Abel's Lemma on Summation by Parts and Basic Hypergeometric Series

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Advances in Applied Mathematics 39 (2007) 490–514 www.elsevier.com/locate/yaama

Abel’s lemma on summation by parts and basic hypergeometric series ✩

Wenchang Chu ∗

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China Received 21 May 2006; accepted 9 February 2007 Available online 6 June 2007

Abstract Basic hypergeometric series identities are revisited systematically by means of Abel’s lemma on summation by parts. Several new formulae and transformations are also established. The author is convinced that Abel’s lemma on summation by parts is a natural choice in dealing with basic hypergeometric series. © 2007 Elsevier Inc. All rights reserved.

MSC: primary 33D15; secondary 05A30

Keywords: Abel’s lemma on summation by parts; Basic hypergeometric series

In 1826, Abel [1] (see Bromwich [7, §20] and Knopp [22, §43] also) found the following ingenious lemma on summation by parts. For two arbitrary sequences {ak}k0 and {bk}k0,if we denote the partial sums by

n An = ak where n = 0, 1, 2,... k=0

then for two natural numbers m and n with m n, there holds the relation:

✩ The work carried out during the visit to Center for Combinatorics, Nankai University (2005). * Current address: Dipartimento di Matematica, Università degli Studi di Lecce, Lecce-Arnesano PO Box 193, 73100 Lecce, Italy. Fax: 39 0832 297594. E-mail address: [email protected].

n n akbk = Anbn+1 − Am−1bm + Ak(bk − bk+1). (1) k=m k=m

As a classical analytic instrument, this transformation formula has been fundamental in convergence test of infinite series (cf. [7, §80], [22, §43] and [25, §7.36] for example). However, it has not been utilized hitherto to evaluate finite and infinite summations. For this purpose, we need to reformulate slightly Abel’s transformation formula on summation by parts. For an arbitrary complex sequence {τk}, define the backward and forward difference operators ∇ and · , respectively, by

∇τk = τk − τk−1 and · τk = τk − τk+1 (2) where · is adopted for convenience in the present paper, which differs from the usual operator only in the minus sign. Then Abel’s lemma on summation by parts for unilateral and bilateral series may be reformu- lated respectively as

+∞ +∞ Bk ∇ Ak =[AB]+ − A−1B0 + Ak · Bk, (3a) k=0 k=0 +∞ +∞ Bk ∇ Ak =[AB]+ −[AB]− + Ak · Bk. (3b) k=−∞ k=−∞

Both formulae just displayed hold for terminating series and nonterminating series, provided, in the latter case, that one of both series is convergent and there exist the limits [AB]± := limn→±∞ AnBn+1.

Proof. Let m and n be two integers. According to the deﬁnition of the backward difference, we have

n n n n Bk ∇ Ak = Bk{Ak − Ak−1}= AkBk − Ak−1Bk. k=m k=m k=m k=m

Replacing k by k + 1 for the last sum, we get the following expression:

n n Bk ∇ Ak = AnBn+1 − Am−1Bm + Ak{Bk − Bk+1} k=m k=m n = AnBn+1 − Am−1Bm + Ak · Bk k=m which is essentially the same as Abel’s original transformation (1) on summation by parts. For n →+∞, the two cases m = 0 and m →−∞of the last equation result in the modiﬁed Abel’s lemma respectively for unilateral and bilateral series. 2 492 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Recently, Abel’s lemma on summation by parts has successfully been used to give new proofs of Bailey’s bilateral 6ψ6-series identity by Chu [12] and terminating well-poised q-series identities by Chu–Jia [13]. The objective of the present work is to explore systematically the applications of Abel’s lemma on summation by parts to basic hypergeometric series identities. Several q-series identities will be exemplified in a unified manner by means of Abel’s lemma on summation by parts such as the q-binomial theorem, Ramanujan’s bilateral 1ψ1-series identity, q-Gauss summation theorem, q-Pfaff–Saalschütz summation formula, q-Dougall–Dixon 6φ5- series identity and Jackson’s very well-poised 8φ7-series identity. In addition to providing new proofs for the identities just mentioned, we shall also establish few new summation and transformation formulae (even though this is not the main concern of the present paper). They will show that as a classical analytic weapon, Abel’s lemma on summation by parts is indeed a very natural and powerful method in dealing with basic hypergeometric series identities. In order to facilitate the readability of the paper, we reproduce the notations of q-shifted factorial and basic hypergeometric series. For two complex x and q, the shifted-factorial of order n with base q is defined by ; = − − ··· − n−1 = ; n; (x q)n (1 x)(1 xq) 1 xq (x q)∞/ xq q ∞ where n is assumed in the last equation, respectively, to be a natural number for the polynomial in the middle and integral for the fraction at the end. Its fractional form is abbreviated compactly to α, β, . . . , γ (α; q) (β; q) ···(γ ; q) q = n n n . ; ; ··· ; A, B, ..., C n (A q)n(B q)n (C q)n Following Bailey [6] and Slater [24], the basic hypergeometric series and the corresponding bilateral series are defined, respectively, by

+∞ a0,a1, ..., am n a0,a1, ..., am 1+mφm q; z = z q , b1, ..., bm q, b1,...,bm n=0 n +∞ a1,a2, ..., am n a1,a2, ..., am mψm q; z = z q b1,b2, ..., bm b1,b2, ..., bm n=−∞ n where the base q will be restricted to |q| < 1 for nonterminating q-series.

1. The terminating q-binomial theorem

Deﬁne the function fm(x) by the terminating q-binomial sum

m k m k k f (x) := (−1) q(2)x . m k k=0

For the two sequences given by + k m − 1 k 1 k A := (−1) q( 2 ) and B := x k k k W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 493 it is trivial to check A−1 = 0 and the differences k m k k ∇A = (−1) q(2) and · B = (1 − x)x . k k k

Then by means of the modiﬁed Abel’s lemma on summation by parts, we can manipulate fm(x) as follows:

m m fm(x) = Bk ∇ Ak = Ak · Bk k=0 k=0 m−1 k m − 1 k k = (1 − x) (−1) q(2)(qx) k k=0 which reads as the following relation

fm(x) = (1 − x)fm−1(qx).

Iterating this equation m-times, we ﬁnd the recurrence relation m fm(x) = (x; q)m × f0 q x .

On account of the fact that f0(x) = 1, we have the q-ﬁnite difference formula.

Proposition 1 (The terminating q-binomial theorem: [17, II-4]).

m k m k k (x; q) = (−1) q(2)x . m k k=0

Now replacing m by m + n and x by xq−m respectively, and then noting the relation − − − m+1 m ; = m ; ; = − m ( 2 ) m ; ; q x q m+n q x q m(x q)n ( 1) q x (q/x q)m(x q)n (4) we can reformulate the q-binomial theorem as

m+n − k−m m + n k m k−m (x; q) (q/x; q) = (−1) q( 2 )x n m k k=0 which becomes, under summation index substitution k → m + k, the following ﬁnite form of the Jacobi triple product identity

n k m + n k k (x; q) (q/x; q) = (−1) q(2)x . (5) n m m + k k=−m 494 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Letting m, n →∞in (5) and recalling the limiting relation

+ ; m n = (q q)m+n −m,n−−−−→∞→ 1 | | + where q < 1 m k (q; q)m+k(q; q)n−k (q; q)∞ we recover the famous Jacobi triple product identity (cf. [4, §10.4] and [17, II-28])

+∞ k k k [q,x,q/x; q]∞ = (−1) q(2) x for |q| < 1. (6) k=−∞

2. Euler’s ﬁrst q-exponential formula

Deﬁne the function g(x) by

∞ xk g(x) := where |x| < 1. (q; q)k k=0

For the two sequences given by

1 k Ak := and Bk := x (q; q)k it is easy to compute two extreme values

A−1B0 =[AB]+ = 0 and the ﬁnite differences

k q k ∇Ak = and · Bk = (1 − x) x . (q; q)k

In view of the modiﬁed Abel’s lemma on summation by parts, we can manipulate g(x) as follows:

∞ ∞ 1 1 g(x) = A · B = B ∇ A 1 − x k k 1 − x k k k=0 k=0 ∞ 1 (qx)k g(qx) = = . 1 − x (q; q)k 1 − x k=0

Iterating this process m-times, we ﬁnd the following recurrence relation

g(qmx) g(x) = . (x; q)m

Letting m →∞and noting that g(0) = 1 in the last relation, we derive the ﬁrst q-exponential expansion formula. W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 495

Proposition 2 (The ﬁrst q-exponential function: [17, II-1]).

∞ 1 xk = where |x| < 1. (x; q)∞ (q; q)k k=0

3. Euler’s second q-exponential formula

Deﬁne the function h(x) by

∞ k (−1)kq(2)xk h(x) := . (q; q)k k=0

For the two sequences given by

+ − k (k 1) ( 1) q 2 k Ak := and Bk := x (q; q)k it is not hard to verify two extreme values

A−1B0 =[AB]+ = 0 and the ﬁnite differences

− k (k) ( 1) q 2 k ∇Ak = and · Bk = (1 − x) x . (q; q)k According to the modiﬁed Abel’s lemma on summation by parts, we can manipulate h(x) as follows: ∞ ∞ h(x) = Bk ∇ Ak = Ak · Bk k=0 k=0 ∞ k k k (−1) q(2)(qx) = (1 − x) = (1 − x) h(qx). (q; q)k k=0

Iterating this process m-times, we ﬁnd the following recurrence relation m h(x) = (x; q)m × h q x .

Letting m →∞and noting that h(0) = 1 in the last relation, we derive the second q-exponential function expansion.

Proposition 3 (The second q-exponential function: [17, II-2]).

∞ k (−1)kq(2)xk (x; q)∞ = . (q; q)k k=0 496 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

4. The nonterminating q-binomial theorem

Deﬁne the two sequences by

; ; (qaz q)k (a q)k k Ak := and Bk := z . (q; q)k (az; q)k

It is not difﬁcult to calculate two extreme values

A−1B0 =[AB]+ = 0 with |z| < 1 and the ﬁnite differences ; − ; (az q)k k 1 z (a q)k k ∇Ak = q and · Bk = z . (q; q)k 1 − az (qaz; q)k

Applying the modiﬁed Abel’s lemma on summation by parts, we can manipulate the 1φ0-series as follows:

∞ a 1 − az φ q; z = A · B 1 0 − 1 − z k k k=0 ∞ 1 − az = B ∇ A 1 − z k k k=0 ∞ 1 − az (a; q) = k (qz)k 1 − z (q; q)k k=0 which results in the following relation a 1 − az a φ q; z = φ q; qz . 1 0 − 1 − z 1 0 −

The reader can ﬁnd an alternative derivation of this relation in [17, §1.3.1]. Iterating the last equation m-times, we ﬁnd the recurrence relation (az; q) a ; = m × a ; m 1φ0 − q z 1φ0 − q q z . (z; q)m

Letting m →∞in the last relation, we derive the q-binomial expansion formula.

Proposition 4 (The nonterminating q-binomial theorem: [17, II-3]). a (az; q)∞ 1φ0 q; z = where |z| < 1. − (z; q)∞ W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 497

5. Ramanujan’s bilateral 1ψ1-series identity

Deﬁne the two sequences by ; (a q)k k Ak := and Bk := z . (c; q)k

We have easily two extreme values

[AB]+ =[AB]− = 0 with |c/a| < |z| < 1 and the ﬁnite differences − ; 1 c/a (a/q q)k k k ∇Ak = q and · Bk = (1 − z) z . 1 − q/a (c; q)k

By means of the modiﬁed Abel’s lemma on summation by parts for nonterminating bilateral series, we can manipulate the bilateral 1ψ1-series as follows:

+∞ +∞ a A · B B ∇ A ψ q; z = k k = k k 1 1 c 1 − z 1 − z k=−∞ k=−∞ +∞ 1 − c/a (a/q; q) = k (qz)k (1 − z)(1 − q/a) (c; q) k=−∞ k 1 − c/a a/q = ψ q; qz . (1 − z)(1 − q/a)1 1 c

Andrews and Askey [3] (see also [17, §5.2]) have established this recurrence relation in a differ- ent manner. But the derivation presented here is more direct and simple. Iterating this process n-times, we ﬁnd the following recurrence relation ; n a (c/a q)n a/q n 1ψ1 q; z = × 1ψ1 q; q z . (7) c (z; q)n(q/a; q)n c

Alternatively, for the two sequences deﬁned by

; k := k := (a q)k c Ck (az/c) and Dk k (c; q)k a we can compute two extreme values

[CD]+ =[CD]− = 0 with |z| < 1 and the ﬁnite differences − ; k ∇ ={ − } k · = 1 c/a (a q)k c Ck 1 c/az (az/c) and Dk k . 1 − c (qc; q)k a 498 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Applying the modiﬁed Abel’s lemma on summation by parts for nonterminating bilateral series, we can similarly reformulate the 1ψ1-series as follows:

+∞ +∞ a D ∇ C C · D ψ q; z = k k = k k 1 1 c 1 − c/az 1 − c/az k=−∞ k=−∞ +∞ 1 − c/a (a; q) = k zk (1 − c)(1 − c/az) (qc; q) k=−∞ k 1 − c/a a = ψ q; z . (1 − c)(1 − c/az) 1 1 qc

Iterating this process m-times, we derive another recurrence relation ; a ; = (c/a q)m × a ; 1ψ1 q z 1ψ1 m q z . (8) c (c; q)m(c/az; q)m q c

The combination of (7) with (8) yields ; + n a ; = (c/a q)m n × a/q ; n 1ψ1 q z 1ψ1 m q q z . (9) c (c; q)m(c/az; q)m(z; q)n(q/a; q)n cq

Letting m, n →∞and recalling Jacobi’s triple product identity (6)

+∞ n a/q n k k k ψ q; q z ⇒ (−1) q(2)(az) =[q,az,q/az; q]∞ 1 1 cqm k=−∞ we derive the bilateral 1ψ1-series identity.

Proposition 5 (Ramanujan’s bilateral 1ψ1-series identity: [14, §18] and [17, II-29]). a q, az, q/az, c/a 1ψ1 q; z = q where |c/a| < |z| < 1. c c, z, c/az, q/a ∞

The last formula can also be derived from (8) directly as follows. Letting m →+∞in (8), we have the relation a (c/a; q)∞ a 1ψ1 q; z = × 1ψ1 q; z . (10) c (c; q)∞(c/az; q)∞ 0

The 1ψ1-series on the right-hand side can be evaluated by ﬁrst putting c = q and then invoking the q-binomial theorem as a (c; q)∞(c/az; q)∞ a 1ψ1 q; z = × 1ψ1 q; z 0 (c/a; q)∞ c (q; q)∞(q/az; q)∞ a = × 1φ0 q; z (q/a; q)∞ − W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 499

(q; q)∞(az; q)∞(q/az; q)∞ = . (z; q)∞(q/a; q)∞

Substituting this into (10) results immediately in Ramanujan’s identity on 1ψ1-series displayed in the last proposition. For other proofs of this important formula, see [2,3], [16, §2] and [18–21].

6. q-Gauss summation theorem

Deﬁne the two sequences by

; ; k := (qb q)k 1 := (a q)k c Ak k and Bk k . (q; q)k b (c; q)k a

It is not hard to have two extreme values

A−1B0 =[AB]+ = 0 with |c/ab| < 1 and the ﬁnite differences

; − ; k ∇ = (b q)k 1 · = 1 c/a (a q)k c Ak k and Bk k . (q; q)k b 1 − c (qc; q)k a

According to the modiﬁed Abel lemma on summation by parts for nonterminating series, we can reformulate the following 2φ1-series:

∞ ∞ a, b c φ q; = B ∇ A = A · B 2 1 c ab k k k k k=0 k=0

− ∞ ; ; k = 1 c/a (a q)k(qb q)k c 1 − c (q; q)k(qc; q)k ab k=0 1 − c/a a,qb c = φ q; . 1 − c 2 1 qc ab

Iterating this process m-times, we ﬁnd the following recurrence relation ; m a, b ; c = (c/a q)m × a, q b ; c 2φ1 q 2φ1 m q . c ab (c; q)m q c ab

Letting m →∞in the last relation and then applying the q-binomial expansion formula, we derive the q-Gauss summation theorem.

Proposition 6 (The q-Gauss summation formula: [17, II-8]). a, b (c/a; q)∞(c/b; q)∞ 2φ1 q; c/ab = where |c/ab| < 1. c (c; q)∞(c/ab; q)∞ 500 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

The formula just displayed can alternatively be proved as follows. For the two sequences given by (qab/c; q) c k a, b C := k D := q . k ; and k (q q)k ab c, ab/c k

It is not difﬁcult to check two extreme values

C−1D0 =[CD]+ = 0 with |c/ab| < 1 and the ﬁnite differences (ab/c; q) c k a, b (1 − c/a)(1 − c/b) ∇C = k · D = qk q . k ; and k − − (q q)k ab qc, qab/c k (1 c)(1 c/ab)

Then we can similarly reformulate the 2φ1-series as follows:

∞ ∞ a, b c φ q; = D ∇ C = C · D 2 1 c ab k k k k k=0 k=0

− − ∞ ; ; k = (1 c/a)(1 c/b) (a q)k(b q)k qc (1 − c)(1 − c/ab) (q; q)k(qc; q)k ab k=0 (1 − c/a)(1 − c/b) a, b qc = φ q; . (1 − c)(1 − c/ab) 2 1 qc ab

Iterating this process m-times, we ﬁnd the following recurrence relation m a, b ; c = c/a, c/b × a, b ; q c 2φ1 q q 2φ1 m q . c ab c, c/ab m q c ab

Letting m →∞in the last relation, we recover again the q-Gauss summation theorem displayed in Proposition 6.

7. q-Pfaff–Saalschütz summation formula

For the shifted factorial fractions with two pairs of balanced parameters, it is not hard to check the following differences.

Lemma 7 (Balanced differences). qbλ, qd/λ bλ, d/λ (1 − λ)(1 − λb/d) ∇ q = qk q , − − qb, qd k bq, dq k (1 λb)(1 λ/d) bλ, d/λ bλ, d/λ (1 − λ)(1 − λb/d) d · q = qk q . − − b, d k bq, dq k (1 b)(1 d) λ W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 501

In particular, with the two sequences given by qx, qa/x qa, qa/bc Ak := q and Bk := q q, qa k qa/b, qa/c k we have the boundary condition qx, qa/x, qa/bc A−1B0 = 0 and [AB]+ = q q, qa/b, qa/c ∞ as well as the ﬁnite differences x, a/x qa, qa/bc (1 − b)(1 − c)(qa/bc) ∇A = qk q · B = qk q . k and k 2 2 − − q, qa k q a/b, q a/c k (1 qa/b)(1 qa/c)

Then the modiﬁed Abel’s lemma on summation by parts enables us to manipulate the following nonterminating balanced series: x, a/x, qa/bc φ q; q 3 2 qa/b, qa/c ∞ ∞ qx, qa/x, qa/bc = Bk ∇ Ak =[AB]+ + Ak · Bk = q q, qa/b, qa/c ∞ k=0 k=0 (1 − b)(1 − c)(qa/bc) qx, qa/x, qa/bc + φ q; q . (1 − qa/b)(1 − qa/c) 3 2 q2a/b, q2a/c

The last balanced 3φ2-series can be derived from the 3φ2-series on the left under parameter replacements a → q2a, b → qb, c → qc and x → qx. Iterating this process m-times, we can proceed the following: x, a/x, qa/bc φ q; q 3 2 qa/b, qa/c m m m = b, c qa q x, q a/x, qa/bc ; q 3φ2 1+m 1+m q q qa/b, qa/c m bc q a/b, q a/c − m1 k 1+k 1+k + b, c qa q x, q a/x, qa/bc q 1+k 1+k q . qa/b, qa/c bc q, q a/b, q a/c ∞ k=0 k

Rewriting the last factorial fraction 1+k 1+k q x, q a/x, qa/bc = qa/b, qa/c × qx, qa/x, qa/bc 1+k 1+k q q q q, q a/b, q a/c ∞ qx, qa/x k q, qa/b, qa/c ∞ we establish the following balanced transformation. 502 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Proposition 8 (Balanced series transformation). x, a/x, qa/bc φ q; q 3 2 qa/b, qa/c m m m = b, c qa q x, q a/x, qa/bc ; q 3φ2 1+m 1+m q q qa/b, qa/c m bc q a/b, q a/c m−1 k qx, qa/x, qa/bc b, c qa + q q . q, qa/b, qa/c ∞ qx, qa/x bc k=0 k

Letting x = q−m, we get the q-Pfaff–Saalschütz summation formula.

Corollary 9 (q-Pfaff–Saalschütz summation formula [9, Eq. 2.2a]). q−m,qma, qa/bc b, c qa m 3φ2 q; q = q . qa/b, qa/c qa/b, qa/c m bc

It is equivalent to the following original form (cf. [17, II-12]): −n q ,a, b ; = c/a, c/b 3φ2 1−n q q q . c, q ab/c c, c/ab n When m →∞, we derive the following transformation formula (cf. [17, III-10]).

Corollary 10 (Balanced series transformation: |qa/bc| < 1). ∞ k x, a/x, qa/bc qx, qa/x, qa/bc b, c qa 3φ2 q; q = q q . qa/b, qa/c q, qa/b, qa/c ∞ qx, qa/x bc k=0 k

8. Very well-poised 6φ5-series identity

For the shifted factorial fractions with two pairs of well-poised parameters, it is not difﬁcult to verify the following differences.

Lemma 11 (Well-poised differences). qb, qd A k 1 − Aq2k b, d A k ∇ q = q − qA/b, qA/d k bd 1 A qA/b, qA/d k bd − − × (1 A)(1 bd/A); (1 − b)(1 − d) b, d A k 1 − Aq2k b, d A k · q = q − A/b, A/d k bd 1 A qA/b, qA/d k bd (1 − A)(1 − A/bd) × . (1 − A/b)(1 − A/d) W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 503

In particular, with the two sequences given by qa, qc 1 k b, d qa k Ak := q and Bk := q q, qa/c k c qa/b, qa/d k bd we have the boundary condition

A−1B0 =[AB]+ = 0 with |qa/bcd| < 1 as well as the ﬁnite differences 1 − aq2k a, c 1 k ∇A = q , k − 1 a q, qa/c k c + 1 − aq2k 1 b, d qa k (1 − qa)(1 − qa/bd) · B = q . k − 2 2 − − 1 qa q a/b, q a/d k bd (1 qa/b)(1 qa/d)

Then by means of the modiﬁed Abel’s lemma on summation by parts, we can reformulate the following nonterminating very well-poised series: √ √ a, q a, −q a, b, c, d qa φ √ √ q; 6 5 a, − a, qa/b, qa/c, qa/d bcd ∞ ∞ (1 − qa)(1 − qa/bd) = B ∇ A = A · B = k k k k (1 − qa/b)(1 − qa/d) k=0 k=0 √ √ qa, q qa, −q qa, b, qc, d qa × φ √ √ q; . 6 5 qa, − qa, q2a/b, qa/c, q2a/d bcd

Observe that the last 6φ5-series can also be obtained from the 6φ5-series displayed in the ﬁrst line with a and c being replaced by qa and qc respectively. Iterating this process m-times, we establish the following relation.

Proposition 12 (Very well-poised transformation).

√ √ a, q a, −q a, b, c, d qa φ √ √ q; 6 5 a, − a, qa/b, qa/c, qa/d bcd √ √ qma, q qma, −q qma, b, qmc, d qa = φ √ √ q; 6 5 qma, − qma, q1+ma/b, qa/c, q1+ma/d bcd qa, qa/bd × q provided |qa/bcd| < 1. qa/b, qa/d m

−m When c = q , we recover from this proposition the terminating very well-poised 6φ5-series identity. 504 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Corollary 13 (Terminating very well-poised 6φ5-series identity: [17, II-21]). √ √ − + a, q a, −q a, b, d, q m q1 ma qa, qa/bd φ √ √ q; = q . 6 5 − 1+m a, a, qa/b, qa/d, q a bd qa/b, qa/d m

Letting m →∞in Proposition 12, we derive the following transformation

√ √ a, q a, −q a, b, c, d qa φ √ √ q; 6 5 a, − a, qa/b, qa/c, qa/d bcd qa, qa/bd b, d qa = q 2φ1 q; . qa/b, qa/d ∞ qa/c bcd

Evaluating the last 2φ1-series by means of the q-Gauss summation theorem (cf. [17, II-8]), we get the following nonterminating very well-poised 6φ5-series identity:

Corollary 14 (Nonterminating very well-poised 6φ5-series identity: [10, §2] and [17, II-20]). √ √ a, q a, −q a, b, c, d qa φ √ √ q; 6 5 a, − a, qa/b, qa/c, qa/d bcd qa, qa/bc, qa/bd, qa/cd = q provided |qa/bcd| < 1. qa/b, qa/c, qa/d, qa/bcd ∞

Alternatively, deﬁne the two sequences by k 2 2 := qa, bcd/a qa := b, c, d, q a /bcd Ck 2 2 q and Dk q . q, q a /bcd k bcd qa/b, qa/c, qa/d, bcd/qa k

We can verify the boundary condition

C−1D0 =[CD]+ = 0 with |qa/bcd| < 1 as well as the ﬁnite differences 1 − aq2k a, bcd/qa qa k ∇C = q , k − 2 2 1 a q, q a /bcd k bcd + 1 − aq2k 1 b, c, d, q2a2/bcd · D = q qk k − 2 2 2 1 qa q a/b, q a/c, q a/d, bcd/a k (1 − qa)(1 − qa/bc)(1 − qa/bd)(1 − qa/cd) × . (1 − qa/b)(1 − qa/c)(1 − qa/d)(1 − qa/bcd)

Applying the modiﬁed Abel’s lemma on summation by parts, we can manipulate the 6φ5-series as follows: W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 505 √ √ a, q a, −q a, b, c, d qa φ √ √ q; 6 5 a, − a, qa/b, qa/c, qa/d bcd ∞ ∞ = Dk ∇ Ck = Ck · Dk k=0 k=0 − − − − = (1 qa)(1 qa/bc)(1 qa/bd)(1 qa/cd) (1 − qa/b)(1 − qa/c)(1 − qa/d)(1 − qa/bcd) √ √ qa, q qa, −q qa, b, c, d q2a × φ √ √ q; . 6 5 qa, − qa, q2a/b, q2a/c, q2a/d bcd

Observe that the last 6φ5-series can also be obtained from the 6φ5-series displayed in the ﬁrst line with a being replaced by qa. Iterating this process m-times, we establish the following relation.

Proposition 15 (Very well-poised transformation). √ √ a, q a, −q a, b, c, d qa φ √ √ q; 6 5 a, − a, qa/b, qa/c, qa/d bcd √ √ qma, q qma, −q qma, b, c, d qm+1a = φ √ √ q; 6 5 qma, − qma, qm+1a/b, qm+1a/c, qm+1a/d bcd qa, qa/bc, qa/bd, qa/cd × q provided |qa/bcd| < 1. qa/b, qa/c, qa/d, qa/bd m

Letting m →∞in the last proposition recovers directly the nonterminating well-poised 6φ5- series identity displayed in Corollary 14. From the proofs of the q-Gauss summation theorem and the nonterminating well-poised 6φ5-series identity, we see that there is much ﬂexibility to choose two sequences when utilizing Abel’s lemma on summation by parts to evaluate inﬁnite series.

9. Jackson’s terminating very well-poised 8φ7-series identity

For the shifted factorial fractions with four pairs of well-poised parameters, we can show the following differences.

Lemma 16 (Well-poised differences: bcde = A2). qb, qc, qd, qe 1 − Aq2k b, c, d, e ∇ q = q qk qA/b, qA/c, qA/d, qA/e 1 − A qA/b, qA/c, qA/d, qA/e k k − − − − × (1 A)(1 A/bc)(1 A/bd)(1 A/cd) bcd ; (1 − b)(1 − c)(1 − d)(1 − e) A b, c, d, e 1 − Aq2k b, c, d, e · q = q qk − A/b, A/c, A/d, A/e k 1 A qA/b, qA/c, qA/d, qA/e k (1 − A)(1 − A/bc)(1 − A/bd)(1 − A/cd) × . (1 − A/b)(1 − A/c)(1 − A/d)(1 − A/bcd) 506 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Proposition 17 (Well-poised nonterminating series transformation: qa2 = bcdef ). √ √ a, q a, −q a, b, c, d, e, f φ √ √ q; q 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, qa/f √ √ qma, q qma, −q qma, b, c, d, qme, qmf = φ √ √ q; q 8 7 qma, − qma, q1+ma/b, q1+ma/c, q1+ma/d, qa/e, qa/f qa, qa/bc, qa/bd, qa/cd qa, b, c, d, qe, qf × q − q qa/b, qa/c, qa/d, qa/bcd m q, qa/b, qa/c, qa/d, qa/e, qa/f ∞ m−1 qa/bcd qa/bc,qa/bd,qa/cd × q qk. 1 − qa/bcd qe,qf,q2a/bcd k=0 k

Proof. For qa2 = bcdef , deﬁne the two sequences by qa, qe, qf, qa/ef b, c, d, qef Ak := q and Bk := q . q, qa/e, qa/f, qef k qa/b, qa/c, qa/d, a/ef k

According to Lemma 16, we have 1 qa,b,c,d,qe,qf A−1B0 = 0 and [AB]+ = q 1 − a/ef q, qa/b, qa/c, qa/d, qa/e, qa/f ∞ as well as the ﬁnite differences 1 − aq2k a, e, f, a/ef ∇A = q qk, k − 1 a q, qa/e, qa/f, qef k (1 − qa)(1 − qa/bc)(1 − qa/bd)(1 − qa/cd) · B = k (1 − qa/b)(1 − qa/c)(1 − qa/d)(1 − qa/bcd) + 1 − aq2k 1 b, c, d, qef × q qk. − 2 2 2 1 qa q a/b, q a/c, q a/d, qa/ef k

Then applying the modiﬁed Abel’s lemma on summation by parts, we can manipulate the series as follows: √ √ a, q a, −q a, b, c, d, e, f φ √ √ q; q 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, qa/f ∞ ∞ = Bk ∇ Ak =[AB]+ + Ak · Bk k=0 k=0 1 qa, b, c, d, qe, qf = q 1 − a/ef q, qa/b, qa/c, qa/d, qa/e, qa/f ∞ − − − − + (1 qa)(1 qa/bc)(1 qa/bd)(1 qa/cd) (1 − qa/b)(1 − qa/c)(1 − qa/d)(1 − qa/bcd) W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 507 √ √ qa, q qa, −q qa, b, c, d, qe, qf × φ √ √ q; q . 8 7 qa, − qa, q2a/b, q2a/c, q2a/d, qa/e, qa/f

Observe that the last 8φ7-series is essentially the same as original 8φ7-series under parameter replacements a → qa, e → qe and f → qf . Iterating this process m-times, we have the equation √ √ a, q a, −q a, b, c, d, e, f φ √ √ q; q 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, qa/f √ √ qma, q qma, −q qma, b, c, d, qme, qmf = φ √ √ q; q 8 7 qma, − qma, q1+ma/b, q1+ma/c, q1+ma/d, qa/e, qa/f m−1 qa, qa/bc, qa/bd, qa/cd qa, qa/bc, qa/bd, qa/cd × q + q qa/b, qa/c, qa/d, qa/bcd qa/b, qa/c, qa/d, qa/bcd m k=0 k 1 1+k 1+k 1+k × q a, b, c, d, q e, q f −k 1+k 1+k 1+k q . 1 − q a/ef q, q a/b, q a/c, q a/d, qa/e, qa/f ∞

Rewriting the last factorial fraction, we establish the transformation formula stated in the proposition. 2

−m Letting f = q in Proposition 17, we ﬁnd Jackson’s very well-poised 8φ7-summation formula.

Corollary 18 (Very well-poised 8φ7-identity: cf. [6, §8.3], [11, Eq. 1.5], [17, II-22] and [24, IV-8]). √ √ a, q a, −q a, b, c, d, e, q−m φ √ √ q; q 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, q1+ma qa, qa/bc, qa/bd, qa/cd + = q provided q1 ma2 = bcde. qa/b, qa/c, qa/d, qa/bcd m

Letting m →∞in Proposition 17, we ﬁnd the following transformation from a very well- poised 8φ7-series to a balanced 3φ2-series (cf. [17, III-36]).

Corollary 19 (Transformation from very well-poised 8φ7-series to balanced 3φ2-series). √ √ a, q a, −q a, b, c, d, e, f φ √ √ q; q 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, qa/f qa, qa/bc, qa/bd, qa/cd b, c, d = q 3φ2 q; q qa/b, qa/c, qa/d, qa/bcd ∞ qa/e, qa/f qa/bcd qa, b, c, d, qe, qf − q 1 − qa/bcd q, qa/b, qa/c, qa/d, qa/e, qa/f ∞ +∞ qa/bc, qa/bd, qa/cd × q qk provided qa2 = bcdef. qe, qf, q2a/bcd k=0 k 508 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

The informed reader may notice the difference of the last relation from Watson’s q-Whipple transformation formula (cf. [17, III-18]):

√ √ a, q a, −q a, b, c, d, e, q−n q2+na2 φ √ √ q; 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, aqn+1 bcde −n = qa, qa/bc q ,b,c,qa/de ; q 4φ3 −n q q . qa/b, qa/c n qa/d, qa/e, q bc/a

In Corollary 19, replacing e and f respectively by q−me and q1+ma2/bcde and then letting m → ∞, we recover the nonterminating very well-poised 6φ5-series identity stated in Corollary 14: √ √ a, q a, −q a, b, c, d qa qa, qa/bc, qa/bd, qa/cd 6φ5 √ √ q; = q a, − a, qa/b, qa/c, qa/d bcd qa/b, qa/c, qa/d, qa/bcd ∞ provided that the parameters are subject to |qa/bcd| < 1 for convergence.

10. 10φ9-Generalization of Jackson’s terminating very well-poised series identity

According to the linear combination

(λ − d)(1 − a/dλ) 1 − qkλ 1 − qka/λ = 1 − qkb 1 − qka/b (b − d)(1 − a/bd) (b − λ)(1 − a/bλ) + 1 − qkd 1 − qka/d (b − d)(1 − a/bd) we derive the following expression

√ √ a, q a, −q a, b, c, d, e, qλ, qa/λ, q−n φ √ √ q; q 10 9 a, − a, qa/b, qa/c, qa/d, qa/e, λ, a/λ, aqn+1 − − − − = (1 b)(1 a/b) (λ d)(1 a/dλ) (b − d)(1 − a/bd) (1 − λ)(1 − a/λ) √ √ a, q a, −q a, qb, c, d, e, q−n × φ √ √ q; q 8 7 a, − a, a/b, qa/c, qa/d, qa/e, aqn+1 − − − − + (1 d)(1 a/d) (b λ)(1 a/bλ) (b − d)(1 − a/bd) (1 − λ)(1 − a/λ) √ √ a, q a, −q a, b, c, qd, e, q−n × φ √ √ q; q . 8 7 a, − a, qa/b, qa/c, a/d, qa/e, aqn+1

n 2 When q a = bcde,two8φ7-series just displayed are both well poised and balanced and can therefore be evaluated by means of Jackson’s terminating very well-poised summation formula as W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 509 √ √ a, q a, −q a, qb, c, d, e, q−n φ √ √ q; q 8 7 a, − a, a/b, qa/c, qa/d, qa/e, aqn+1 qa, a/bc, a/bd, qa/cd = q ; a/b, qa/c, qa/d, a/bcd n √ √ a, q a, −q a, b, c, qd, e, q−n φ √ √ q; q 8 7 a, − a, qa/b, qa/c, a/d, qa/e, aqn+1 qa, qa/bc, a/bd, a/cd = q . qa/b, qa/c, a/d, a/bcd n

Substituting these evaluations into 10φ9-series expression and then simplifying the result, we obtain the following very well-poised series identity.

n 2 Theorem 20 (Very well-poised and balanced 10φ9-series identity: q a = bcde). √ √ a, q a, −q a, b, c, d, e, qλ, qa/λ, q−n φ √ √ q; q 10 9 a, − a, qa/b, qa/c, qa/d, qa/e, λ, a/λ, aqn+1 qa, qa/bc, qa/bd, qa/cd = q × Ω(b,d)+ Ω(d,b) qa/b, qa/c, qa/d, qa/bcd n where Ω(b,d) is deﬁned by

(λ − d)(1 − a/dλ) (1 − b)(1 − qna/b) (1 − a/bc)(1 − qna/bcd) Ω(b,d)= (1 − λ)(1 − a/λ) (b − d)(1 − qna/bd) (1 − qna/bc)(1 − a/bcd) and its symmetric sum reads as

Ω(b,d)+ Ω(d,b) − − − − n − − n = (1 λ/d)(1 a/dλ)(1 b)(1 q a/b) (1 a/bc)(1 q a/bcd) (1 − λ)(1 − a/λ)(1 − b/d)(1 − qna/bd) (1 − qna/bc)(1 − a/bcd) (λ − b)(1 − a/bλ) (1 − d)(1 − a/cd) (1 − qna/bc)(1 − qna/d) × 1 − . (λ − d)(1 − a/dλ) (1 − b)(1 − a/bc) (1 − qna/b)(1 − qna/cd)

Letting λ →∞in the theorem and reformulating the fractional difference

(1 − d)(1 − a/cd) (1 − qna/bc)(1 − qna/d) 1 − (1 − b)(1 − a/bc) (1 − qna/b)(1 − qna/cd) − − − = (d b)(1 a/bd)(1 a/cd) (1 − b)(1 − qna/b)(1 − qna/cd) (1 − qn)(1 − a/bcd)(1 − qna2/bcd) × qn + (1 − a/bc)(1 − a/bd)(1 − a/cd) we derive the following identity. 510 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

Proposition 21 (Very well-poised and 3-balanced 8φ7-series identity: Lakin [23, Eq. 7], see also [5] and [8, Eq. 3.3b]). √ √ a, q a, −q a, b, c, d, e, q−n φ √ √ q; q2 8 7 a, − a, qa/b, qa/c, qa/d, qa/e, aqn+1 (1 − qn)(1 − a/bcd)(1 − qna2/bcd) = qn + (1 − a/bc)(1 − a/bd)(1 − a/cd) qa, a/bc, a/bd, a/cd × q where qna2 = bcde. qa/b, qa/c, qa/d, a/bcd n

11. Bibasic transformation and symmetric formula

Deﬁne two sequences by factorial fractions pb, pc, pd, pe Ak := p , pa/b, pa/c, pa/d, pa/e k B, C, D, E Bk := q . A/B, A/C, A/D, A/E k

When a2 = bcde and A2 = BCDE, we can compute, by means of Lemma 16, the following differences 1 − ap2k b, c, d, e ∇A = p pk k − 1 a pa/b, pa/c, pa/d, pa/e k − − − − × bcd (1 a)(1 a/bc)(1 a/bd)(1 a/cd); a (1 − b)(1 − c)(1 − d)(1 − e) 1 − Aq2k B, C, D, E · B = q qk k − 1 A qA/B, qA/C, qA/D, qA/E k (1 − A)(1 − A/BC)(1 − A/BD)(1 − A/CD) × . (1 − A/B)(1 − A/C)(1 − A/D)(1 − A/BCD)

We can also evaluate the following two limits B, C, D, E pb, pc, pd, pe [AB]+ = q p , A/B, A/C, A/D, A/E ∞ pa/b, pa/c, pa/d, pa/e ∞ qB/A, qC/A, qD/A, qE/A b/a, c/a, d/a, e/a [AB]− = q p . q/B, q/C, q/D, q/E ∞ 1/b, 1/c, 1/d, 1/e ∞

According to the modiﬁed Abel’s lemma on summation by parts for bilateral series, we establish the following bilateral bibasic series transformation.

Theorem 22 (Bilateral bibasic series transformation). For the indeterminate p and q with 0 < |p| < 1 and with 0 < |q| < 1, there holds transformation W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 511

+∞ 1 − ap2n b, c, d, e B, C, D, E p pn q 1 − a pa/b, pa/c, pa/d, pa/e A/B, A/C, A/D, A/E n=−∞ n n − − − − = (a/bcd)(1 b)(1 c)(1 d)(1 e) (1 − a)(1 − a/bc)(1 − a/bd)(1 − a/cd) B, C, D, E pb, pc, pd, pe × q p A/B, A/C, A/D, A/E ∞ pa/b, pa/c, pa/d, pa/e ∞ qB/A, qC/A, qD/A, qE/A b/a, c/a, d/a, e/a − q p q/B, q/C, q/D, q/E ∞ 1/b, 1/c, 1/d, 1/e ∞ +∞ 1 − Aq2k B, C, D, E + q qk 1 − A qA/B, qA/C, qA/D, qA/E k=−∞ k pb, pc, pd, pe × p pa/b, pa/c, pa/d, pa/e k − − − − × (a/bcd)(1 b)(1 c)(1 d)(1 e) (1 − a)(1 − a/bc)(1 − a/bd)(1 − a/cd) − − − − × (1 A)(1 A/BC)(1 A/BD)(1 A/CD) (1 − A/B)(1 − A/C)(1 − A/D)(1 − A/BCD) provided that a2 = bcde and A2 = BCDE, which imply that the ﬁrst series is 2-balanced with respect to p and the second to q. Proof. We need only to verify the convergence. For the ﬁrst series, the truncated part n0 along the positive direction is convergent for |p| < 1 and |q| < 1, which make the series to be n comparable with the geometric series n0 p . By means of transformation

; n ; (x q)−n = y × (q/y q)n (y; q)−n x (q/x; q)n the truncated part n<0 of the ﬁrst series along the negative direction can be expressed under replacement n →−n as

∞ 1 − p2n/a b/a, c/a, d/a, e/a qB/A, qC/A, qD/A, qE/A p pn q . 1 − 1/a p/b, p/c, p/d, p/e q/B, q/C, q/D, q/E n=1 n n

This is again a convergent series for |p| < 1 and |q| < 1. Therefore the ﬁrst bilateral series with respect to n is convergent. Symmetrically, the second bilateral series with respect to k is convergent either. 2

In particular, letting e = a, we derive the following transformation between bibasic unilateral series.

Proposition 23 (Bibasic series transformation). For the indeterminate p and q with 0 < |p| < 1 and with 0 < |q| < 1, there holds transformation 512 W. Chu / Advances in Applied Mathematics 39 (2007) 490–514

∞ 1 − ap2n a, b, c, d B, C, D, E p pn q 1 − a p, pa/b, pa/c, pa/d A/B, A/C, A/D, A/E n=0 n n B, C, D, E pa, pb, pc, pd = q p A/B, A/C, A/D, A/E ∞ p, pa/b, pa/c, pa/d ∞ − − − − + (1 A)(1 A/BC)(1 A/BD)(1 A/CD) (1 − A/B)(1 − A/C)(1 − A/D)(1 − A/BCD) ∞ 1 − Aq2k B, C, D, E pa, pb, pc, pd × q qk p 1 − A qA/B, qA/C, qA/D, qA/E p, pa/b, pa/c, pa/d k=0 k k provided that a = bcd and A2 = BCDE, which imply that the ﬁrst series is 2-balanced with respect to p and the second to q.

Performing parameter replacements in the last transformation

− A → q 2m/A, − B → q mB/A, − C → q mC/A, − D → q mD/A, − E → q m and then reversing the ﬁnite series on the right-hand side by k → m − k, we get the following symmetric transformation.

Corollary 24 (Bibasic symmetric transformation). For complex parameters {a,b,c,d} and {A,B,C,D} satisfying a = bcd and A = BCD respectively, there holds bibasic symmetric transformation

m − 2n −m −m −m −m 1 ap a, b, c, d n q ,q B/A, q C/A, q D/A p p − − − − q 1 − a p, pa/b, pa/c, pa/d q m/A, q m/B, q m/C, q m/D n=0 n n m 1 − Aq2k A, B, C, D = q qk 1 − A q, qA/B, qA/C, qA/D k=0 k −m −m −m −m × p ,p b/a, p c/a, p d/a −m −m −m −m p p /a, p /b, p /c, p /d k pa, pb, pc, pd q, qA/B, qA/C, qA/D × p q . p, pa/b, pa/c, pa/d m qA, qB, qC, qD m W. Chu / Advances in Applied Mathematics 39 (2007) 490–514 513

In fact, if letting : p a, b, c, d := p, pa/b, pa/c, pa/d Ξ : p q A, B, C, D pa, pb, pc, pd m m 1 − ap2n a, b, c, d × p pn 1 − a p, pa/b, pa/c, pa/d n=0 n −m −m −m −m × q B/A, q C/A, q D/A, q −m −m −m −m q q /B, q /C, q /D, q /A n then we can reformulate the transformation in Corollary 24 as the following symmetric expression: p : a, b, c, d q : A, B, C, D Ξ = Ξ . q : A, B, C, D p : a, b, c, d

When p = q, the last symmetric relation reduces to the “split-poised” transformation due to Gasper [15, Eq. 2.11]: √ √ a, q a, −q a, b, c, d, q−m,q−mB/A, q−mC/A, q−mD/A φ √ √ q; q 10 9 a, − a, qa/b, qa/c, qa/d, q−m/A, q−m/B, q−m/C, q−m/D √ √ A, q A, −q A, B, C, D, q−m,q−mb/a, q−mc/a, q−md/a = φ √ √ q; q 10 9 A, − A, qA/B, qA/C, qA/D, q−m/a, q−m/b, q−m/c, q−m/d qa, qb, qc, qd, qA/B, qA/C, qA/D × q qA, qB, qC, qD, qa/b, qa/c, qa/d m where a = bcd and A = BCD are assumed as before.

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