Bilateral inversions and terminating basic hypergeometric series identities

Discrete Mathematics 309 (2009) 3888–3904

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Bilateral inversions and terminating basic hypergeometric series identities Wenchang Chu ∗ , Chenying Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China

article

info

a b s t r a c t

Article history: Received 26 May 2008 Received in revised form 5 November 2008 Accepted 6 November 2008 Available online 11 December 2008

The q-analogue of Legendre inversions is established and generalized to bilateral sequences. They are employed to investigate the dual relations of three basic formulae due to Jackson and Bailey, on balanced 3 φ2 -series, well-poised 8 φ7 -series and bilateral 6 ψ6 -series. Several terminating well-poised series identities are consequently derived, including the q-Dixon formulae on terminating 3 ψ3 -series and two terminating wellpoised 5 ψ5 -series identities due to [F.H. Jackson, Certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 167–172; W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318–320]. © 2008 Elsevier B.V. All rights reserved.

Keywords: Basic hypergeometric series Bilateral q-series inversions Terminating balanced series Terminating well-poised series

For two indeterminate x and q, define the shifted factorial by

(x; q)0 = 1 and (x; q)n = (1 − x)(1 − xq) · · · (1 − xqn−1 ) for n ∈ N. When |q| < 1, the shifted factorial of infinite order is well-defined

(x; q)∞ =

∞ Y

(1 − xqk ) and (x; q)n =

k=0

(x; q)∞ (xqn ; q)∞

for n ∈ Z.

Its product and fraction forms are abbreviated compactly to [a, b, . . . , c ; q]n = (a; q)n (b; q)n · · · (c ; q)n , a, α,

b, β,



..., ...,

(0.1a)

 (a; q)n (b; q)n · · · (c ; q)n c q = . γ n (α; q)n (β; q)n · · · (γ ; q)n

(0.1b)

Following Bailey [3] and Slater [23], the unilateral and bilateral basic hypergeometric series are defined, respectively, by 1+r

r

φs

ψs

a0 ,



a1 , b1 ,



a1 , b1 , a2 , b2 ,

..., ..., ..., ...,



ar q; z bs



ar q; z bs

 =

∞ X

z

n

n=0

 =

∞ X n=−∞

z

n

a0 , q,

a1 , b1 ,

..., ...,

ar q bs

a1 , b1 ,

a2 , b2 ,

..., ...,

ar q bs







,

(0.2a)

.

(0.2b)

n

 n

Obviously, the unilateral series may be considered as a special case of the corresponding bilateral one with one of the denominator parameters equal to q. When one of the numerator parameters has the form q−m with m ∈ N0 , then the

∗

Corresponding address: Dipartimento di Matematica, Università del Salento, Lecce-Arnesano, P.O. Box 193 Lecce, Italy. E-mail addresses: [email protected] (W. Chu), [email protected] (C. Wang).

0012-365X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2008.11.002

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

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corresponding r φs -series consists of only finite number of terms. In this case, we say that the r φs -series is terminating. Similarly, the r ψs -series is terminating if there is a numerator parameter and a denominator parameter having forms q−m and qn , respectively, with m ∈ N0 and n ∈ N. In addition, the base q will be confined, throughout the paper, to |q| < 1 for non-terminating q-series. The inversion techniques have been efficient tools to prove terminating basic hypergeometric series identities (cf. Chu [10–13]). This paper will present new inverse series relations and derive further terminating series identities. The Section 1 will establish the q-analogue of Legendre inversions and give new proofs of the q-Dixon formulae on terminating 3 ψ3 -series as dual formulae of the q-Saalschütz theorem on balanced 3 φ2 -series. Section 2 will prove a general pair of bilateral inverse series relations and then utilize it to derive, as the dual formulae of Jackson’s well-poised 8 φ7 -series identity, eight terminating well-poised series identities including two well-poised 5 ψ5 -series identities due to Jackson [19] and Bailey [4], which seem to have been unrelated with Jackson’s 8 φ7 -series identity up to now. Finally in Section 3, the reduced bilateral inversions of particular interest are employed to investigate the dual formulae of Bailey’s bilateral 6 ψ6 series identity, where the q-analogues of Watson and Whipple formulae on 3 F2 -series due to Andrews [1] and Jain [21] as well as the polynomial analogues of Euler’s pentagonal number theorem due to Berkovich–Garvan [7] and Warnaar [27] are reviewed. 1. q-Legendre inversions and the terminating q-Dixon formulae For a real number x and two non negative integers n ≥ k ≥ 0, define the Gaussian binomial coefficients by

(qx−k+1 ; q)k x := k (q; q)k

(q; q)n n := . k (q; q)k (q; q)n−k

 

 

and

Then for Legendre inversions (cf. Riordan [22, Table 2.5; P 68] and Chu–Wei [17]), we have the following unilateral qanalogue. Lemma 1 (q-Legendre Inversions). Let λ be a fixed real number. Then for all n ∈ N0 , the following inverse series relations hold: f ( n) =

  λ + 2n 1 − qλ+2k g (k), (−1) n − k 1 − qλ+2n

(1.1a)

    n X n−k λ+n+k−1 (−1)k q 2 f (k). n−k

(1.1b)

n X k=0

g (n) =

k

k=0

Making the replacements in (1.1a) and (1.1b) f (k) =

F (k) 1 − qλ+2k

and

g (k) =

G(k) 1 − qλ+2k

we derive also another equivalent pair of inversions:

  n X k λ + 2n F (n) = (−1) G(k), n−k

(1.2a)

k=0

G(n) =

n X k=0

(−1)k

    λ + n + k 1 − qλ+2n n−2 k q F (k); n−k 1 − qλ+n+k

(1.2b) 1−qλ+2n

which will also be used in this paper. In particular when λ = 0, one should be aware that the fraction 1−qλ+n+k displayed in (1.2b) is set to be one for n = k = 0 in view of the initial condition F (0) = G(0) for both (1.2a) and (1.2b). We remark that a stronger version of Lemma 1 was established by Andrews [2, Lemma 3]. In order for the reader to follow Lemma 1 without difficulty, its proof is produced below. Proof. One implication of this inverse pair is that for every identity of the form (1.1a) or (1.1b), there is a companion of the dual identity. To prove each is to prove both. It is sufficient to show that (1.1a) implies (1.1b). Substituting (1.1a) into (1.1b) and then interchanging the summation order, we can reformulate the right member of (1.1b) as follows:

   X   n k λ+2i X n−k k λ+n+k−1 i λ + 2k 1 − q 2 RHS (1.1b) = (−1) q (−1) g (i) n−k k − i 1 − qλ+2k k=0 i=0     n n X X 1 − qλ+2i n − i (q; q)λ+n+k−1 n−2 k = g (i) (−1)k+i q . n−k (q; q)n−i (q; q)λ+k+i k=i i =0

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

The last sum with respect to k reduces to 1/(1 − qλ+2n ) for i = n and zero otherwise. This is justified by observing that (q;q)λ+n+k−1 (q;q)λ+k+i

is a polynomial of degree n − i − 1 in qk and then applying the q-binomial theorem

(x; q)m =

    m X k m (−1)k q 2 xk k

k=0

to each term of the polynomial. This completes the proof of the lemma.



From this lemma, we are now going to show that two terminating q-Dixon formulae are dual ones of the q-Saalschütz theorem [18, II-12]:

3

φ2

 −n q

,

b, qa/bd q; q = qa/b, qa/d

q n a, qa/b,





   qa n



d q qa/d

bd

n

.

(1.3)

1.1 When a = 1, rewrite the last equation as the following q-binomial sum 1 + qn 1 + δ0,n

q( 2 ) n

b, q/b,

   q n



d q q/d

bd

n

=

     n 2n  n−k  X 2k q, q/bd n+k 1−q 2 q (−1)k q k q/b, q/d n − k 1 − qn+k k k=0

where δm,n is the usual Kronecker symbol. This relation matches perfectly with (1.2b) for λ = 0. Then the dual relation (1.2a) gives another binomial sum



2n n

q, q/bd q q/b, q/d





n

    n X k 1 + qk 2n b, k = (−1) q 2 n − k 1 + δ0,k q/b,

   q k

d q q/d

k=0

bd

k

.

According to the linear factor 1 + qk , splitting the last sum into two and then making the replacement k → −k for the sum corresponding to qk , we get the following simplified bilateral sum:



2n n

q, q/bd q q/b, q/d

n X





= n

(−1)

k=−n

k



2n



n−k

 

q

k 2

b, q/b,

   q k



d q q/d

k

bd

.

This is the terminating well-poised 3 ψ3 -series identity due to Jackson [19, Eq. 2] (see also Carlitz [9, Eq. 1.2] and Bressoud [8, Eq. 2]):

3

ψ3

q −n , q1+n ,



b, q/b,

d q, q/bd q q; q1+n /bd = q/b, q/d q/d









.

(1.4)

n

Performing the replacement k → −k, we get the reversal of the last identity 3 ψ3

q −n , q1+n ,



b, q/b,

d q, q/bd q; q2+n /bd = q q /b, q/d q/d







 (1.5) n

which can be found in Bailey [5, Eq. 2] and Verma–Joshi [26, Eq. 3.10]. The nonterminating forms of the last two identities have been discovered by Bailey [4, Eqs. 2.2 and 1.2] (see Chu [14] also) 3

3

ψ3 ψ3

b, q/b,

c, q/c ,

b, q/b,

c, q/c ,

 



q d q; q/d bcd



q, q/bc , q/bd, q/cd = q q/b, q/c , q/d, q/bcd





, ∞    q2 d q, q/bc , q/bd, q/cd q; = q ; q/d q/b, q/c , q/d, q/bcd bcd ∞

where the former can also be found in Verma–Jain [24, Eq. 5.5]. For the sake of brevity, we fix δ = 0, 1 from now on. In terms of unilateral series, (1.4) and (1.5) can be restated together as a single one (cf. Gasper–Rahman [18, II-15]): 3

φ2

 −2n q

,

b,

q1−2n /b,

     q, bd 1+δ−n n(δ−1) b, d q; q /bd = q q q . q, bd b, d q1−2n /d n 2n d

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

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1.2 When a = q, we can rewrite the Saalschütz formula displayed in (1.3) as q( 2 ) n

b, q /b,

 



d q q /d

2

2

n

n

q2 bd

=

       n X n−k n+k 2k q, q2 /bd (−1)k q 2 q . n−k k q2 /b, q2 /d k k=0

This relation matches perfectly with (1.1b) for λ = 1. Then the dual relation (1.1a) gives another binomial sum



2n n

q, q2 /bd q q2 /b, q2 /d





= n

   n 2k+1  k  X b, 2n + 1 1 − q 2 q (−1)k n − k 1 − q2n+1 q2 /b,

q2

 

d q 2 q /d

k=0

k

k

bd

.

According to the linear factor 1 − q2k+1 , splitting the last sum into two and then making the replacement k → −k − 1 for the sum corresponding to q2k+1 , we get the following simplified bilateral sum:

  2     n X k (q; q)2n+1 b, q /bd k 2n + 1 q = (− 1 ) q 2 2 2 2 n−k q /b, (q; q)n q /b, q /d n k=−n−1

 

d q 2 q /d

k

q2 bd

k

.

This is another terminating well-poised 3 ψ3 -series identity 3 ψ3

q−n , q2+n ,

b, q2 /b,



d q2 , q2 /bd q; q2+n /bd = (1 − q) 2 q 2 q /d q /b, q2 /d







 (1.6) n

which has been given explicitly by Carlitz [9, Eq. 2.12] and Verma–Joshi [26, Eq. 3.13]. Similarly, the reversal of this bilateral sum results in the following slightly different formula:

3

ψ3

q−n , q2+n ,

b, q2 /b,



q − 1 q2 , q2 /bd d q q; q4+n /bd = 2 q /d q2 /b, q2 /d q









.

(1.7)

n

The nonterminating versions of these identities have also been discovered by Bailey [4, Eq. 2.3] (cf. [14] for proofs through Abel’s lemma on summation by parts): 3 ψ3

3 ψ3

b, q2 /b,

c, q2 /c ,

b, q2 /b,

c, q2 /c ,

 

q2 d q; 2 q /d bcd

q, q2 /bc , q2 /bd, q2 /cd q , q2 /b, q2 /c , q2 /d, q2 /bcd ∞    q4 −1 q, q2 /bc , q2 /bd, q2 /cd d q ; q . = 2 2 2 2 q2 /d bcd q q /b, q /c , q /d, q /bcd ∞









=

The last identity results from the reversal of the first one. We therefore attribute it to Bailey [4] although it did not appear there explicitly. In terms of unilateral series, (1.6) and (1.7) can also be unified as follows 3

φ2

 −1−2n q

,

b,

    
q, qbd 2δ−n (2n+1)(2δ−1) qb, qd q; q /bd = 1 − q q q . q, qbd qb, qd q−2n /d n 2n d

q−2n /b,

2. Bilateral inverse q-series relations and well-poised series The unilateral q-Legendre inversions illustrated in Section 3 admit the following bilateral generalization. Theorem 2 (Bilateral q-Inverse Series Relations). Let λ be a fixed nonnegative integer. Then for all n ∈ N0 , there hold the following inverse series relations λ+2n

(1 − q

)f (n) =

n X

    k λ+n n−k x (q /x; q)k λ + 2n (−1) q 2 g (k), n−k (q1−n x; q)k k

(2.1a)

k=−n−λ

g (n) + g (−λ − n) 1 − qλ+2n

=

  n X λ + n + k − 1 (1 − x)(1 − qλ+2k /x)(q1−k x; q)n−1 (−1)k f (k); n−k xn (qλ+k /x; q)n+1

(2.1b)

k=0

provided that λ = 0 or 1. Furthermore, these inversions are also valid for λ > 1 under the additional conditions g (−1) = g (−2) = · · · = g (1 − λ) = 0.

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

In order to facilitate the subsequent applications, we highlight the following equivalent form obtained under the replacement x → 1/y: λ+2n

(1 − q

n X

)F (n) =

    n−k (qλ+n y; q)k λ + 2n (−1) q 2 G(k), n−k yk (q1−n /y; q)k k

(2.2a)

k=−n−λ

G(n) + G(−λ − n) (q1+λ y; q)n 1 − qλ+2n

yn (1/y; q)n

=

  n X λ + n + k − 1 1 − qλ+2k y (−1)k F (k) n−k 1 − qλ y k=0   y, q λ y × 1+λ+n 1−n q qk−kn . q y, q y k

(2.2b)

(2.2c)

Proof. Substituting (2.1a) into the right member of (2.1b) and then interchanging the summation order, we have RHS (2.1b) =

  n X λ + n + k − 1 (1 − x)(1 − qλ+2k /x)(q1−k x; q)n−1 (−1)k n−k xn (1 − qλ+2k )(qλ+k /x; q)n+1 k=0     i λ+k k X k−i x (q /x; q)i λ + 2k (−1)i q 2 × g (i) 1 − k k−i (q x; q)i i=−k−λ n X

=

(1 − x)

i=−n−λ

xi x

g (i) n

n X

(−1)k+i

k=max{0,i,−i−λ}

λ+2k

×

(q1−k x; q)n−1 (qλ+k /x; q)n+1

(1 − q /x)(q; q)λ−1+n+k (qλ+k /x; q)i q (q; q)n−k (q; q)k−i (q; q)λ+k+i (q1−k x; q)i



k−i 2



.

Denote by Ω (n, i) the last sum with respect to k. By making the replacement k → n − k, we can reverse the summation order as follows: min{n,n−i,n+i+λ}

X

Ω ( n, i ) =

(−1)

(q1−n+k x; q)n−1 (1 − qλ+2n−2k /x)(q; q)λ−1+2n−k (qλ+n−k /x; q)i q (qλ+n−k /x; q)n+1 (q; q)k (q; q)n−k−i (q; q)λ+n−k+i (q1−n+k x; q)i



n+k+i

k=0

n−k−i 2



.

By invoking the following three relations

(q1−n+k x; q)n−1 (q1−n x; q)n−1 (x; q)k = , 1 − n + k (q x; q)i (q1−n x; q)i (q1−n+i x; q)k (qλ+n−k /x; q)i (qλ+n /x; q)i (q−λ−2n x; q)k (1+n−i)k = λ+n q , λ+ n − k (q /x; q)n+1 (q /x; q)n+1 (q1−λ−n−i x; q)k     k (q; q)λ−1+2n−k (−1)k (q; q)λ−1+2n qi−n , q−λ−n−i k− 2 = q q ; 1−λ−2n q (q; q)n−k−i (q; q)λ+n−k+i (q; q)n−i (q; q)λ+n+i k we can reformulate the last sum as

(1 − qλ+2n /x)(q; q)λ−1+2n (qλ+n /x; q)i (q1−n x; q)n−1 n−2 i q (q; q)n−i (q; q)λ+n+i (q1−n x; q)i (qλ+n /x; q)n+1   −λ−2n min{n,nX −i,n+i+λ} 1 − q−λ−2n+2k x q x, x, qi−n , q−λ−n−i × q qk . 1−λ−2n , q1−λ−n−i x, q1−n+i x 1 − q−λ−2n x q, q 

Ω (n, i) = (−1)n−i



k

k=0

Recalling the well-poised 6 φ5 -series identity [18, II-21] 6

φ5

a,



√

q√a, a,

√ −q√a, − a,

b, qa/b,

d, qa/d,

1 +m a q−m q 1+m q; q a bd





qa, qa/bd = q qa/b, qa/d



 (2.3) m

we can evaluate the last sum with respect to k as follows

" 6

φ5

q−λ−2n x,

p

qpq−λ−2n x,

p −qpq−λ−2n x, − q−λ−2n x,

x,

qi−n ,

q−λ−2n x, q1−λ−2n , q1−λ−n−i x,    1−λ−2n 1−n+i q x, q   = δi,n , 0 ≤ i ≤ n;  q1−n+i x, q1−λ−2n q i =  1−λ−2n 1−λ−n−i n− q x, q   = δi,−n−λ , −λ − n ≤ i ≤ −λ.  1−λ−n−i 1−λ−2n q q x, q λ+n+i

# q; q q1−n+i x

q−λ−n−i

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Consequently, Ω (n, i) reduces to

Ω (n, i) =

 1   ,    (1 − x)(1 − qλ+2n )

i = n;

xλ+2n

 , i = −n − λ;  λ+2n )    (1 − x)(1 − q 0, −λ − n < i ≤ −λ and 0 ≤ i < n;

which confirms for λ = 0 and 1 that RHS (2.1b) =

g (n) + g (−λ − n)

= LHS (2.1b).

1 − qλ+2n

When λ > 1, observe that Ω (n, i) does not vanish for −λ < i < 0. However this gap is covered by the additional conditions g (−1) = g (−2) = · · · = g (1 − λ) = 0.  By means of Theorem 2, we shall investigate the dual relations of Jackson’s q-analogue of the very well-poised Dougall formula (cf. [3, Section 8.3], [18, II-22] and [23, IV-8]): a,



8 φ7

√

√ −q√a, − a,

q√a, a,

b, qa/b,

qa, qa/bc , qa/bd, qa/cd q qa/b, qa/c , qa/d, qa/bcd



 =

c, qa/c ,

d, qa/d,

e, qa/e,



q−n q; q q1+n a

 (2.4a)

where q1+n a2 = bcde.

(2.4b)

n

These dual formulae not only recover two well-known terminating well-poised 5 ψ5 -series identities due to Jackson [19] and Bailey [4], but also lead to four other well-poised series identities, which do not seem to have appeared previously. Surprisingly, we can even apply Theorem 2 to these 5 ψ5 -series identities just mentioned, which will result in two further well-poised 8 φ7 -series identities. 2.1 The two examples in this and the next subsections connect surprisingly with Jackson’s summation formula on very well-poised 8 φ7 -series to the two terminating well-poised 5 ψ5 -series identities, both due to Jackson [19,20], which seem unrelated up to now. First by specializing (2.4a) and (2.4b) to the following series q/bcd,

qpq/bcd, q/bcd,

p −qpq/bcd, 8 φ7 − q/bcd,   2 q /bcd, b, c , d = q q/b, q/c , q/d, bcd/q n

q/bc , q/d,

p



q/bd, q/c ,

q/cd, q/b,

q−n , q2+n /bcd,

 q; q q2−n /bcd qn

we can rewrite the resulting equation explicitly as

 (q2 /bcd; q)n b, (1 − qn )(bcd/q; q)n q/b,

c, q/c ,



d q q/d

= n

n X

(−1)

k

1+2k /bcd n+k−1 1−q n−k 1 − q/bcd



k=0





q/bcd, q/bcd × 2+n q q /bcd, q2−n /bcd

q, q/bc , q/bd, q/cd q q/b, q/c , q/d, q/bcd

  



2k k

k



q

k+1 2

 −nk

1 − q2k

k

.

This matches exactly with (2.2b) and (2.2c) for λ = 0 and the specifications



2n F (n) := n G(n) :=

b, q/b,



q, q/bc , q/bd, q/cd q q/b, q/c , q/d, q/bcd

q( 2 ) n





c, q/c ,

1 − q2n

,

n   q n d q ; q/d n bcd

where for G(n), there holds the reflection property G(−n) = qn G(n). Then the dual relation displayed in (2.2a) reads as



2n n

q, q/bc , q/bd, q/cd q q/b, q/c , q/d, q/bcd



 n

q( 2 ) = n

n X

(−1)k

k=−n



2n



n−k



q

n−k 2



 (q1+n /bcd; q)k b, (q−n bcd; q)k q/b,

c, q/c ,



d q q/d

. k

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

Expressing this relation in terms of a well-poised 5 ψ5 -series, we recover the following identity due to Jackson [19, Eq. 1] and Bailey [4, Eq. 3.1]: 5

q −n , q1+n ,

b, q/b,



ψ5

c, q/c ,

d, q/d,

q1+n /bcd q, q/bc , q/bd, q/cd q; q = q q/b, q/c , q/d, q/bcd q−n bcd









.

(2.5)

n

In this inversion process, if replacing G(n) by G0 (n) defined by b, G (n) := q/b,

c, q/c ,



0

 

d q q/d

q2

n

bcd

n

with the reflection property G0 (−n) = q−n G0 (n), then the corresponding F (n) remains unchanged. However the resulting identity yields the reversal of (2.5) q−n , q1+n ,

b, q/b,



5 ψ5

c, q/c ,

d, q/d,

q1+n /bcd q; q2 q−n bcd





q, q/bc , q/bd, q/cd q q/b, q/c , q/d, q/bcd



 =

(2.6) n

which can be found explicitly in Bressoud [8, Eq. 1] and Joshi–Verma [26, Eq. 3.8]. In terms of unilateral series, (2.5) and (2.6) can be restated together as a single one (cf. Chu [15, Section 2]): 5

φ4

 −2n q

,

b,

q1−2n /b,

c,

d,

q1−2n /c ,

q1−3n /bcd q; q1+δ qn bcd



q1−2n /d,



= qn(δ−1)

b, c , d, bcd q q, bc , bd, cd



q, bc , bd, cd q b, c , d, bcd



  n

.

2n

2.2 Considering another special case of (2.4a) and (2.4b) q2 pq/bcd, q q/bcd,

p −q2p q/bcd, 8 φ7 −q q/bcd,   4 q /bcd, b, c , d = 2 2 2 2 q q /b, q /c , q /d, bcd/q n q /bcd,

q2 /bc , q2 /d,

p

3

q2 /bd, q2 /c ,

q2 /cd, q2 /b,

q−n , 4 +n q /bcd,



q1+n q; q q3−n /bcd



we can rewrite it as the following equation:

    n 3+2k X /bcd (q4 /bcd; q)n b, c, d k n+k 1−q q = (− 1 ) 2 2 2 2 3 /bcd n − k (bcd/q ; q)n q /b, q /c , q /d 1 − q n k=0        2 k+1 2k q, q2 /bc , q2 /bd, q2 /cd q /bcd, q3 /bcd −nk q q 2 q . × 4+n 2 2 2 2 3 −n k q /b, q /c , q /d, q /bcd q /bcd, q /bcd k k This matches with (2.2b) and (2.2c)) for λ = 1 and the sequence specifications



2n F (n) := n G(n) :=

q, q2 /bc , q2 /bd, q2 /cd q q2 /b, q2 /c , q2 /d, q2 /bcd





b, q2 /b,

c, q2 /c ,



q( 2 ) , n

n

  2 n q d q ; 2 q /d n bcd

where for G(n), there holds the reflection property G(−n − 1) = −q2n+1 G(n). Then the dual relation displayed in (2.2a) reads as

(1 − q1+2n ) n X

=



2n n

q, q2 /bc , q2 /bd, q2 /cd q q2 /b, q2 /c , q2 /d, q2 /bcd





(−1)

k



k=−n−1





2n + 1 q n−k

n−k 2



q( 2 ) n

n

 (q3+n /bcd; q)k b, 2 (q−1−n bcd; q)k q /b,

c, q2 /c ,



d q q2 /d

. k

Its reformulation in terms of well-poised 5 ψ5 -series leads us to Bailey’s identity [4, Eq. 3.2] (cf. Carlitz [9, Eq. 3.4] also): 5

ψ5

q −n , q2+n ,

b, q2 /b,



= (1 − q)

c, q2 /c ,

d, q2 /d,

q3+n /bcd q; q q−1−n bcd



q2 , q2 /bc , q2 /bd, q2 /cd q q2 /b, q2 /c , q2 /d, q2 /bcd



 n

.

 (2.7a)

(2.7b)

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

3895

Again in this inversion process, if G(n) is specified instead by b, G (n) := 2 q /b,



0

c, q2 /c ,

 

d q q2 /d

n

q4 bcd

n

with the reflection property G0 (−n − 1) = −q−2n−1 G0 (n), then the corresponding F (n) remains unchanged. The resulting identity reads as the reversal of (2.7a) and (2.7b)): 5

ψ5

q−n , q2+n ,

b, q2 /b,



c, q2 /c ,

d, q2 /d,

q3+n /bcd q; q3 q−1−n bcd



q − 1 q2 , q2 /bc , q2 /bd, q2 /cd q q2 /b, q2 /c , q2 /d, q2 /bcd q





=

 (2.8a)

.

(2.8b)

n

In terms of unilateral series, (2.7a), (2.7b) and (2.8a), (2.8b) are unified as follows: 5

φ4

 −1−2n q

,

b,

q−2n /b,

1+2n δ−1

= (−q

)

c,

q−2n /c ,

(q; q)2n+1

d,

q−1−3n /bcd q; q1+2δ q1+n bcd



q−2n /d,

qb, qc , qd, qbcd q q, qbc , qbd, qcd

qbc , qbd, qcd q qb, qc , qd, qbcd



 



n



. 2n

Refer to Chu [15, Section 2] and Joshi-Verma [26, Eq. 3.12] for different proofs. 2.3 The following special case of (2.4a) and (2.4b)

√

√ −q√a, a2 , qn/2 , q(n+1)/2 , 1−n/2 − a, q/a, aq , aq(1−n)/2 ,     aq1/2 , aq−n 1/2 aq(1−n)/2 , aq−n/2 = q q aq(1−n)/2 , aq−n/2 aq1/2 , aq−n n n

8 φ7

a,



q−(n−1)/2 , aq(1+n)/2 ,

q√a, a,

q−n/2 q; q aq1+n/2





can be rewritten, under the base replacement q → q2 , as

     [X n/2]  4k (−1/a)n a, qa (q/a; q2 )n a, a n + 2k − 1 1 − q a = q 1−n 1+n q n 2 1/a, q/a n − 2k 1−q 1 − a q a, q a 2k n (qa; q )n k=0 

×

4k 2k



2k+1  −2nk 2 q, q, q , a 2 q q a, qa, qa, q2 /a 1 − q4k k

 

2

2

where [x] denotes the integer part of the real number x. The last equation matches with (2.2b) and (2.2c) exactly for λ = 0 and

   2k      4k q 2 q, q, q2 , a2 2 q , F (n) := 4k 2k a, qa, qa, q2 /a  k 1−q  0, G(n) := (−1)n

n = 2k; n = 2k + 1;

(a; q)n (q/a; q )n ; (q/a; q)n (qa; q2 )n 2

as well as the reflection property G(−n) = qn G(n). The dual relation displayed in (2.2a) reads as

      2m    n   4m q, q, q2 , a2 2 n 2 X n−k ( q a ; q ) ( a ; q ) ( q / a ; q ) 2n q q 2 , k k k −k 2m a, qa, qa, q2 /a q 2 a = m n−k  (q1−n /a; q)k (q/a; q)k (qa; q2 )k k=−n 0,

n = 2m; n = 2m + 1;

which can be reformulated as the following bilateral q-series identity: 5

ψ5

q−n , q1+n ,



qn a, q1−n /a,

a, q/a,

   q, q, q2 , a2 2 q , = a, qa, qa, q2 /a m  0,

p q/a, √ qa,

p  − q/a √ q; −q/a − qa

n = 2m; n = 2m + 1.

(2.9a)

(2.9b)

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

In the proof of the last identity, if taking

(a; q)n (q/a; q2 )n (q/a; q)n (qa; q2 )n with the reflection property G0 (−n) = q−n G0 (n), then the corresponding F (n) remains unchanged. We therefore recover the G0 (n) := (−q)n

reversal of the last identity as follows: 5

q −n , q1+n ,



ψ5

qn a, 1−n q /a,

a, q/a,

p  − q/a √ q; −q2 /a − qa

p q/a, √ qa,

   q, q, q2 , a2 2 q , = a, qa, qa, q2 /a m  0,

(2.10a)

n = 2m;

(2.10b)

n = 2m + 1.

In terms of unilateral series, (2.9a), (2.9b) and (2.10a), (2.10b) are unified as follows: 5

φ4

 −2n q

 

=



,

a,

q−n a, q1−n /a,

q1−2n /a,

2m(δ−1)

q

0,

a , qa,

2

q/a 2 q q2



q−n q/a, √ q−n qa,

p

p  −q−n q/a 1+δ √ q; −q /a −q−n qa

(q; q)4m , [a, q/a; q2 ]2m

 m

n = 2m; n = 2m + 1.

2.4 The following special case of (2.4a), (2.4b)

√

√ −q qa, √ 8 φ7 − qa,   aq3/2 , aq−n 1/2 = q aq(1−n)/2 , aq1−n/2 qa,

q qa, √ qa,



n

q(n+1)/2 , aq(3−n)/2 ,

qa2 , q/a,

q1+n/2 , aq1−n/2 ,

aq(1−n)/2 , aq1−n/2 q aq3/2 , aq−n

q−(n−1)/2 , aq(3+n)/2 ,

q−n/2 q; q aq2+n/2









n

can be rewritten, under the replacements q → q2 and a → a/q, as

 [X n/2]  1+4k (q2 /a; q2 )n a n + 2k 1 − q = (−1/a) 2 a; q2 ) n − 2k ( q 1 − qa n n k=0        2 2k+1 a, qa 4k q, q, q , a2 2 −2nk 2 q q × 2+n 1−n q 2k a, qa, q2 a, q3 /a q a, q a 2k k which matches with (2.2b) and (2.2c) exactly for λ = 1 and       2k  4k q, q, q2 , a2 2 q q 2 , n = 2k; 2 3 2k F (n) := a, qa, q a, q /a k  0, n = 2k + 1; n

q2 a, a q 1/a, q2 /a





(a; q)n (q2 /a; q2 )n ; (q2 /a; q)n (q2 a; q2 )n as well as the reflection property G(−n − 1) = −q2n+1 G(n). G(n) := (−1)n

Then the dual relation displayed in (2.2a) reads as

    1+n n X n−k (q a; q)k (a; q)k (q2 /a; q2 )k −k 2n + 1 q 2 a n−k (q1−n /a; q)k (q2 /a; q)k (q2 a; q2 )k k=−n−1       2m  4m q, q, q2 , a2 2 4m+1 (1 − q ) q q 2 , n = 2m; 2m a, qa, q2 a, q3 /a = m  0, n = 2m + 1. This can be expressed as the following new bilateral q-series identity: 5

ψ5

q −n , q2+n ,



=

  

q1+n a, q1−n /a,

(1 − q) 0,

a, q2 /a,

√

q, q2 , q3 , a2 2 q a, qa, q2 a, q3 /a



 √ −q/√ a q ; − q / a −q a

q/√ a, q a,



 m

,

n = 2m; n = 2m + 1.

(2.11a)

(2.11b)

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

3897

In this inversion process, if replacing G(n) by G0 (n) defined by G0 (n) := (−1)n

(a; q)n (q2 /a; q2 )n 2n q (q2 /a; q)n (q2 a; q2 )n

with the reflection property G0 (−n − 1) = −q−2n−1 G0 (n), then the corresponding F (n) remains unchanged. We find that the resulting identity is the reversal of the last identity: 5

q−n , q2+n ,

q1+n a, q1−n /a,



ψ5

a, q2 /a,

√

 √ −q/√ a 3 q; −q /a −q a

q/√ a, q a,

    q − 1 q, q2 , q3 , a2 2 q , = a, qa, q2 a, q3 /a q m  0,

(2.12a)

n = 2m;

(2.12b)

n = 2m + 1.

The unilateral series forms of (2.11a), (2.11b) and (2.12a), (2.12b) are unified as follows: 5 φ4

 −1−2n q

=

 

,

a,

q−2n /a,

(−q4m+1 )(δ−1)

0,

√

q−n /√ a, q−n a,

q−n−1 a, q1−n /a,

a , a,

q2 /a 2 q q2



2

 m

 √ −q−n /√ a 1+2δ q ; − q / a −q−n a

(q; q)4m+1 , [qa, q2 /a; q2 ]2m



n = 2m; n = 2m + 1.

2.5 Consider the following special case of (2.4a), (2.4b) 8 φ7

a,



√

√ −q√a, − a,

q√a, a,

qa2 , 1/a,

qn , 1−n q a,

−qn , −q1−n a,

−q−n , −q1+n a,

qa, −qa, q−n /a, −qn /a q−n q 1+n q; q = 1/a, −1/a, −q1+n a, q1−n a q a









.

n

Rewrite it, under the base replacement q → q1/2 , as a−2n q−n/2 qa2 , qa2 , −1/a, −q1/2 /a q 2 2 1/2 1 − qn 1/a , 1/a , −q a, −qa





a2 , a2 × 1+n 2 1−n 2 q q a ,q a

k

2k k

(−1)

k

k=0

2k 2 n+k−1 1−q a n−k 1 − a2





   k2 /2−nk −q1/2 , q1/2 a2 1/2 q q 2k −a, 1/a k k 1−q

q, −a q a2 , −qa

  



= n

n X

which matches with (2.2b) and (2.2c) exactly for λ = 0 and

   n −n/2 −q1/2 , q1/2 a2 1/2 q( 2 ) , q 2n −a, 1/a n n 1−q  2  qa , −1/a, −q1/2 /a G(n) := q q−n/2 ; 1/a2 , −q1/2 a, −qa n 

2n F (n) := n

q, −a q a2 , −qa



as well as the reflection property G(−n) = qn G(n). Then the dual relation displayed in (2.2a) reads as



   n −q1/2 , q1/2 a2 1/2 q( 2 )−n/2 q −a, 1/a n n   2     n n 2 X n−k ( q a ; q)k qa , −1/a, −q1/2 /a 2n k q (q1/2 a2 )−k = (−1) q 2 1−n /a2 ; q) n−k 1/a2 , −q1/2 a, −qa k ( q k k=−n

2n n

q, −a q a2 , −qa



which can be reformulated, under the replacement a → −a1/2 , as the following bilateral q-series identity:

p p  1/a, q/a 1/2 √ √ q; q /a 5 ψ5 q a, qa  1/2   1/2 1/2  q, a −q , q a 1/2 = q q−n/2 . 1/2 q a, qa a1/2 , −a−1/2 q−n , q1+n ,



qn a, 1−n q /a,

qa, 1/a,

n

n

In the proof of the last identity, if G(n) is replaced by qa2 , −1/a, −q1/2 /a q G (n) := 1/a2 , −q1/2 a, −qa 0





qn/2 n

(2.13a)

(2.13b)

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

with the reflection property G0 (−n) = q−n G0 (n), then the corresponding F (n) remains unchanged. The reversal of the last identity is consequently recovered as follows: 5

p p  1/a, q/a √ √ q; q3/2 /a qa q a,  1/2   1/2 1/2  q, a q −1/q2 , q−1/a2 q1/2 q−n/2 . = a, qa1/2 a , −a

ψ5

q −n , q1+n ,

qn a, 1−n q /a,



qa, 1/a,

n

(2.14a)

(2.14b)

n

In terms of unilateral series, (2.13a), (2.13b) and (2.14a), (2.14b) are unified as: 5

φ4

 −2n q

,

a,

q1−2n /a,

 1/2 −q , = 1/2 a ,

√

q1−n a, q−n /a,

q1/2 a 1/2 q −a−1/2



q−n /√ a, q1−n a, q , qn a,

  1+n n

1/a, qa,



q−n q/a √ q; qδ+1/2 /a q−n qa

p

a1/2 , a−1/2 ,



 (qa)1/2 qn(δ−1) . 1/2 q (q/a) n

2.6 The following special case of (2.4a) and (2.4b) 8

φ7

√

qa,

√ −q qa, √ − qa,

q qa, √ qa,



q a, = q/a,

q−n /a, q1−n a,

2

qa2 , q/a,

−q1+n /a, −q2+n a,

q1+n , q1−n a,

−q1+n , −q1−n a,

−q−n , −q2+n a,



q−n q; q q2+n a



 −qa q −1/a n

can be rewritten, under the base replacement q → q1/2 , as qa2 , q2 a2 , −q1/2 /a, −q/a q 1/a2 , q/a2 , −qa, −q3/2 a

  n 1+2k 2 X a n+k 1−q (−1)k 2 n−k 1 − qa n k=0   1/2 1/2 2      1/2 2 2 2 −q , q a 1/2 2k q, −q a a , qa q qk /2−nk q × 2+n 2 1−n 2 q 1/2 1/2 k a2 , −q3/2 a − q a , q / a q a ,q a k k k 



(q1/2 a2 )−n =

which matches with (2.2b) and (2.2c) exactly for λ = 1 and

 n −q1/2 , q1/2 a2 1/2 q q( 2 )−n/2 , 1/2 1/2 − q a , q / a n n   2 qa , −q1/2 /a, −q/a q q−n/2 ; G(n) := q/a2 , −qa, −q3/2 a n 

2n F (n) := n

q, −q1/2 a q a2 , −q3/2 a

 



with the reflection property G(−n − 1) = −q2n+1 G(n). The dual relation displayed in (2.2a) reads as

 n −q1/2 , q1/2 a2 1/2 q q( 2 )−n/2 1/2 1/2 − q a , q / a n n      1+n 2  2 n X n−k ( q a ; q ) qa , − q1/2 /a, −q/a 2n + 1 k k 2 = (−1) q q (q1/2 a2 )−k . n−k (q1−n /a2 ; q)k q/a2 , −qa, −q3/2 a k k=−n−1

(1 − q2n+1 )



2n n

q, −q1/2 a q a2 , −q3/2 a

 



Writing this relation in terms of a 5 ψ5 -series and then relabeling a by −a1/2 , we find the following bilateral series identity:

# p p 2 /a q / a , q 1 / 2 p p q; q /a 5 ψ5 q2+n , q1−n /a, q/a, q3 a, q2 a   2  q , (qa)1/2 −q1/2 , q1/2 a 1/2 = (1 − q) q q q−n/2 . a, (q3 a)1/2 n (qa)1/2 , −(q/a)1/2 n "

q−n ,

qn+1 a,

qa,

In the proof of the last identity, if G(n) is specified by qa2 , −q1/2 /a, −q/a q G (n) := q/a2 , −qa, −q3/2 a 0



 n

q3n/2

(2.15a)

(2.15b)

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

3899

with the reflection property G0 (−n − 1) = −q−2n−1 G0 (n), then the corresponding F (n) remains unchanged. The resulting identity gives the reversal of the last one:

# p 2 /a q 5 / 2 p q; q /a 5 ψ5 q2+n , q1−n /a, q/a, q2 a     q − 1 q2 , (qa)1/2 −q1/2 , q1/2 a 1/2 = q q q−n/2 . a, (q3 a)1/2 n (qa)1/2 , −(q/a)1/2 q n "

q−n ,

qn+1 a,

qa,

p pq/a, q3 a,

(2.16a)

(2.16b)

In terms of unilateral series, (2.15a), (2.15b) and (2.16a), (2.16b) are unified as follows: 5

 −1−2n q

φ4

,

a,

q−2n /a,

= 1 − q(2n+1)(2δ−1)




√

q−n a, q−n /a,

q−n / qa, √ q−n qa,

√

q−n /√ a q; q2δ+1/2 /a q−n a



   1+n −q1/2 , q1/2 a 1/2 q , q/a, qa1/2 , (qa)1/2 q q . (qa)1/2 , −(q/a)1/2 q1+n a, qa, q/a1/2 , (q/a)1/2 n n



In addition, we can study the dual formulae from bilateral series to unilateral ones. 2.7 By applying Theorem 2 to Jackson’s identity (2.4a) and (2.4b), we have recovered the two 5 ψ5 -series identities displayed in (2.5), (2.7a) and (2.7b). Interestingly enough, Theorem 2 can again be applied even to these 5 ψ5 -series identities, which will lead us to further well-poised 8 φ7 -series identities. In fact, consider the following special case of (2.5)

√

√

q−n , −q−n , qn √ a, −qn a, q√ /a q2 , q2 /a, −a, −qa 2 q q; q = 1 +n 1 +n 1 −n 1−n a, a2 , −q, −q2 q , −q , q / a , −q / a, a



5 ψ5









  n a

q

n

.

Rewrite it, under the base replacement q → q1/2 , as



2n n

q, a,

q/a, a2 ,



     n X n−k n (qn a; q)k (q1/2 /a; q1/2 )k −k/2 2n −q1/2 a k n ( 2 )−n/2 2 (− 1 ) q q a q = q . n−k −q n (q1−n /a; q)k (a; q1/2 )k k=−n

−a , −q1/2 ,

This equation matches with (2.2a) for λ = 0 and an q( 2 )−n/2 n

F (n) := G(n) :=

1 − q2n



2n n

q, a,



q/a, a2 ,

−a , −q1/2 ,

 −q1/2 a q , −q n

(q1/2 /a; q1/2 )n −1/2 n (aq ) ; (a; q1/2 )n

as well as the following reduction G(n) + G(−n) 1 − q2n

=

1 + q−n/2 (q1/2 /a; q1/2 )n 1 − q2n

(a; q1/2 )n

an .

Then the dual relation displayed in (2.2b) and (2.2c) reads as 1 + q−n/2 (qa; q)n (q1/2 /a; q1/2 )n

(a; q1/2 )n     a, a 2k q, × 1+n 1−n q k a, q a, q a k

1 − q2n (1/a; q)n

  n 2k X n+k−1 1−q a (−1)k n−k 1−a k=0  2 ak qk /2−kn q/a, −a , −q1/2 a q a2 , −q1/2 , −q k 1 − q2k =

which leads us to the following well-poised series identity: 8 φ7

a,



√

q√a, a,

√ −q√a, − a,

q/a, a2 ,

1 + q−n/2 (qa; q)n (q1/2 /a; q1/2 )n

=

1 + qn

−q1/2 a, −q1/2 ,

−a , −q,

(1/a; q)n (a; q1/2 )n

qn , 1 −n q a,



q−n q; q1/2 a q1+n a

 (2.17a)

.

(2.17b)

This identity can also be derived from the following special case of (2.6) 5 ψ5

√

√

q−n , −q−n , qn √ a, −qn a, q√ /a q; q2 1 +n 1 +n 1 −n 1−n q , −q , q / a , −q / a, a





taking into account of the relation G(−n) = qn/2 G(n).

q , q2 /a, −a, −qa 2 q a, a2 , −q, −q2

2

 =



  n a

n

q

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

Recall Watson’s q-analogue of the Whipple transformation [28] (cf. [18, III-18] also)

 2+n 2 −qa1/2 , b, c, d, e, q−n q a q ; −a1/2 , qa/b, qa/c , qa/d, qa/e, aqn+1 bcde    −n  qa, qa/bc q , b, c, qa/de = q 4 φ3 q;q . −n qa/b, qa/c qa / d , qa / e , q bc /a n

8 φ7

qa1/2 , a1/2 ,

a,



(2.18a)

(2.18b)

Applying this transformation to (2.17a) and (2.17b), we find the following balanced q-series identity

" 4

q−n ,

φ3

#   2   1 + qn/2 q 21 /a 1 a q2 q q ; q = n 2 q/a2 n a 1+q q/a n

1

qn ,

q 2 /a,

−q,

−q 2 ,

q/a

1

1

q2

!n (2.19)

a

which does not seem to have appeared previously in the literature. 2.8 Analogously, the following special case of (2.7a) and (2.7b) 5

√

√

q−n , −q−n , q1+n √ a, −q1+n a√ , q/a q4 , q2 /a, −qa, −q2 a 2 q q; q = (1 − q) 2+n 2+n 1−n 1 −n −q2 , −q3 , a, q2 a2 q , −q , q / a, −q / a, qa



ψ5





1/2

can be rewritten, under the base replacement q → q

(1 − q2n+1 ) n X

=

an q( )−n/2 n 2



1 + q1/2

(−1)k



k=−n−1

2n n

q, −q,





2n + 1 q n−k



n−k 2

  n a

n

q

, as

−q1/2 a, a,

q/a, −q3/2 ,







 −qa q qa2

n

(q1+n a; q)k (q1/2 /a; q1/2 )k −k/2 q (q1−n /a; q)k (q1/2 a; q1/2 )k

which matches with (2.2a) exactly for λ = 1 and an q( 2 )−n/2 n

F (n) :=



2n n

1 + q1/2

q, −q,

q/a, −q3/2 ,



−q1/2 a, a,

 −qa q , qa2 n

(q1/2 /a; q1/2 )n −1/2 n (aq ) ; G(n) := (q1/2 a; q1/2 )n as well as the following reduction G(n) + G(−n − 1) 1 − q2n+1

=

an q−n/2 1 + qn+1/2

(q1/2 /a; q1/2 )n . (q1/2 a; q1/2 )n

Then the dual relation displayed in (2.2b) and (2.2c) reads as q−n/2

  n 1+2k X (q2 a; q)n (q1/2 /a; q1/2 )n a k n+k 1−q = (− 1 ) n + 1 / 2 1 / 2 1 / 2 n − k 1+q (1/a; q)n (q a; q )n 1 − qa k=0     k k2 /2−kn  a q a, qa 2k q, q/a, −q1/2 a, −qa × 2+n 1−n q . 2 q 1/2 k −q, −q3/2 , q a, q a a , qa k k 1+q This results in the following well-poised series identity: 8

φ7

qa,



√

q qa, √ qa,

√ −q qa, √ − qa,

q/a, qa2 ,

−q1/2 a, −q3/2 ,

−qa, −q,

q1+n , q1−n a,



q−n q; q1/2 a q2+n a

 (2.20a)

1 + q1/2 (q2 a; q)n (q1/2 /a; q1/2 )n −n/2 q . 1 + qn+1/2 (1/a; q)n (q1/2 a; q1/2 )n

=

(2.20b)

Similarly, the last identity can also be confirmed by inverting the following special case of (2.8a) and (2.8b) 5

ψ5

q −n , q2+n ,



−q−n , −q2+n ,

√

q1+n √a, q1−n / a,

√ −q1+n √a, 1 −n −q / a ,

q/a q; q3 qa

n+1/2

in view of the reflection property G(−n − 1) = −q



G(n).



q − 1 q4 , q2 /a, −qa, −q2 a 2 q = −q2 , −q3 , a, q2 a2 q





  n a

n

q

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

3901

Furthermore, by applying Watson’s transformation (2.18a), (2.18b) to (2.20a), (2.20b), we find another terminating balanced q-series identity

" 4

φ3

q−n ,

"1 #  #  1 + q1/2 q 2 /a 1 qa2 2 q q; q = q 1 1 q/a2 n q/a2 1 + qn+ 2 q 2 a n

1

qn+1 ,

q 2 /a,

−q,

−q 2 ,

1

q/a

3

q2

!n .

a

(2.21)

As q-analogues of 3 F2 -series due to Watson [29] and Whipple [30], there are several terminating balanced q-series identities in the literature, which are mainly found by Andrews [1, Theorems 1 and 2], Jain [21, Eqs. 3.17 and 3.19] and Verma–Jain [25, Eqs. 1.1 and 1.2] (see also (3.3) and (3.4) in Section 3 as well as Gasper–Rahman [18, II-17 and 19]). However, we have failed to locate both (2.19) and (2.21) in these articles. 3. Reduced bilateral inversions and Bailey’s 6 ψ6 -series identity When x → 0, we find from Theorem 2 the following bilateral version of (1.2a) and (1.2b). Proposition 3 (Bilateral q-Inverse Series Relations). Let λ be a fixed nonnegative integer. Then for all n ∈ N0 , there hold the following inverse series relations λ+2n

(1 − q

n X

)F (n) =

  λ + 2n (−1) G(k), n−k k

(3.1a)

k=−n−λ

G(n) + (−1)λ G(−λ − n)

=

1 − qλ+2n

    n X n−k λ+n+k−1 (−1)k q 2 F (k); n−k

(3.1b)

k=0

provided that λ = 0 and 1. Furthermore, these inversions are also valid for λ > 1 under the additional conditions G(−1) = G(−2) = · · · = G(1 − λ) = 0. This proposition will be employed to derive the dual formulae of the very well-poised 6 ψ6 -series identity due to Bailey [6] (see also [18, II-33] and [16])

 √ √ qa2 b, c, d, e q√a, −q√a, q; 6 ψ6 a, − a, qa/b, qa/c , qa/d, qa/e bcde   q, qa, q/a, qa/bc , qa/bd, qa/be, qa/cd, qa/ce, qa/de q = qa/b, qa/c , qa/d, qa/e, q/b, q/c , q/d, q/e, qa2 /bcde ∞ provided that qa2 /bcde < 1 for convergence.

(3.2a)

(3.2b)

3.1 It is not hard to check from the last identity the following terminating sum: q, qab; q2

 (1 − q2n+1 )

[X n/2]

 n

:=

[qa, qb; q]n

q/a, 2n + 1 (1 − q4k+1 ) 2 n − 2k q a,



k=−[(n+1)/2]





q/b 2 q q2 b





This equation matches with (3.1a) for λ = 1 and the specifications q, qab; q2

 F (n) := G(n) :=

 n

[qa, qb; q]n

  

(1 − q

4k+1

0,

, 

q/a, ) 2 q a,

q/b 2 q q2 b





2

q2k (ab)k ,

k

n = 2k; n = 2k + 1;

as well as the following reduced expression

 q/a,    q2 a, G(n) − G(−n − 1) =   1 − q2n+1   1/a, qa,  1 −n 2

= qn

q/b 2 q q2 b





2

× q2m (ab)m ,

n = 2m m 1/b 2 q × q1+2m(m−1) (ab)m , n = 2m − 1 qb m  q a, q1−n b; q2 n . [qa, qb; q]n

2

q2k (ab)k . k

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W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

Then the dual relation displayed in (3.1b) reads as

 1−n 1−n       n X n−k q, qab; q2 k q a, q b; q2 n k n+k n2 2 (−1) q =q . n−k [qa, qb; q]k [qa, qb; q]n k=0 This leads directly to the following terminating q-analogue of Whipple’s summation formula (cf. Bailey [3, Section 3.4]) due to Andrews [1, Theorem 2] (cf. [24, Eq. 1.2] also): n X (q−n ; q)k (qn+1 ; q)k (qab; q2 )k

(qa; q)k (qb; q)k (q2 ; q2 )k

k=0

 k

q =q

n+1 2

  1−n

q

a, q1−n b; q2

 n

[qa, qb; q]n

.

(3.3)

We remark that its reversal gives the q-analogue of Watson’s summation formula (cf. Bailey [3, Section 3.3]) due to Jain [21, Eq. 3.17]: n X (a; q)k (b; q)k (q−2n ; q2 )k k (qa; q2 )n (qb; q2 )n q = . (q; q)k (qab; q2 )k (q−2n ; q)k (q; q2 )n (qab; q2 )n k=0

(3.4)

3.2 Consider the following terminating case of (3.2a) and (3.2b)

(1 − q2n+1 )

    [X n/3] (e; q3 )n (q2 /e; q3 )k 3k 2n + 1 := (−1)k (1 − q6k+1 ) 2 3 q 2 ek . n − 3k (e; q)n (q e; q )k k=−[(n+1)/3]

This matches with (3.1a) for λ = 1 and the specifications

(e; q3 )n F (n) := (e; q)n

and

G(n) :=

 

(1 − q6k+1 )



0,

(q2 /e; q3 )k q (q2 e; q3 )k



3k 2



ek ,

n = 3k; otherwise.

Simplifying the difference    2 (q /e; q3 )m 3m  2  q em ,  2 e; q3 )  ( q m G(n) − G(−n − 1) = 0,    1 − q2n+1 3 3m−1    (q/e; q )m q 2 em , (qe; q3 )m

n = 3m; n = 3m + 1; n = 3m − 1;

we find from the dual relation displayed in (3.1b) the following interesting identity:    2 (q /e; q3 )m 3m  2  q em ,      n  (q2 e; q3 )m X n−k (e; q3 ) n+k k (−1)k = 0, q 2   n−k  ( e ; q ) 3 k 3m−1  k=0   (q/e; q )m q 2 em , (qe; q3 )m

n = 3m; n = 3m + 1; n = 3m − 1.

3.3 Instead, consider another terminating case of (3.2a) and (3.2b)

(1 − q

2n+2

    [X n/3] 3 3 3k (qe; q3 )n 2n + 2 +k k 6k+2 (q /e; q )k ) := (1 − q ) 2 3 q 2 e . n − 3k (e; q)n ( q e ; q ) k k=−[(n+2)/3]

This matches with (3.1a) for λ = 2 and the following specifications

(qe; q3 )n F (n) := (e; q)n

and

G(n) :=

 

(1 − q6k+2 )



0,

(q3 /e; q3 )k q (q2 e; q3 )k



3k 2



+k k

e ,

In view of the following reduction    3 (q /e; q3 )m 3m  2 +m m  q e ,  2 e; q3 )  ( q m G(n) + G(−n − 2) = 0,    1 − q2n+2   (q/e; q3 )m 3m 2 −5m+2 m  − q e , (e; q3 )m

n = 3m; n = 3m − 1; n = 3m − 2;

n = 3k; otherwise.

(3.5)

W. Chu, C. Wang / Discrete Mathematics 309 (2009) 3888–3904

3903

the dual relation displayed in (3.1b) reads as another interesting identity:

    n 3 X 1 + n + k (qe; q )k n−2 k q (−1)k n−k (e; q)k k=0

   3 (q /e; q3 )m 3m  2 +m m  q e ,  (q2 e; q3 )  m = 0,    3 3m   − (q/e; q )m q 2 −5m+2 em , (e; q3 )m

n = 3m; n = 3m − 1;

(3.6)

n = 3m − 2.

3.4. Limiting cases Letting e → 0 in (3.5), we get the following formula:

     q6m2 −m , n X k 2n − k (−1)k q 2 = 0, k  6m2 −5m+1 k=0 −q ,

n = 3m; n = 3m + 1;

(3.7)

n = 3m − 1.

Similarly from (3.6), we have another formula:

     q6m2 +m , n X k 1 + 2n − k (−1)k q 2 = 0, k  6m2 −7m+2 k=0 −q ,

n = 3m; n = 3m − 1;

(3.8)

n = 3m − 2.

They can be unified as a single interesting identity [X m/2]

(−1)k



k=0





  k 2

m−k q k

=

(−1)[m/3] qm(m−1)/6 , 0,

m 6≡ 2 (mod 3); m ≡ 2 (mod 3).

(3.9)

Under the base replacement q → q−1 , this leads to another similar identity [X m/2]

(−1)k

k=0





 

m−k 3 q k

k 2

+k(2−m)

 =

(−1)[m/3] qm(1−m)/6 , 0,

m 6≡ 2 (mod 3); m ≡ 2 (mod 3).

(3.10)

Both identities can be found in Warnaar [27], who derived them from limiting cases of a cubic summation formula. Letting m = 3n + ε with ε = 0, 1, 2 and then shifting the summation index by k → k + n, we can reformulate the last identity further to the following one: [(nX +ε)/2]

(−1)k



k=−n

    ε + 2n − k 3 2k +k(2−ε) 1, q = n+k 0,

ε = 0, 1; ε = 2.

(3.11)

Berkovich and Garvan [7] (cf. [27, Eq. 3] also) discovered the case ε = 0 in their finite analogue of Dyson’s proof for Euler’s identity by the rank of partitions. Warnaar [27, Eq. 7] recorded the case ε = 1 separately. Both formulae may be considered as polynomial analogues of Euler’s pentagonal number theorem. Acknowledgements The authors are grateful to the anonymous referees for having pointed out and fixed the lacuna of Theorem 2 that appeared in an earlier version of the manuscript. References [1] G.E. Andrews, On q-analogues of the Watson and Whipple summations, SIAM J. Math. Anal. 7 (1976) 332–336. [2] G.E. Andrews, Connection coefficient problems and partitions, in: Relations Between Combinatorics and Other Parts of Mathematics, Ohio State Univ., Columbus, Ohio, 1978, in: Proc. Sympos. Pure Math., vol. 34, Amer. Math. Soc., Providence, R.I, 1979, pp. 1–24. [3] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935. [4] W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318–320. [5] W.N. Bailey, A note on certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 173–175. [6] W.N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. (Oxford) 7 (1936) 105–115. [7] A. Berkovich, F. Garvan, Some observations on Dyson’s new symmetries of partitions, J. Combin. Theory Ser. A 100 (1) (2002) 61–93. [8] D.M. Bressoud, Almost poised basic hypergeometric series, Proc. Indian Acad. Sci. Math. Sci. 97 (1–3) (1987) 61–66. [9] L. Carlitz, Some formulas of F.H. Jackson, Monatshefte für Math. 73 (1969) 193–198. [10] W. Chu, Inversion techniques and combinatorial identities, Boll. Unione. Mat. Ital. B- 7 (1993, SerieVII) 737–760. MR95e:33006. [11] W. Chu, Inversion techniques and combinatorial identities: Basic hypergeometric identities, Publicationes Math. Debrecen 44 (3–4) (1994) 301–320. [12] W. Chu, Basic hypergeometric identities: An introductory revisiting through the Carlitz inversions, Forum Math. 7 (1995) 117–129. MR95m:33015 & Zbl815:05009.

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[13] W. Chu, Inversion techniques and combinatorial identities: Balanced hypergeometric series, Rocky Mountain J. Math. 32 (2) (2002) 561–587. [14] W. Chu, Abel’s method on summation by parts and bilateral well-poised 3 ψ3 -series identities, in: S. Elaydi, et al. (Eds.), Difference Equations Special Functions and Orthogonal Polynomials, in: Proceedings of the International Conference, World Scientific Publishers, 2007, pp. 752–761. [15] W. Chu, Basic almost-poised hypergeometric series, Mem. Amer. Math. Soc. 135 (642) (1998) x+99 pp.; MR99b:33035. [16] W. Chu, Bailey’s very well-poised 6 ψ6 -series identity, J. Combin. Theory Ser. A 113 (6) (2006) 966–979. [17] W. Chu, C.A. Wei, Legendre Inversions and Balanced Hypergeometric Series Identities, Discrete Math. 308 (4) (2008) 541–549. [18] G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd ed., Cambridge University Press, Cambridge, 2004. [19] F.H. Jackson, Certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 167–172. [20] F.H. Jackson, Summation of q-hypergeometric series, Messenger Math. 50 (1921) 101–112. [21] V.K. Jain, Some transformations of basic hypergeometric functions II, SIAM J. Math. Anal. 12 (6) (1981) 957–961. [22] J. Riordan, Combinatorial Identities, John Wiley & Sons Inc., New York, 1968. [23] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966. [24] A. Verma, V.K. Jain, Certain summation formulae for q-series, J. Indian Math. Soc. 47 (1983) 71–85. [25] A. Verma, V.K. Jain, Some summation formulae for nonterminating basic hypergeometric series, SIAM J. Math. Anal. 16 (3) (1985) 647–655. [26] A. Verma, C.M. Joshi, Some remarks on summation of basic hypergeometric series, Houston J. Math. 5 (2) (1979) 277–294. [27] S.O. Warnaar, q-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem, Ramanujan J. 8 (4) (2004) 467–474. [28] G.N. Watson, A new proof of the Rogers–Ramanujan identities, J. London Math. Soc. 4 (1929) 4–9. [29] G.N. Watson, A note on generalized hypergeometric series, Proc. London Math. Soc. 23 (1925) xiii–xv. [30] F.J.W. Whipple, A group of generalized hypergeometric series: Relations between 120 allied series of type F [a, b, c , d, e], Proc. London Math. Soc. 23 (1925) 104–114.