q xxrs ; q) yr

r , s =1

xn

i =1

c y; q yi



(by , dy ; q) y 1 +···+ yn a, byq y 1 +···+ yn y 1 +2 y 2 +···+nyn × ; q, dx q . 2 φ1 adyq y 1 +···+ yn (ady ; q) y 1 +···+ yn

(4.2)

Proof. Replacing (b, c ) by (bx, cx) and (by , c y ), respectively, in Proposition 4.1, we have

 (bx) N 1 +···+ Nn

n  ( xxi c /b; q) N i

 =

n

( xxni cx; q) N i

i =1



 0 y i  N i i =1,2,...,n

×

xr y r − y s q xs 1 − xxr s

1−

1r
n   xi

xn

i =1





cx; q

−1 

   n  xr − N s (x q ; q) yr s

(q xxrs ; q) yr

r , s =1

 (bx; q) y 1 +···+ yn q

y 1 +2 y 2 +···+nyn

(4.3)

yi

and

 (by )

n  ( xxi c /b; q) N i

N 1 +···+ N n



( xxni c y ; q) N i

i =1





=

n

0 y i  N i i =1,2,...,n

×





xr y r − y s q xs 1 − xxr s

1−

1r
n   xi

xn

i =1

c y; q

−1 

   n  xr − N s (x q ; q) yr s

(q xxrs ; q) yr

r , s =1

 (by ; q) y 1 +···+ yn q

y 1 +2 y 2 +···+nyn

(4.4)

.

yi

Comparing (4.3) and (4.4), we immediately obtain



 0 y i  N i i =1,2,...,n

=





xr y r − y s q xs 1 − xxr s

1−

1r
N 1 +···+ Nn  n x

   n  n  xr − N s (x q ; q) yr  xi s

r , s =1





( xxni c y ; q) N i

( xi cx; q) N i 0 y  N i =1 xn i i i =1,2,...,n

y

 n  n  xr − N s  (x q ; q) yr  xi

×

(q xxrs ; q) yr

s

r , s =1

(q xxrs ; q) yr

xn

i =1

c y; q



i =1



cx; q

xr y r − y s q xs 1 − xxr s

1−

1r
−1 

xn

−1 

 (bx; q) y 1 +···+ yn q

y 1 +2 y 2 +···+nyn

yi



 (by ; q) y 1 +···+ yn q

y 1 +2 y 2 +···+nyn

(4.5)

.

yi

We rewrite (4.5) as



 0 y i  N i i =1,2,...,n

×

=

xr y r − y s q xs 1 − xxr s

1−

1r
n   xi i =1





xn

cx; q

−1  yi

N 1 +···+ Nn  n x

y

×

   n  xr − N s ( xs q ; q) yr

i =1

r , s =1

(q xxrs ; q) yr 

1

(by , bxq y 1 +···+ yn ; q)∞

( xxni c y ; q) N i ( xxni cx; q) N i

1

(bx, byq y 1 +···+ yn ; q)∞

q





0 y i  N i i =1,2,...,n

y 1 +2 y 2 +···+nyn

q

y 1 +2 y 2 +···+nyn

 1r


xr y r − y s q xs 1 − xxr s

1−

   n  n  xr − N s (x q ; q) yr  xi s

r , s =1

(q xxrs ; q) yr

i =1

xn

c y; q

−1  yi

 .

(4.6)

D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

273

Applying the operator T(a, d; D q ) with respect to the variable b to both sides of the equation and using (2.2), we get



 0 y i  N i i =1,2,...,n

×

xr y r − y s q xs 1 − xxr s

1−

1r
n   xi

xn

i =1



cx; q

−1  yi

N 1 +···+ Nn  n x

y

×

   n  xr − N s (x q ; q) yr s

(q xxrs ; q) yr

r , s =1

(adxq y 1 +···+ yn ; q)∞ (dxq y 1 +···+ yn , by , bxq y 1 +···+ yn ; q)∞ 

a, bxq y 1 +···+ yn ; q, dy q y 1 +2 y 2 +···+nyn adxq y 1 +···+ yn

× 2 φ1 =





i =1

( xxni cx; q) N i

0 y i  N i i =1,2,...,n

 n  n  xr − N s  (x q ; q) yr  xi s

(q xxrs ; q) yr

r , s =1







( xxni c y ; q) N i

xn

i =1

c y; q







xr y r − y s q xs 1 − xxr s

1−

1r
−1 

(adyq y 1 +···+ yn ; q)∞

(dyq y 1 +···+ yn , bx, byq y 1 +···+ yn ; q)

∞

yi



a, byq y 1 +···+ yn ; q, dx q y 1 +2 y 2 +···+nyn . adyq y 1 +···+ yn

× 2 φ1

2

We obtain the theorem after using (1.1).

Remark 4.1. If we take a = 0 in Theorem 4.1, we get Theorem 3.2 of [16]. Proposition 4.2 (The U (n + 1) generations of the q-Chu–Vandermonde summation theorem). (See [13, Theorem 5.26].) Let b, c and x1 , . . . , xn be indeterminate, let N i be nonnegative integers for i = 1, 2, . . . , n with n  1. Suppose that none of the denominators in the following identity vanishes. Then

 1 (c ; q)− N 1 +···+ N n





=

0 y i  N i i =1,2,...,n

×

i =1

xn





1r
n   xn

xi

i =1

n  xi

bq

 c /b; q

b

N 1 +···+ N n

qe 2 ( N 1 , . . . , N n )

Ni xr y r − y s q xs 1 − xxr s

1−

y 1 +···+ yn − y i

 ;q yi

   n  xr − N s (x q ; q) yr

N i  n  xn xi

i =1

s

r , s =1

(q xxrs ; q) yr 

1 y 1 +2 y 2 +···+nyn (c ; q)− y 1 +···+ yn q

,

(4.7)

where e 2 ( N 1 , . . . , N n ) is the second elementary symmetric function of { N 1 , . . . , N n }.

2

Proof. See [13].

Theorem 4.2 (The U (n + 1) generalization of the first Kalnins–Miller transformation). Let b, c, x, y and x1 , . . . , xn be indeterminate, let N i be nonnegative integers for i = 1, 2, . . . , n with n  1. Suppose that none of the denominators in the following identity vanishes and that max{|dxq N 1 +···+ Nn |, |dyq N 1 +···+ Nn |, |dyq y 1 +···+ yn |, |dxq y 1 +···+ yn |} < 1. Then



 0 y i  N i i =1,2,...,n

×



1r
n   xn i =1

× 2 φ1

xi





xr y r − y s q xs 1 − xxr s

1−

   n  xr − N s (x q ; q) yr

bxq y 1 +···+ yn − y i ; q

s

r , s =1

(q xxrs ; q) yr

 yi

q y 1 +2 y 2 +···+nyn

1

(cx, ady ; q) y 1 +···+ yn



a, c yq y 1 +···+ yn ; q, dxq N 1 +···+ Nn adyq y 1 +···+ yn

274

D.-q. Lu / J. Math. Anal. Appl. 359 (2009) 265–274

N 1 +···+ Nn =

x

y

×

 n  xr − N s  (x q ; q) yr s

r , s =1

×

(c y , dy ; q) N 1 +···+ Nn (ady , cx; q) N 1 +···+ Nn

(q xxrs ; q) yr 1

(c y , dy ; q) y 1 +···+ yn

 0 y i  N i i =1,2,...,n

q y 1 +2 y 2 +···+nyn

2 φ1



 1r
n   xn i =1

xi



xr y r − y s q xs 1 − xxr s

1−

bq y 1 +···+ yn − y i ; q

 a, c yq N 1 +···+ Nn y 1 +···+ yn . N 1 +···+ N n ; q, dxq



 yi

adyq

Proof. Similar to the proof of Theorem 3.1.

2

Acknowledgments The author would like to thank the referees and editors for their many valuable comments and suggestions. The author is also grateful to Professor Liu and Cao Jian for many helpful suggestions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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