THE ABEL LEMMA AND THE q-gosper ALGORITHM

Transcription

1 MATHEMATICS OF COMPUTATION Volume 77 Number 262 April 2008 Pages S ) Article electronically published on October THE ABEL LEMMA AND THE q-gosper ALGORITHM VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU Abstract Chu has recently shown that the Abel lemma on summation by parts reveals the telescoping nature of Bailey s 6 ψ 6 bilateral summation formula 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 Introduction We follow the notation and terminology in 0 For q < the q-shifted factorial is defined by a; q) = aq k )anda; q) n = a; q) aq n for n Z ; 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 where n is an integer or infinity In particular for a nonnegative integer k wehave ) a; q) k = aq k ; q) k The unilateral) basic hypergeometric series r φ s is defined by a a 2 a r a a 2 a r ; q) k rφ s ; q z = ) k q b b 2 b k 2) +s r z k s q b b 2 b s ; q) k k=0 while the bilateral basic hypergeometric series r ψ s is defined by a a 2 a r rψ s ; q z = b b 2 b s a a 2 a r ; q) k ) k q 2) s r k z k b b 2 b s ; q) k Recently Chu 9 used the Abel lemma on summation by parts to give an elementary proof of Bailey s very well-poised 6 ψ 6 -series identity 5 see also 0 Received by the editor July and in revised form August Mathematics Subject Classification Primary 33D5; Secondary 33F0 Key words and phrases The Abel lemma Abel pairs basic hypergeometric series the q-gosper algorithm Gosper pairs 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 057 c 2007 American Mathematical Society Reverts to public domain 28 years from publication License or copyright restrictions may apply to redistribution; see

2 058 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU Appendix II33): qa 2 qa 2 b c d e 6ψ 6 a ; q qa2 2 a 2 aq/b aq/c aq/d aq/e bcde 2) = qaqq/aaq/bcaq/bdaq/beaq/cdaq/ceaq/de; q) aq/b aq/c aq/d aq/e q/b q/c q/d q/e qa 2 /bcde; q) where qa 2 /bcde < The Abel lemma on summation by parts is stated as 3) A k B k B k )= B k A k A k+ ) provided that the series on both sides are convergent and k A kb k is absolutely convergent Based on the Abel lemma Chu found an Abel pair A k B k ): b c d q 2 a 2 /bcd; q) k A k = aq/b aq/c aq/d bcd/aq; q) k ) qe bcd/a; q) k qa 2 k B k = aq/e q 2 a 2 /bcd; q) k bcde which leads to the following iteration relation: 4) Ωa; b c d e) =Ωaq; b c d eq) a e) aq) aq/bc) aq/bd) aq/cd) e a) aq/b) aq/c) aq/d) a 2 q/bcde) where 5) qa 2 qa 2 b c d e Ωa; b c d e) = 6 ψ 6 ; q qa2 a 2 a 2 aq/b aq/c aq/d aq/e bcde Because of the symmetries in b c d ande applying the identity 4) 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 ; bqcqdqeq) a4 q 6 bcde aq4 b) c) d) e) 6) 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 Again iterating the above relation m times we get Ωa; b c d e) =Ωaq 4m ; bq m cq m dq m eq m ) 7) a4m q 6m2 bcde) m aq4m a b c d e; q) m 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 License or copyright restrictions may apply to redistribution; see

3 THE ABEL LEMMA AND THE q-gosper ALGORITHM 059 Replacing the summation index k by k 2m we obtain the transformation formula: 8) 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 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 2) by Jacobi s triple product identity 0 Appendix II28: 9) q k2 z k =q 2 qz q/z; q 2 ) since 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 ; q a2 q 4m+ a 2 a 2 aq m+ /b aq m+ /c aq m+ /d aq m+ bcde /e aq 2k = 2 a q2k k a 2k = ) k q k 2) a k =aqq/aq; q) a This paper is motivated by the question of how to systematically compute the Abel pairs for bilateral summation formulas We find that the q-gosper algorithm 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 and Petkovšek developed the package qhyper for finding all q-hypergeometric solutions of linear homogeneous recurrences with polynomial coefficients Later Böing and Koepf 7 gave an algorithm for the same purpose The Maple package qsummpl was described by Böing and Koepf 7 In 8 Riese presented a generalization of the q-gosper algorithm to indefinite bibasic hypergeometric summations Recall that a function t k is said to be a basic hypergeometric term if t k+ /t k is a rational function of q k Theq-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 0) t k = z k+ z k We observe that for an iteration relation of a summation formula the difference of the kth terms of the two sides is a basic hypergeometric term so that 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 start with an iteration relation derived from the original identity Then we use the q-gosper algorithm to generate a Gosper pair g k h k ) if it exists This step can be regarded as a verification of the iteration relation Finally we employ the iteration relation to prove the desired identity by computing a certain limit value In fact once a Gosper pair g k h k ) is obtained one can easily compute the corresponding Abel pair It turns out that the Abel pair for the 6 ψ 6 sum discovered by Chu 9 coincides 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 License or copyright restrictions may apply to redistribution; see

4 060 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU As examples we give Gosper pairs and Abel pairs of several well-known bilateral summation formulas including Ramanujan s ψ summation formula and two formulas due to Schlosser 20 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 a Gosper pair and an Abel pair for the 6 φ 5 summation formula In comparison to a recent approach presented by Chen Hou and Mu 8 for proving nonterminating basic hypergeometric series identities by using the q-zeilberger algorithm one sees that the method in this paper does not rely on the introduction of the parameter n in order to establish recurrence relations and it only makes use of the q-gosper algorithm 2 The Gosper pairs for bilateral summations In spite of its innocent appearance 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 Abel pairs to justify iteration relations for bilateral summations Furthermore it is easily seen that one can compute Abel pairs from Gosper pairs Suppose that we have a bilateral series F ka a 2 a n ) which has a closed product 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 in the following form: 2) F k a a 2 a n )= G k a a 2 a n ) A 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 ) We assume that F ka a 2 a n )and G ka a 2 a n )are both convergent We also assume that 22) lim h k = lim h k k k Note that there are many bilateral summations with the above limit property Evidently once a Gosper pair is derived the identity 2) immediately follows from the telescoping relation: 23) g k h k )= g k h k+ ) We are now ready to describe our approach Let us take Ramanujan s ψ sum 0 Appendix II29 as an example: a 24) ψ b ; q z = q b/a az q/az; q) b q/a z b/az; q) License or copyright restrictions may apply to redistribution; see

5 THE ABEL LEMMA AND THE q-gosper ALGORITHM 06 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 Theorem 2 The following is a Gosper pair for Ramanujan s ψ sum: g k = azk+ a; q) k az b) b; q) k h 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 24) Substituting b to bq in 24) we get 25) ψ a bq ; q z Define = q bq/a az q/az; q) bqq/azbq/az; q) a 26) fa b z) = ψ b ; q z Comparing the right hand sides of 24) and 25) gives the following iteration relation see also 4): a 27) ψ b ; q z b/a) a = b) b/az) ψ bq ; q z Notice that both sides of the above identity are convergent Let F k a b z) andg k a b z) denotethekth terms of the left hand side and the right hand side summations in 27) respectively that is 28) F k a b z) = a; q) k z k b/a) and G k a b z) = b; q) k b) b/az) F ka bq z) Step 2 Apply the q-gosper algorithm 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 with respect to the equation 29) F k a b z) G k a b z) =h k+ h k and we find a solution 20) h k = bzk a; q) k az b) b; q) k which also satisfies the limit condition lim h k = lim h k =0 k k License or copyright restrictions may apply to redistribution; see

6 062 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU As far as the verification of 27) is concerned the existence of a solution h k and the limit condition 22) would guarantee that the identity holds Now it takes one more step to compute the Gosper pair: 2) g k = h k + F k a b z) = azk+ a; q) k 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 27) we may reduce the evaluation of the bilateral series ψ to the special case 22) fa b z) = b/a; q) fa 0z) b b/za; q) Setting b = q in 22) we get fa 0z)= 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 II3 23) we get the evaluation k=0 a; q) k q; q) k z k = az; q) z; q) 24) fa 0z)= q az q/az; q) q/a z; q) Hence the identity 24) follows from 22) and 24) It should be warned that it is not always the case that there is a solution h k to the equation 29) If one encounters such a scenario one still has an alternative to try another iteration relation as is done for the 3 ψ 3 sum in Example 24 An Abel pair A k B k ) can be easily constructed from a Gosper pair g k h k ) Setting 25) g k = A k B k and h k = A k B k then we see that B k 26) = g k B k h k Without loss of generality we may assume that B 0 = Iterating 26) yields an Abel pair A k B k ) For Ramanujan s ψ sum 24) we can compute an Abel pair by using the q-gosper algorithm Theorem 22 The following is an Abel pair for Ramanujan s ψ sum: A k = az ) k a; q) k b az b b; q) k a az ) k B k = b License or copyright restrictions may apply to redistribution; see

7 THE ABEL LEMMA AND THE q-gosper ALGORITHM 063 It is routine to verify that A k B k ) is indeed an Abel pair for the ψ sum First we have az ) az ) k B k B k = b b ) k b/a) a; q) k b A k A k+ = b) b/az) bq; q) k a Then the iteration relation 27) is deduced from the Abel lemma 27) = az a; q) k az b b; q) k b a ) k az az ) k b/a) b b) b/az) ) az b b a; q) k bq; q) k Next we give some examples for bilateral summations ) k ) k b a Example 23 The sum of a well-poised 2 ψ 2 series 0 Appendix II30): 28) b c 2ψ 2 aq/b aq/c where aq/bc < ; q aq bc = aq/bc; q) aq 2 /b 2 aq 2 /c 2 q 2 aq q/a; q 2 ) aq/b aq/c q/b q/c aq/bc; q) Write the kth term of the left hand side of 28) as b c; q) k 29) F k a b c) = aq ) k aq/b aq/c; q) k bc Substituting b with b/q in 28) we are led to the iteration relation aq/bc) aq 2 /b 2 ) 220) F k a b c) = F k a b/q c) + aq/bc) q/b) aq/b) Let aq/bc) aq 2 /b 2 ) 22) G k a b c) = + aq/bc) q/b) aq/b) F ka b/q c) Implementing the q-gosper algorithm we obtain a Gosper pair g k = b2 cq k aq 2 ) aq + bc)bq k q) F ka b c) bqc aq k ) h k = aq + bc)bq k q) F ka b c) The companion Abel pair is given below: 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 ) B k = b2 c/aq; q) k aq ) k aq/c; q) k bc License or copyright restrictions may apply to redistribution; see

8 064 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU Noticing that 28) is symmetric in b and c wehave F k a b c) = F k a b/q c/q) 222) 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) Finally we can reach 28) by iterating 222) infinitely many times and by using Jacobi s triple product identity 9) as the limit case Example 24 Bailey s sum of a well-poised 3 ψ 3 0 Appendix II3): b c d 223) 3ψ 3 q/b q/c q/d ; q q = q q/bc q/bd q/cd; q) bcd q/b q/c q/d q/bcd; q) where q/bcd < Substituting d with d/q in 223) one obtains the iteration relation b c d 3ψ 3 q/b q/c q/d ; q q bcd 224) = q/bd) q/cd) q/d) q/bcd) 3 ψ 3 q c d/q q/b q/c q 2 /d q2 ; q bcd 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 223) to get 225) 3ψ 3 b c d q/b q/c q/d q2 ; q bcd Let F k b c d) be the average of the kth summands of 223) and 225) namely b c d; q) k q ) k +q k 226) F k b c d) = q/b q/c q/d; q) k bcd 2 and let q/bd) q/cd) 227) G k b c d) = q/d) q/bcd) F kb c d/q) 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) db q k )c q k ) h k = + q k )q bcd) dq k ) F kb c d) License or copyright restrictions may apply to redistribution; see

9 THE ABEL LEMMA AND THE q-gosper ALGORITHM 065 which implies the iteration relation 224) Invoking the symmetric property of the parameters b c andd wehave 228) 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 q 2 /b q 2 /c q 2 /d bcd The above relation enables us to reduce the summation formula 223) to Jacobi s triple product identity Example 25 A basic bilateral analogue of Dixon s sum 0 Appendix II32 qa 2 b c d 4ψ 4 ; q qa 3 2 a 2 aq/b aq/c aq/d bcd 229) = aqaq/bcaq/bdaq/cdqa 2 /b qa 2 /c qa 2 /dqqa; q) aq/b aq/c aq/d q/b q/c q/d qa 2 qa 2 qa 3 2 /bcd; q) where qa 3 2 /bcd < For the above formula we may consider the substitution d d/q in 229) 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) 230) 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 2 bcd Let ) k qa 2 bcd; q) k qa ) F k a b c d) = a 2 aq/baq/caq/d; q) k bcd and let 232) 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) By computation we obtain a Gosper pair abdq k+ acdq k+ + q 2 a a 2 q k+2 bcda 2 q k+ da 3 2 q k+ g k = dq k q) + a 2 q k )bcd a 3 2 q) ) bcd 2 q k + bcd 2 a 2 q 2k + F dq k q) + a 2 q k )bcd a 3 k a b c d) 2 q) h k = daq k c)aq k b) dq k ) + a 2 q k )bcd qa 3 2 ) F ka b c d) License or copyright restrictions may apply to redistribution; see

10 066 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU So the iteration relation 230) holds From the symmetric property of the parameters b c andd wehave 233) 4ψ 4 qa 2 b c d a 2 aq/b aq/c aq/d 3 qa 2 ; q 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 2 bcd By iteration it follows that qa 2 b c d 4ψ 4 ; q qa 3 2 a 2 aq/b aq/c aq/d bcd 234) where 235) Ha) = = aq/bc aq/bd aq/cd qa 2 /b qa 2 /c qa 2 /d; q) Ha) aq/b aq/c aq/d q/b q/c q/d qa 3 2 /bcd; q) qa 2 ; q) k q 3k 2) ) k qa 3 a 2 2 ; q) k Taking b = a 2 and c d in 234) and by Jacobi s triple product identity 9) it can be verified that 236) Ha) = q aq q/a; q) qa 2 qa 2 ; q) which leads to 229) Example 26 Bailey s very well-poised 6 ψ 6 -series identity 2) Let us denote the kth term of 5) by ) qa 2 qa 2 bcde; q) k qa 2 k 237) Ω k a; b c d e) = a 2 a 2 aq/baq/caq/daq/e; q) k bcde Set F k =Ω k a; b c d e) and a e) aq)bc aq)bd aq)cd aq) 238) G k =Ω k aq; b c d eq) a)b aq)c aq)d aq)bcde a 2 q) By computation we find the following Gosper pair: 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) We observe that the Abel pair derived from the above Gosper pair coincides with the Abel pair given by Chu 9 License or copyright restrictions may apply to redistribution; see

11 THE ABEL LEMMA AND THE q-gosper ALGORITHM 067 Next we give another Gosper pair which leads to a different iteration relation of the 6 ψ 6 series by setting e e/q in 2) that is aq/be) aq/ce) aq/de) 239) Ωa; b c d e) =Ωa; b c d e/q) aq/e) q/e) a 2 q/bcde) We will see that the above iteration has the advantage that it directly points to the identity 8) by taking into account the symmetries in b c d ande 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 aq/be) aq/ce) aq/de) 240) G k =Ω k a; b c d e/q) aq/e) q/e) a 2 q/bcde) From the above iteration relation 239) we obtain the Gosper pair a 2 eq k+ + a 3 q 2k+2 a 2 q 2 a 2 beq 2k+ a 2 ceq 2k+ a 2 deq 2k+ bcde 2 q k g k = bcde qa 2 )eq k q)aq 2k ) + abceqk+ +abdeq k+ +acdeq k+ +abcde 2 q 3k abcdeq 2k+ bcde qa 2 )eq k q)aq 2k ) ) Ω k a; b c d e) h k = qeb aqk )c aq k )d aq k ) bcde qa 2 ) aq 2k )eq k q) Ω ka; b c d e) Since the parameters b c d ande are symmetric in 2) we obtain 24) 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) Again the limit value can be given by Jacobi s triple product identity so that we arrive at 2) in view of 24) The following is Shukla s very-well-poised 8 ψ 8 summation 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 27 Shukla s very-well-poised 8 ψ 8 summation formula: 242) qa 2 8ψ 8 qa 2 b c d e f aq 2 /f a 2 a 2 aq/b aq/c aq/d aq/e aq/f f/q = bc/a) bd/a) be/a) bq/f) bf/aq) bcde/a 2 ) ; q ) f/bq) bf/aq) f/aq) f/q) qaqq/aaq/bcaq/bdaq/beaq/cdaq/ceaq/de; q) aq/b aq/c aq/d aq/e q/b q/c q/d q/e a 2 q/bcde; q) where a 2 /bcde < a 2 bcde License or copyright restrictions may apply to redistribution; see

12 068 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU Observe that the right hand side of 242) can be rewritten as bc/a) bd/a) be/a) bq/f) bf/aq) bcde/a 2 ) ) f/bq) bf/aq) Ωa; b c d e) f/aq) f/q) where Ωa; b c d e) denotes the 6 ψ 6 series as given by 5) Let F k denote the kth term of the left hand side of 242) namely qa 2 qa 2 bcdefaq 2 ) /f; q) k a 2 k 243) F k = a 2 a 2 aq/baq/caq/daq/eaq/ff/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 ) 244) f/bq) bf/aq) f/aq) f/q) where Ω k a; b c d e) isthekth term of the 6 ψ 6 series Ωa; b c d e) as given by 237) With the aid of the q-gosper algorithm for F k G k we find 245) 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) Thus 242) can be deduced from the telescoping relation 23) and the 6 ψ 6 formula 2) We now turn to two identities due to Schlosser 20 derived by matrix inverse Example 28 Let a b c d e and u be indeterminates Then 246) 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 ) = qaqq/aaq/bcaq/bdaq/beaq/cdaq/cea/de; q) q/b aq/b q/c aq/c /d aq/d /e aq/e a 2 q/bcde; q) provided a 2 /bcde < ) k a/cu) bu/a) b a/c) As in the preceding example the right hand side of 246) contains the product of the 6 ψ 6 sum We may proceed in the same manner Write the kth term of the left hand side of 246) as F k thatis F k = ) aq2k b c; q) k dq eq; q) k a 2 k 247) and let a) aq/b aq/c; q) k aq/d aq/e; q) k de/a) uqk ) a 2 q k /bcu) a/bc) dq k ) eq k ) 248) G k = bcde ) a/cu) bu/a) a/de) Ω k a; b c d e) b a/c) /d) /e) License or copyright restrictions may apply to redistribution; see

13 THE ABEL LEMMA AND THE q-gosper ALGORITHM 069 where Ω k is the same notation as in the preceding example Employing the q-gosper algorithm with respect to F k G k weobtain 249) h k = b aqk )c aq k )d aq k )e aq k ) aq k d) e)a bc) aq 2k ) Ω ka; b c d e) Then 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 29 Let a b c andd be indeterminates Let j be an arbitrary integer and N a nonnegative integer Then 250) 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 qcq/bd/adbcq N /a cdq +N /a; q) j qqaqq/acdq/aaq/cdcq/ddq/c; q) cqq/cdqq/dcq/aaq/cdq/aaq/d; q) Let Λa b c d) denote the above 8 ψ 8 summation and let Λ k a b c d) bethekth term of this sum namely 25) Λ k a b c d) = qa 2 qa 2 bcdq j aq j /c aq +N /b aq N /d; q) k a 2 a 2 aq/baq/caq j /d cq +j bq N dq +N ; q) k q k Substituting the parameter b with b/q leads to an iteration relation: 252) 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) 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) By the q-gosper algorithm for F k G k weget 253) 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 ) License or copyright restrictions may apply to redistribution; see

14 070 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU The limit identity can be verified as follows: 254) lim Λa M bq M cd) = lim Λa b c d) b = 6 ψ 6 qa 2 qa 2 aq N /d c dq j aq j /c ; q q N+ a 2 a 2 dq +N aq/c aq j /d cq +j = qaqq/adqn+ /c q N j+ cdq N+j+ /a aq j /cd q j+ cq/d; q) dq N+ aq/caq 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 qqaqq/acdq/aaq/cdcq/ddq/c; q) cqq/cdqq/dcq/aaq/cdq/aaq/d; q) Thus 250) can be deduced from 252) and 254) 3 The Gosper 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 suggested by a closed product formula which can be stated in the following form: 3) F k a a 2 a n )= G k a a 2 a n ) k=0 k=0 For the same reason as in the bilateral case we see that 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: 32) F k a a 2 a n ) G k a a 2 a n )=h k+ h k k 0 We assume that k=0 F ka a 2 a n )and k=0 G ka a 2 a n )areconvergent Moreover we assume that the following limit condition holds: 33) lim h k = h 0 k A Gosper pair g k h k ) is defined by the equations g k h k = F k a a 2 a n ) g k h k+ = G k a a 2 a n ) Therefore the identity 3) can be demonstrated by the Gosper pair g k h k )because of the telescoping relation 34) g k h k )= g k h k+ ) k=0 and the limit condition lim k h k = h 0 k=0 License or copyright restrictions may apply to redistribution; see

15 THE ABEL LEMMA AND THE q-gosper ALGORITHM 07 Given a Gosper pair it is easy to compute the corresponding Abel pair which implies iteration relation 3) by the following unilateral form of the Abel lemma: 35) A k B k B k )= B k A k A k+ ) k=0 Note that we also need the limit condition lim A kb k = A 0 B k As will be seen we will encounter only the case B = 0 due to the fact /q; q) = 0by) The above approach is suitable for many classical unilateral summation formulas including the q-gauss sum 0 Appendix II8 the q-kummer Bailey-Daum) sum 0 Appendix II9 the q-dixon sum 0 Appendix II3 a q-analogue of Watson s 3F 2 sum 0 Appendix II6 and a q-analogue of Whipple s 3 F 2 sum 0 Appendix II8 just to name a few Here we give only two examples to demonstrate this technique Example 3 The q-gauss sum a b 36) 2φ c ; q c ab where c/ab < k=0 = c/a c/b; q) c c/ab; q) Set a b 37) fa b c) = 2 φ c ; q c ab Write the kth term of 37) as 38) F k a b c) = a b; q) k c ) k q c; q) k ab The substitution c cq in 36) implies 39) fa b c) = Let c/a) c/b) fa b cq) c) c/ab) c/a) c/b) 30) G k a b c) = c) c/ab) F ka b cq) Applying the q-gosper algorithm to F k a b c) G k a b c) we arrive at a Gosper pair g k = c abqk c ab F ka b c) h k = ab qk ) F k a b c) c ab License or copyright restrictions may apply to redistribution; see

16 072 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU So we have an Abel pair A k = abqk /c) a b; q) k ab/c) c abq/c; q) k B k = abq/c; q) k c ) k q; q) k ab We see that identity 36) is true because of 35) and the limit value fa b 0) = Example 32 The sum of Rogers nonterminating very-well-poised 6 φ 5 series 0 Appendix II20: a qa 2 qa 2 b c d 6φ 5 ; q aq a 2 a 2 aq/b aq/c aq/d bcd 3) where aq/bcd < Let us write = aqaq/bcaq/bdaq/cd; q) aq/b aq/c aq/d aq/bcd; q) a qa 2 qa 2 b c d 32) fa b c d) = 6 φ 5 a 2 a 2 aq/b aq/c aq/d Denote the kth term of 32) by a qa 2 qa 2 bcd; q)k aq ) k 33) F k a b c d) = q a 2 a 2 aq/baq/caq/d; q) k bcd The substitution a aq in 3) leads to the iteration relation 34) fa b c d) = Let aq ; q bcd aq) aq/cd) aq/bc) aq/bd) aq/b) aq/c) aq/d) aq/bcd) faqbcd) aq) aq/cd) aq/bc) aq/bd) 35) G k a b c d) = aq/b) aq/c) aq/d) aq/bcd) F kaqbcd) Then we get a Gosper pair g k = aqk )q k aq/bcd) aq 2k ) aq/bcd) F ka b c d) h k = qk ) a 2 q k+ /bcd) aq 2k ) aq/bcd) F ka b c d) The corresponding Abel pair is as follows: A k = bcdqk aq) b c d a 2 q 2 /bcd; q) k bcd aq) aq/b aq/c aq/d bcd/a; q) k B k = aq bcd/a; q) k aq ) k q a 2 q 2 /bcd; q) k bcd Therefore the identity 3) is a consequence of the unilateral Abel lemma 35) and the limit value f0bcd)= License or copyright restrictions may apply to redistribution; see

17 THE ABEL LEMMA AND THE q-gosper ALGORITHM 073 Acknowledgments We are grateful to George Andrews Michael Schlosser and Doron Zeilberger for valuable comments References S A Abramov P Paule and M Petkovšek q-hypergeometric solutions of q-difference equations Discrete Math ) 3 22 MR f:3900) 2 G E Andrews On Ramanujan s summation of ψ a b z) Proc Amer Math Soc ) MR :3042) 3 G E Andrews On a transformation of bilateral series with applications Proc Amer Math Soc ) MR :2064) 4 G E Andrews and R Askey A simple proof of Ramanujan s summation of the ψ Aequationes Math 8 978) MR h:33004) 5 W N Bailey Series of hypergeometric type which are infinite in both directions Quart J Math Oxford) 7 936) B C Berndt Ramanujan s theory of theta-functions In: Theta Functions from the Classical to the Modern M R Murty ed CRM Proceedings and Lecture Notes AmerMathSoc Providence 993 pp 63 MR m:054) 7 H Böing and W Koepf Algorithm for q-hypergeometric summation in computer algebra J Symbolic Comput ) MR j:3309) 8 W Y C Chen Q H Hou and Y P Mu Nonterminating basic hypergeometric series and the q-zeilberger algorithm arxiv:mathco/ W C Chu Bailey s very well-poised 6 ψ 6 -series identity J Combin Theory Ser A ) MR G Gasper and M Rahman Basic Hypergeometric Series second ed Cambridge University Press Cambridge 2004 MR d:33028) W Hahn Über Polynome die gleichzeitig zwei verschiedenen Orthogonalsystemen angehören Math Nachr 2 949) MR :356f) 2 MEHIsmailA simple proof of Ramanujan s ψ sum Proc Amer Math Soc ) MR :22695) 3 M Jackson On Lerch s transcendant and the basic bilateral hypergeometric series 2 ψ 2 J London Math Soc ) MR :78f) 4 T H Koornwinder On Zeilberger s algorithm and its q-analogue J Comput Appl Math ) 9 MR b:330) 5 M Mohammed and D Zeilberger Sharp upper bounds for the orders of the recurrences outputted by the Zeilberger and q-zeilberger Algorithms J Symbolic Comput ) MR P Paule Short and easy computer proofs of the Roger-Ramanujan identities and of identities of similar type Electron J Combin 994) R0 MR g:099) 7 P Paule and M Schorn A Mathematica version of Zeilberger s algorithm for proving binomial coefficients identities JSymbolicComput20 995) MR j:39006) 8 A Riese A generalization of Gosper s algorithm to bibasic hypergeometric summation Electron J Combin 3 996) R9 MR h:3303) 9 M Schlosser Abel-Rothe type generalizations of Jacobi s triple product identity in Theory and Applications of Special Functions A Volume Dedicated to Mizan Rahman M E H Ismail and E Koelink eds) Dev Math ) MR b:33032) 20 M Schlosser Inversion of bilateral basic hypergeometric series Electron J Combin ) R0 MR d:3307) 2 HSShuklaA note on the sums of certain bilateral hypergeometric series Proc Cambridge Phil Soc ) MR :3590) 22 H S Wilf and D Zeilberger An algorithmic proof theory for hypergeometric ordinary and q ) multisum/integral identities Invent Math ) MR k:3300) License or copyright restrictions may apply to redistribution; see

18 074 VINCENT Y B CHEN WILLIAM Y C CHEN AND NANCY S S GU Center for Combinatorics LPMC Nankai University Tianjin P R China address: ybchen@mailnankaieducn Center for Combinatorics LPMC Nankai University Tianjin P R China address: chen@nankaieducn Center for Combinatorics LPMC Nankai University Tianjin P R China address: gu@nankaieducn License or copyright restrictions may apply to redistribution; see