The Abel Lemma and the q-gosper Algorithm

Transcription

1 The Abel Lemma and the q-gosper Algorithm Vincent Y. B. Chen, William Y. C. Chen 2, and Nancy S. S. Gu 3 Center for Combinatorics, LPMC Nankai University, Tianjin P. R. China ybchen@mail.nankai.edu.cn, 2 chen@nankai.edu.cn, 3 gu@nankai.edu.cn Abstract Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey s 6 ψ 6 bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the 6 ψ 6 sum. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the q-gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas. Keywords: the Abel lemma, Abel pairs, basic hypergeometric series, the q-gosper algorithm, Gosper pairs. AMS Classification: 33D5, 33F0 Introduction We follow the notation and terminology in 0. For q <, the q-shifted factorial is defined by a; q) = aq k ) and a; q) n = a; q), for n Z. aq n ; q) k=0 For convenience, we shall adopt the following notation for multiple q-shifted factorials: a, a 2,..., a m ; q) n = a ; q) n a 2 ; q) n a m ; q) n,

2 where n is an integer or infinity. In particular, for a nonnegative integer k, we have a; q) k = The unilateral) basic hypergeometric series r φ s is defined by rφ s a, a 2,..., a r ; q, z = b, b 2,..., b s k=0 aq k ; q) k..) a, a 2,..., a r ; q) k ) k q k 2) +s r z k,.2) q, b, b 2,..., b s ; q) k while the bilateral basic hypergeometric series r ψ s is defined by a, a 2,..., a r a, a 2,..., a r ; q) k rψ s ; q, z = ) k q b, b 2,..., b k 2) s r z k..3) s b, b 2,..., b s ; q) k Recently Chu 9 used the Abel lemma on summations by parts to give an elementary proof of Bailey s very well-poised 6 ψ 6 -series identity 5, see also, 0, Appendix II.33: qa 2, qa 2, b, c, d, e qa2 6ψ 6 ; q, a 2, a 2, aq/b, aq/c, aq/d, aq/e bcde where qa 2 /bcde <. = q, aq, q/a, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de; q) aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, qa 2 /bcde; q),.4) Let us give a brief review of Chu s approach. The Abel lemma on summation by parts is stated as A k B k B k ) = B k A k A k+ ).5) provided that the series on both sides are convergent and k A kb k is absolutely convergent. Based on the Abel lemma, Chu found a pair A k, B k ): and A k = B k = which leads to the following iteration relation: Ωa; b, c, d, e) = Ωaq; b, c, d, eq) b, c, d, q 2 a 2 /bcd; q) k aq/b, aq/c, aq/d, bcd/aq; q) k,.6) qe, bcd/a; q) ) k qa 2 k,.7) aq/e, q 2 a 2 /bcd; q) k bcde a e) aq) aq/bc) aq/bd) aq/cd) e a) aq/b) aq/c) aq/d) a 2 q/bcde),.8) 2

3 where Ωa; b, c, d, e) = 6 ψ 6 qa 2, qa 2, b, c, d, e a 2, a 2, aq/b, aq/c, aq/d, aq/e qa2 ; q,..9) bcde Because of the symmetries in b, c, d, e, applying the identity.8) three times with respect to the parameters a and d, a and c, a and b, we arrive at the following iteration relation: Ωa; b, c, d, e) = Ωaq 4 ; bq, cq, dq, eq) Again, iterating the above relation m times, we get a4 q 6 bcde aq4 b) c) d) e) a a 2 q/bcde; q) 4 aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de; q) 2 aq/b, aq/c, aq/d, aq/e; q) 3..0) Ωa; b, c, d, e) = Ωaq 4m ; bq m, cq m, dq m, eq m ) a4m q 6m2 bcde) aq4m b, c, d, e; q) m m a a 2 q/bcde; q) 4m aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de; q) 2m aq/b, aq/c, aq/d, aq/e; q) 3m..) Replacing the summation index k with k 2m, we obtain the transformation formula Ωa; b, c, d, e) = Ωa; bq m, cq m, dq m, eq m ) aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de; q) 2m..2) q/b, q/c, q/d, q/e, aq/b, aq/c, aq/d, aq/e; q) m a 2 q/bcde; q) 4m Setting m, Chu obtained the 6 ψ 6 summation formula.4) by Jacobi s triple product identity 0, Appendix II.28 since q k2 z k = q 2, qz, q/z; q 2 ),.3) lim Ωa; m bq m, cq m, dq m, eq m ) = lim m 6 ψ 6 = qa 2, qa 2, bq m, cq m, dq m, eq m a 2, a 2, aq m+ /b, aq m+ /c, aq m+ /d, aq m+ /e ; q, a2 q 4m+ bcde aq 2k a q2k2 k a 2k = a 3 ) k q k 2) a k = aq, q/a, q; q)..4)

4 This paper is motivated by the question of how to systematically compute Abel pairs for bilateral summation formulas. We find that the q-gosper is an efficient mechanism for this purpose. The q-gosper algorithm has been extensively studied. Koorwinder gave a rigorous description of the q-gosper algorithm in 4. Abramov- Paule-Petkovšek developed the algorithm qhyper for finding all q-hypergeometric solutions of linear homogeneous recurrences with polynomial coefficients. Later Böing- Koepf 7 gave an algorithm for the same purpose. The Maple package qsum.mpl was described by Böing-Koepf 7. In 8, Riese presented an generalization of the q-gosper algorithm to indefinite bibasic hypergeometric summations. Recall that a function t k is called a basic hypergeometric term if t k+ /t k is a rational function of q k. The q-gosper algorithm is devised to answer the question if there is a basic hypergeometric term z k for a given basic hypergeometric term t k such that t k = z k+ z k..5) We observe that for an iteration relation of a summation formula, the difference of the kth term of the both sides is a basic hypergeometric term for which the q-gosper algorithm can be employed. The main result of this paper is to present a general framework to deal with basic hypergeometric identities based on the q-gosper algorithm. We always start with an iteration relation. Then we use the q-gosper algorithm to generate a Gosper pair g k, h k ) if it exists. We next turn to the iteration relation and derive the desired identity by computing the limit value. Actually, once a Gosper pair g k, h k ) is obtained, one can easily compute the corresponding Abel pair. Indeed, the Abel pair for the 6 ψ 6 sum discovered by Chu 9 agrees with the Abel pair derived from the Gopser pair by using our approach. In general, our method is efficient for many classical summation formulas with parameters. As examples, we give Gosper pairs and Abel pairs of several well-known bilateral summation formulas including Ramanujan s ψ summation formula. In the last section we demonstrate that the idea of Gosper pairs can be applied to unilateral summation formulas as well. We use the q-gauss 2 φ summation formula as an example to illustrate the procedure to compute Gosper pairs. As another example, we derive the Gosper pair and the Abel pair of the 6 φ 5 summation formula. In comparison with a recent approach presented by Chen-Hou-Mu 8 for proving nonterminating basic hypergeometric series identities by using the q-zeilberger algorithm 4, 5, 7, 22, one sees that the approach we undertake in this paper does not rely on the introduction of the parameter n in order to establish recurrence relations, and only makes use of the q-gosper algorithm. 4

5 2 The Gosper Pairs for Bilateral Summations In spite of its innocent looking, the Abel lemma is intrinsic for some sophisticated bilateral basic hypergeometric identities. In this section, we introduce the notion of Gosper pairs and show that one may apply the q-gosper algorithm to construct Gosper pairs which can be regarded as certificates like the Abel pairs to justify iteration relations for bilateral summations. Furthermore, it is easily seen that one can compute the Abel pairs from Gosper pairs. Suppose that we have a bilateral series F ka, a 2,..., a n ) which has a closed form. Making the substitutions a i a i q or a i a i /q for some parameters a i, the closed product formula induces an iteration relation for the summation which can be stated as an identity of the form: F k a, a 2,..., a n ) = G k a, a 2,..., a n ). 2.) We assume that F ka, a 2,..., a n ) and G ka, a 2,..., a n ) are both convergent. We also assume that lim h k = lim h k. 2.2) k k We note that there are many bilateral summations with the above limit property. An Gosper pair g k, h k ) is a pair of basic hypergeometric terms such that g k h k = F k a, a 2,..., a n ), g k h k+ = G k a, a 2,..., a n ). Evidently, once a Gosper pair is derived, the identity 2.) immediately justified by the following telescoping relation: g k h k ) = g k h k+ ). 2.3) We are now ready to describe our approach. Let us take Ramanujan s ψ sum 0, Appendix II.29 as an example: a ψ b ; q, z = q, b/a, az, q/az; q), 2.4) b, q/a, z, b/az; q) where b/a < z <. There are many proofs of this identity, see, for example, Hahn, Jackson 3, Andrews 2, 3, Ismail 2, Andrews and Askey 4, Berndt 6 and Schlosser 9. 5

6 Proposition 2. The following is a Gosper pair for Ramanujan s ψ sum: g k = h k = azk+ a; q) k, az b) b; q) k bzk a; q) k. az b) b; q) k Step. Construct an iteration relation from the closed product form, namely, the right hand side of 2.4). Setting b to bq in 2.4), we get ψ a bq ; q, z = q, bq/a, az, q/az; q) bq, q/a, z, bq/az; q). 2.5) Define fa, b, z) = ψ a b ; q, z. 2.6) Comparing the right hand sides of 2.4) and 2.5) gives the following iteration relation see also 4) a ψ b ; q, z b/a) a = b) b/az) ψ bq ; q, z. 2.7) Notice that both sides of the above identity are convergent. Let F k a, b, z) and G k a, b, z) denote the kth terms of the left hand side and the right hand side summations in 2.7) respectively, that is, F k a, b, z) = a; q) k b; q) k z k and G k a, b, z) = b/a) b) b/az) F ka, bq, z). 2.8) Step 2. Apply the q-gosper algorithm to try to find a Gosper pair g k, h k ). It is essential to observe that F k a, b, z) G k a, b, z) is a basic hypergeometric term. In fact it can be written as bq k b/a ) a; q)k z k. b/az b; q) k+ Now we may employ the q-gosper algorithm for the following equation and we find a solution of simple form F k a, b, z) G k a, b, z) = h k+ h k, 2.9) h k = bzk a; q) k, 2.0) az b) b; q) k 6

7 which also satisfied the limit condition lim h k = lim h k = 0. k k As far as the verification of 2.7) is concerned, the existence of a solution h k and the limit condition 2.2) would guarantee that the identity holds. Now it takes one more step to compute the Gosper pair: g k = h k + F k a, b, z) = azk+ a; q) k. 2.) az b) b; q) k Step 3. Based on the iteration relation and the limit value, one can verify the summation formula. From the iteration relation 2.7), we may reduce the evaluation of the bilateral series ψ to a special case fa, b, z) = b/a; q) fa, 0, z). 2.2) b, b/za; q) Setting b = q in 2.2), we get fa, 0, z) = q, q/za; q) q/a; q) a; q) k q; q) k z k. Invoking the relation.), we see that /q, q) k = 0 for any positive integer k. Consequently, the above bilateral sum reduces to a unilateral sum. Exploiting the q-binomial theorem 0, Appendix II.3 we get the evaluation k=0 Hence the identity 2.4) follows from 2.2) and 2.4). a; q) k q; q) k z k = az; q) z; q), 2.3) fa, 0, z) = q, az, q/az; q) q/a, z; q). 2.4) It should be warned that it is not always the case that there is a solution h k to the equation 2.9) in general case. If one encounters this scenario, one should still have alternatives to try another iteration relations, as is done for the 3 ψ 3 sum in Example 2.4. Let us now examine how to generate an Abel pair A k, B k ) from a Gosper pair g k, h k ). Setting g k = A k B k and h k = A k B k, 2.5) then we see that B k B k = g k h k. 2.6) 7

8 Without loss of generality, we may assume that B 0 =. Iterating 2.6) yields an Abel pair A k, B k ). For the Ramanujan s ψ sum 2.4), we can compute the Abel pair by using the q-gosper algorithm. Proposition 2.2 The following is an Abel pair for Ramanujan s ψ sum: A k = B k = az a; q) k az b b; q) k az ) k. b ) k b, a It is a routine to verify A k, B k ) is indeed an Abel pair for the ψ sum. First we have az ) az ) k B k B k = b, 2.7) b A k A k+ = b/a) a; q) k b) b/az) bq; q) k ) k b. 2.8) a Then the iteration relation 2.7) is deduced from the Abel lemma: az a; q) k az b b; q) k ) k b az ) az ) k a b 2.9) b = az ) k b/a) b b) b/az) a; q) k bq; q) k ) k b. 2.20) a We next give some examples for bilateral summations. Example 2.3 The sum of a well-poised 2 ψ 2 series 0, Appendix II.30): b, c 2ψ 2 ; q, aq = aq/bc; q) aq 2 /b 2, aq 2 /c 2, q 2, aq, q/a; q 2 ), 2.2) aq/b, aq/c bc aq/b, aq/c, q/b, q/c, aq/bc; q) where aq/bc <. Write the kth term of the left hand side of 2.2) as b, c; q) k F k a, b, c) = aq ) k. 2.22) aq/b, aq/c; q) k bc 8

9 Substituting b with b/q in 2.2), we are led to the iteration relation aq/bc) aq 2 /b 2 ) F k a, b, c) = F k a, b/q, c). 2.23) + aq/bc) q/b) aq/b) Let G k a, b, c) = aq/bc) aq 2 /b 2 ) + aq/bc) q/b) aq/b) F ka, b/q, c). 2.24) Implementing the q-gosper algorithm, we obtain a Gosper pair g k = h k = The companion Abel pair is given below: b 2 cq k aq 2 ) aq + bc)bq k q) F ka, b, c), 2.25) bqc aq k ) aq + bc)bq k q) F ka, b, c). 2.26) b, c; q) k b 2 cq k aq 2 ) A k = aq/b, b 2 c/aq; q) k aq + bc) q + bq k ), 2.27) B k = b2 c/aq; q) k aq ) k. 2.28) aq/c; q) k bc Noticing that 2.2) is symmetric in b and c, we have F k a, b, c) = F k a, b/q, c/q) aq/bc) aq 2 /bc) aq 2 /b 2 ) aq 2 /c 2 ) + aq/bc) + aq 2 /bc) q/b) q/c) aq/b) aq/c). 2.29) Finally, we can reach 2.2) by iterating 2.29) infinitely many times along with Jacobi s triple product identity.3) as the limit case. Example 2.4 Bailey s sum of a well-poised 3 ψ 3 0, Appendix II.3): b, c, d 3ψ 3 q/b, q/c, q/d ; q, q = q, q/bc, q/bd, q/cd; q), 2.30) bcd q/b, q/c, q/d, q/bcd; q) where q/bcd <. Substituting d with d/q in 2.30), one obtains the iteration relation b, c, d 3ψ 3 q/b, q/c, q/d ; q, q bcd = q/bd) q/cd) q/d) q/bcd) 3 ψ 3 9 q, c, d/q q/b, q/c, q 2 /d q2 ; q,. 2.3) bcd

10 We remark that this sum is in fact an example for which the q-gosper algorithm does not succeed for the iteration relation derived from a straightforward substitution such as d dq or d d/q. Instead, using an idea of Paule 6 of symmetrizing a bilateral summation, we replace k by k on the left hand side of 2.30) to get 3ψ 3 b, c, d q/b, q/c, q/d q2 ; q, bcd. 2.32) Let F k b, c, d) be the average of the kth summands of 2.30) and 2.32), namely, and let b, c, d; q) k q ) k + q k F k b, c, d) =, 2.33) q/b, q/c, q/d; q) k bcd 2 G k b, c, d) = q/bd) q/cd) q/d) q/bcd) F kb, c, d/q). 2.34) With regard to F k b, c, d) G k b, c, d), the q-gosper algorithm generates a Gosper pair: g k = bdqk+ + cdq k+ bcd 2 q k q 2 + dq k+ + bcdq k+ bcd 2 q 2k q k+2 + q k )bcd q)q dq k ) F k b, c, d), 2.35) h k = db q k )c q k ) + q k )q bcd) dq k ) F kb, c, d), 2.36) which implies the iteration relation 2.3). Invoking the symmetric property of the parameters b, c and d, we have b, c, d 3ψ 3 q/b, q/c, q/d ; q, q = q/bc) q2 /bc) q/bd) q 2 /bd) bcd q/b) q/c) q/d) q/bcd) q/cd) q2 /cd) b/q, c/q, d/q q4 q 2 /bcd) q 3 /bcd) 3 ψ 3 ; q,. 2.37) q 2 /b, q 2 /c, q 2 /d bcd The above relation enables us to reduce the summation formula 2.30) to Jacobi s triple product identity. Example 2.5 A basic bilateral analogue of Dixon s sum 0, Appendix II.32: qa 3 2, b, c, d qa 2 4ψ 4 ; q, a 2, aq/b, aq/c, aq/d bcd where qa 3 2 /bcd <. = aq, aq/bc, aq/bd, aq/cd, qa 2 /b, qa 2 /c, qa 2 /d, q, qa; q) aq/b, aq/c, aq/d, q/b, q/c, q/d, qa 2, qa 2, qa 3 2 /bcd; q), 2.38) 0

11 For the above formula, we may consider the substitution d d/q in 2.38) which suggests the iteration relation qa 3 2, b, c, d qa 2 4ψ 4 ; q, a 2, aq/b, aq/c, aq/d bcd = aq/bd) aq/cd) qa 2 /d) aq/d) q/d) qa 3 2 /bcd) qa 2, b, c, d/q 4 ψ 4 a 2, aq/b, aq/c, aq 2 /d ; q, q2 3 a ) bcd Let and let F k a, b, c, d) = qa 2, b, c, d; q) k a 2, aq/b, aq/c, aq/d; q) k ) k qa ) bcd G k a, b, c, d) = aq/bd) aq/cd) qa 2 /d) aq/d) q/d) qa 3 2 /bcd) F ka, b, c, d/q). 2.4) By computation we obtain the Gosper pair: g k = abdqk+ acdq k+ + q 2 a a 2 q k+2 bcda 2 q k+ da 3 2 q k+ + bcd 2 q k + bcd 2 a 2 q 2k dq k q) + a 2 q k )bcd a 3 2 q) h k = F k a, b, c, d), 2.42) daq k c)aq k b) dq k ) + a 2 q k )bcd qa 3 2 ) F ka, b, c, d). 2.43) So the iteration relation 2.39) holds. From the symmetric property of the parameters b, c and d, we have qa 3 2, b, c, d qa 2 4ψ 4 ; q, a 2, aq/b, aq/c, aq/d bcd = aq/bc) aq2 /bc) aq/bd) aq 2 /bd) aq/cd) aq 2 /cd) aq/b) aq/c) aq/d) q/b) q/c) q/d) qa 3 2 /bcd) qa 2 /b) qa 2 /c) qa 2 /d) q 2 a 3 2 /bcd) q 3 a 3 2 /bcd) qa 2, b/q, c/q, d/q 4 ψ 4 a 2, aq 2 /b, aq 2 /c, aq 2 /d ; q, q4 3 a ) bcd

12 By iteration, it follows that 4ψ 4 qa 2, b, c, d a 2, aq/b, aq/c, aq/d 3 qa 2 ; q, bcd = aq/bc, aq/bd, aq/cd, qa 2 /b, qa 2 /c, qa 2 /d; q) aq/b, aq/c, aq/d, q/b, q/c, q/d, qa 3 2 /bcd; q) Ha), 2.45) where Ha) = qa 2 ; q) k q 3k 2) ) k qa 3 a ) 2 ; q) k Taking b = a 2 and c, d in 2.45) and by Jacobi s triple product identity.3), it can be verified that Ha) = q, aq, q/a; q) qa 2, qa 2 ; q), 2.47) which leads to 2.38). Example 2.6 Bailey s very well-poised 6 ψ 6 -series identity.4). Let us denote the kth term of.9) as Ω k a; b, c, d, e) = ) qa 2, qa 2, b, c, d, e; q) k qa 2 k. 2.48) a 2, a 2, aq/b, aq/c, aq/d, aq/e; q) k bcde Set F k = Ω k a; b, c, d, e) and G k = Ω k aq; b, c, d, eq) By computation, we find the following Gosper pair: a e) aq)bc aq)bd aq)cd aq) a)b aq)c aq)d aq)bcde a 2 q). 2.49) g k = abcdqk aq) eq k ) bcde a 2 q) aq 2k ) Ω ka; b, c, d, e), h k = e aqk )bcd a 2 q k+ ) bcde a 2 q)aq 2k ) Ω ka; b, c, d, e). 2.50) We note that the Abel pair derived from the above Gosper pair coincides with the Abel pair given by Chu 9. We next give the Gosper pair of a different iteration relation of the 6 ψ 6 series by setting e e/q in.4), that is, Ωa; b, c, d, e) = Ωa; b, c, d, e/q) aq/be) aq/ce) aq/de) aq/e) q/e) a 2 q/bcde). 2.5) 2

13 We will see that the above iteration has the advantage that it directly points to the identity.2) by taking into account the symmetries in b, c, d, e. On the other hand, the Gosper pair does not have a simple expression in this case. Set F k = Ω k a; b, c, d, e) and G k = Ω k a; b, c, d, e/q) aq/be) aq/ce) aq/de) aq/e) q/e) a 2 q/bcde). 2.52) From the above iteration relation, we obtain the Gosper pair: abceq k+ + abdeq k+ a 2 beq 2k+ + acdeq k+ a 2 ceq 2k+ a 2 deq 2k+ bcde 2 q k g k = bcde qa 2 )eq k q)aq 2k ) ) + abcde2 q 3k abcdeq 2k+ a 2 q 2 + a 2 eq k+ + a 3 q 2k+2 Ω bcde qa 2 )eq k q)aq 2k k a; b, c, d, e), 2.53) ) h k = qeb aqk )c aq k )d aq k ) bcde qa 2 ) aq 2k )eq k q) Ω ka; b, c, d, e). 2.54) Since the parameters b, c, d, e are symmetric in.9), we obtain Ωa; b, c, d, e) = Ωa; b/q, c/q, d/q, e/q) aq/bc) aq2 /bc) aq/bd) aq 2 /bd) aq/b) aq/c) aq/d) aq/e) aq/be) aq2 /be) aq/cd) aq 2 /cd) q/b) q/c) q/d) q/e) aq/ce) aq 2 /ce) aq/de) aq 2 /de) a 2 q/bcde) a 2 q 2 /bcde) a 2 q 3 /bcde) a 2 q 4 /bcde). 2.55) Again, the limit value can be given by Jacobi s triple product identity, so that we arrive at.4) in view of 2.55). The following is a 8 ψ 8 summation formula of Shukla 2. Note that the relation we aim to verify is not an iteration relation. Instead, we establish the identity based on the observation that the product formula contains the factors in Bailey s 6 ψ 6 identity. Example 2.7 Shukla s very-well-poised 8 ψ 8 summation: qa 2, qa 2, b, c, d, e, f, aq 2 /f 8ψ 8 a 2, a 2, aq/b, aq/c, aq/d, aq/e, aq/f, f/q = where a 2 /bcde <. bc/a) bd/a) be/a) bq/f) bf/aq) bcde/a 2 ) ) ; q, a 2 bcde f/bq) bf/aq) f/aq) f/q) q, aq, q/a, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de; q) aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, a 2 q/bcde; q), 2.56) 3

14 Observe that the right hand side of 2.56) can be rewritten as bc/a) bd/a) be/a) bq/f) bf/aq) bcde/a 2 ) where Ωa; b, c, d, e) denotes the 6 ψ 6 series as given by.9). ) f/bq) bf/aq) Ωa; b, c, d, e), f/aq) f/q) 2.57) Let F k denote the kth term of the left hand side of 2.56), namely, F k = ) qa 2, qa 2, b, c, d, e, f, aq 2 /f; q) k a 2 k, 2.58) a 2, a 2, aq/b, aq/c, aq/d, aq/e, aq/f, f/q; q) k bcde and let ) bc/a) bd/a) be/a) G k = Ω k a; b, c, d, e) bq/f) bf/aq) bcde/a 2 ) f/bq) bf/aq) f/aq) f/q), 2.59) where Ω k a; b, c, d, e) is the kth term of the 6 ψ 6 series Ωa; b, c, d, e), as given by 2.48). With the aid of the q-gosper algorithm for F k G k, we find h k = fb aqk )c aq k )d aq k )e aq k ) q k f q)f aq)bcde a 2 ) aq 2k ) Ω ka; b, c, d, e). 2.60) Thus 2.56) can be deduced from the telescoping relation 2.3) and the 6 ψ 6 formula.4). We now turn to two identities due to Schlosser derived by matrix inverse 20. Example 2.8 Let a, b, c, d, e, and u be indeterminates. Then provided a 2 /bcde <. aq 2k b, c; q) k dq, eq; q) k a 2 a) aq/b, aq/c; q) k aq/d, aq/e; q) k bcde de/a) ) uqk ) a 2 q k /bcu) a/bc) dq k ) eq k ) = q, aq, q/a, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, a/de; q) q/b, aq/b, q/c, aq/c, /d, aq/d, /e, aq/e, a 2 q/bcde; q) a/cu) bu/a), 2.6) b a/c) ) k As in the preceding example, the right hand side of 2.6) contains the product for the 6 ψ 6 sum. We may proceed in the same manner. Write the kth term of the left 4

15 hand side of 2.6) as F k, that is, and let F k = aq2k a) ) b, c; q) k dq, eq; q) k a 2 k aq/b, aq/c; q) k aq/d, aq/e; q) k bcde de/a) uqk ) a 2 q k /bcu) a/bc) dq k ) eq k ) ), 2.62) G k = a/cu) bu/a) a/de) Ω k a; b, c, d, e), 2.63) b a/c) /d) /e) where Ω k is the same notation as in the proceeding example. Employing the q-gosper algorithm for F k G k, we obtain h k = b aqk )c aq k )d aq k )e aq k ) a d) e)a bc) aq 2k )q k Ω ka; b, c, d, e). 2.64) So we get the desired identity. The next example is a 8 ψ 8 summation formula which can be verified by using our method. It turns out that the limit identity is a special case of Bailey s 6 ψ 6 sum. Example 2.9 Let a, b, c, and d be indeterminates, let j be an arbitrary integer and N a nonnegative integer. Then qa 2, qa 2, b, c, dq j, aq j /c, aq +N /b, aq N /d 8ψ 8 ; q, q a 2, a 2, aq/b, aq/c, aq j /d, cq +j, bq N +N, dq = aq/bc, cq/b, dq, dq/a; q) N cdq/a, dq/c, q/b, aq/b; q) N cd/a, bd/a, cq, cq/a, dq +N /b, q N ; q) j q, cq/b, d/a, d, bcq N /a, cdq +N /a; q) j q, q, aq, q/a, cdq/a, aq/cd, cq/d, dq/c; q) cq, q/c, dq, q/d, cq/a, aq/c, dq/a, aq/d; q). 2.65) Let Λa, b, c, d) denote the above 8 ψ 8 summation, and let Λ k a, b, c, d) be the kth term of this sum, namely, Λ k a, b, c, d) = qa 2, qa 2, b, c, dq j, aq j /c, aq +N /b, aq N /d; q) k a 2, a 2, aq/b, aq/c, aq j /d, cq +j, bq N, dq +N ; q) k q k. 2.66) Substituting the parameter b by b/q leads to the iteration relation: Λa, b, c, d) = Λa, b/q, c, d) aq/bc) cq/b) qn+ /b) aq N+ /b) q/b) aq/b) aq N+ /bc) cq N+ /b) bdqj /a) dq N+ /b) bcq N /a) cq j+ /b) bd/aq) cq/b) dq N+j+ /b) bcq N+j /a). 2.67) 5

16 Let F k a, b, c, d) = Λ k a, b, c, d), and let G k a, b, c, d) = Λ k a, b/q, c, d) aq/bc) cq/b) qn+ /b) aq N+ /b) q/b) aq/b) aq N+ /bc) cq N+ /b) bdqj /a) dq N+ /b) bcq N /a) cq j+ /b) bd/aq) cq/b) dq N+j+ /b) bcq N+j /a). 2.68) By the q-gosper algorithm for F k G k, we get h k = Λ k a, b, c, d) b dqn+k )bq k q N+ ) cq k+j ) q k bcq j aq N+ )b dq N+j+ )aq bd) dqj aq k )c aq k )b 2 aq N+2 ) b cq N+ ) aq 2k ) bq k ). 2.69) The limit identity can be verified as follows: lim Λa, M bq M, c, d) N+ = lim Λa, b, c, d) b = 6 ψ 6 qa 2, qa 2, aq N /d, c, dq j, aq j /c a 2, a 2, dq +N, aq/c, aq j /d, cq +j ; q, q = = q, aq, q/a, dq N+ /c, q N j+, cdq N+j+ /a, aq j /cd, q j+, cq/d; q) dq N+, aq/c, aq j /d, cq j+, dq N+ /a, q/c, q j /d, cq j+ /a, q N+ ; q) ) dq, dq/a; q) N cd/a, cq, cq/a, q N ; q) j dq N j cdq/a, dq/c; q) N q, d/a, d, cdq +N /a; q) j c q, q, aq, q/a, cdq/a, aq/cd, cq/d, dq/c; q) cq, q/c, dq, q/d, cq/a, aq/c, dq/a, aq/d; q). 2.70) Thus 2.65) can be deduced from 2.67) and 2.70). 3 The Abel Pairs for Unilateral Summations The idea of Gosper pairs can be adapted to unilateral summation formulas with a slight modification. We also begin with an iteration relation guided by the closed product formula which can be stated in the following form: F k a, a 2,..., a n ) = k=0 G k a, a 2,..., a n ). 3.) k=0 6

17 We assume that k=0 F ka, a 2,..., a n ) and k=0 G ka, a 2,..., a n ) are convergent. Moreover, we assume that the following limit condition holds: For the same reason as in the bilateral case, we see that lim h k = h ) k F k a, a 2,..., a n ) G k a, a 2,..., a n ) is a basic hypergeometric term so that we can resort to the q-gosper algorithm to solve the following equation: F k a, a 2,..., a n ) G k a, a 2,..., a n ) = h k+ h k, k ) A Gosper pair g k, h k ) is then given by g k h k = F k a, a 2,..., a n ), g k h k+ = G k a, a 2,..., a n ). Therefore, one can use the Gosper pair g k, h k ) to justify 3.) by the relation g k h k ) = k=0 g k h k+ ) 3.4) k=0 and the limit condition lim k h k = h 0. Given a Gosper pair it is easy to compute the corresponding Abel pair which implies iteration relation 3.) by the following unilateral Abel sum: A k B k B k ) = k=0 B k A k A k+ ), 3.5) k=0 which we call the unilateral Abel lemma. We also need the corresponding limit condition lim k A kb k = A 0 B. Note that we will only encounter the case B = 0 because of the fact /q; q) = 0 by.). The above approach is suitable for many classical unilateral summation formulas including the q-gauss sum, the q-kummer Bailey-Daum) sum 0, Appendix II.9, the q-dixon sum 0, Appendix II.3, a q-analogue of Watson s 3 F 2 sum 0, Appendix II.6, and a q-analogue of Whipple s 3 F 2 sum 0, Appendix II.8, just to name a few. Here we only give two examples to demonstrate this technique. 7

18 Example 3. The q-gauss sum: 2φ a, b c ; q, c ab = c/a, c/b; q) c, c/ab; q), 3.6) where c/ab <. Set fa, b, c) = 2 φ a, b c ; q, c ab. 3.7) Write the kth term of 3.7) as F k a, b, c) = a, b; q) k c ) k. 3.8) q, c; q) k ab The iteration c cq in 3.6) implies fa, b, c) = c/a) c/b) fa, b, cq). 3.9) c) c/ab) Let c/a) c/b) G k a, b, c) = c) c/ab) F ka, b, cq). 3.0) Applying the q-gosper algorithm to F k a, b, c) G k a, b, c), we arrive at the Gosper pair: So we have the Abel pair: g k = c abqk c ab F ka, b, c), 3.) h k = ab qk ) F k a, b, c). 3.2) c ab A k = abqk /c) a, b; q) k, ab/c) c, abq/c; q) k 3.3) B k = abq/c; q) k c ) k. q; q) k ab 3.4) Now we see that identity 3.6) is true because of the unilateral Abel lemma 3.5) and the limit value fa, b, 0) =. Example 3.2 The sum of Rogers nonterminating very-well-poised 6 φ 5 series 0, Appendix II.20: a, qa 2, qa 2, b, c, d aq 6φ 5 ; q, a 2, a 2, aq/b, aq/c, aq/d bcd where aq/bcd <. = aq, aq/bc, aq/bd, aq/cd; q) aq/b, aq/c, aq/d, aq/bcd; q), 3.5) 8

19 Let us write fa, b, c, d) = 6 φ 5 a, qa 2, qa 2, b, c, d a 2, a 2, aq/b, aq/c, aq/d Denote the kth term of 3.6) by F k a, b, c, d) = aq ; q,. 3.6) bcd a, qa 2, qa 2, b, c, d; q) k aq ) k. 3.7) q, a 2, a 2, aq/b, aq/c, aq/d; q) k bcd The substitution a aq in 3.5) leads to the iteration relation Let fa, b, c, d) = G k a, b, c, d) = Then we get the Gosper pair: aq) aq/cd) aq/bc) aq/bd) faq, b, c, d). 3.8) aq/b) aq/c) aq/d) aq/bcd) aq) aq/cd) aq/bc) aq/bd) aq/b) aq/c) aq/d) aq/bcd) F kaq, b, c, d). 3.9) g k = aqk )q k aq/bcd) aq 2k ) aq/bcd) F ka, b, c, d), 3.20) h k = qk ) a 2 q k+ /bcd) aq 2k ) aq/bcd) F ka, b, c, d). 3.2) The corresponding Abel pair is as follows: A k = bcdqk aq) b, c, d, a 2 q 2 /bcd; q) k, 3.22) bcd aq) aq/b, aq/c, aq/d, bcd/a; q) k B k = aq, bcd/a; q) k aq ) k. 3.23) q, a 2 q 2 /bcd; q) k bcd Therefore, the identity 3.5) is a consequence of the unilateral Abel lemma 3.5) and the limit value f0, b, c, d) =. Acknowledgments. We are grateful to George Andrews, Michael Schlosser and Doron Zeilberger for valuable comments. This work was supported by the 973 Project on Mathematical Mechanization, the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China. References S. A. Abramov, P. Paule, and M. Petkovšek, q-hypergeometric solutions of q- difference equations, Discrete Math ),

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